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1,320 terms

Probability & Statistics

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Bar graph
Displays the distribution of a categorical variable
Binary variable
Categorical variable with 2 choices such as gender- male or female
Categorical variable
Records a group designation such as gender
Data
Numbers or categories recorded for the observational units in a study
Distribution (of a variable)
Refers to it's pattern of variation. With a categorical variable, distribution means the variable's possible categories and the proportion of responses in each
Dot plot
Useful for displaying the distribution of a relatively small data set of a quantitative variable
Observational unit
Person or thing assigned a number or category
Quantitative variable
Measures a numerical characteristic such as height
Variability
Phenomenon of a variable taking on different values or categories from observational unit to observational unit.
Variable
Any characteristic of a person or thing that can be assigned a number or category
compound
a sequence of simple events
counting principle
the number of possible outcomes in an experiment
event
a subset of a sample
experimental probability
the ratio of the number of times an outcome occurs to the total amount of trials performed
Independent events
events for which the occurrence of one has no impact on the occurrence of the other
Outcome
a possible result of an experiment
Probability
a measure of the likelihood of an event
relative frequency
the number of times an outcome occurs divided by total number of trials
sample space
all possible outcomes of given experiment
simple event
an event consisting of just one outcome.
theoretical probability
The mathematical calculation that an event will happen in theory
tree diagram
a tree-shaped diagram that illustrates sequentially the possible outcomes of a given event
binomial distribution
a theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success
causation
A cause and effect relationship in which one variable controls the changes in another variable.
central limit theorem
Regardless of the population distribution, The sampling distribution is normal IF n is large enough (>30).
cluster sampling
divide population into sections then randomly select some of those clusters and then choose ALL members from selected clusters
confounding
a situation where the effect of one variable on the response variable cannot be separated from the effect of another variable on the response variable.
control group
the group that does not receive the experimental treatment.
correlation
measuring the strength and direction of the relationship between two numerical variables
degrees of freedom
A concept used in tests of statistical significance; the number of observations that are free to vary to produce a known outcome.
density curve
describes the overall pattern of a distribution, area = 1
discrete random variables
A random variable that assumes countable values
disjoint events
mutually exclusive, events that have no outcomes in common
double blind experiments
experiments in which neither the participants nor the people analyzing the results know who is in the control group
Empirical Rule
The rules gives the approximate % of observations w/in 1 standard deviation (68%), 2 standard deviations (95%) and 3 standard deviations (99.7%) of the mean when the histogram is well approx. by a normal curve
explanatory variables
the treatment (ex. studying or not studying); factors
factor
an independent variable in statistics
five-number summary
minimum, 1st quartile, median, 3rd quartile, maximum
geometric distribution
Success / Failure, trials continue until successful, each outcome is independent, constant probability of success
independent events
The outcome of one event does not affect the outcome of the second event
inference
drawing conclusions that go beyond the data at hand
influential observations
Individual points that change the regression line. Often outliers in the x direction, but require large residuals.
interquartile range
The difference between the upper and lower quartiles.
law of large numbers
as an experiment is repeated over and over, the empirical probability of an event approaches the actual probability of the event
lurking variable
A lurking variable is a variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.
margin of error
The +- value added to and subtracted from a point estimate in order to develop an interval estimate of a population parameter
matched pairs design
A matched pairs design is a special case of the randomized block design. It is used when the experiment has only two treatment conditions; and subjects can be grouped into pairs, based on some blocking variable. Then, within each pair, subjects are randomly assigned to different treatments.
mean
an average of n numbers computed by adding some function of the numbers and dividing by some function of n
median
the value below which 50% of the cases fall
mutually exclusive
Events that cannot occur at the same time.
normal distribution
A function that represents the distribution of variables as a symmetrical bell-shaped graph.
p-value
measure of how rare the sample results would be if Ho were true
parameters
numbers that describe a population
randomization
the best defense against bias, in which each individual is given a fair, random chance of selection
replication
the repetition of an experiment in order to test the validity of its conclusion
residual
the difference between the observed value and the predicted value of a regression equation; y - y-hat
response bias
people answer questions the way they think you want them answered. There are some questions they simply don't want to answer truthfully.
sampling distribution
a distribution of statistics obtained by selecting all the possible samples of a specific size from a population
sampling variability
the natural tendency of randomly drawn samples to differ
scatterplots
a graphed cluster of dots, each of which represents the values of two variables. The slope of the points suggests the direction of the relationship between the two variables. The amount of scatter suggests the strength of the correlation.
scope of inference
to whom the generalization of the inference may be directed
simple random sample
abbreviated SRS, this requires that every item in the population has an equal chance to be chosen and that every possible combination of items has an equal chance to exist. No grouping can be involved.
simpson's paradox
conclusions drawn from two or more separate crosstabulations that can be reveresed when the data are aggregated between two quantitative variables
single blind experiments
an experiment in which the participants are unaware of which participants received the treatment
skewed
a distribution is this if it's not symmetric and one tail stretches out farther than the other
slope
the average change in the response variable as the explanatory variable increases by one
spread
how varaiable the data is; measured by standard deviation, IQR, variance, range
standard deviation
a measure of variability that describes an average distance of every score from the mean
standard error
the standard deviation of a sampling distribution
standard normal curve
A normal distribution with mean of zero and standard deviation of one. Probabilities are given in Table A for values of the standard Normal variable.
statistically significance
said to exist when the probability that the observed findings are due to chance is very low
statistic
A numerical measurement describing some characteristic of a sample
stratified random sample
a sample in which the population is first divided into similar, nonoverlapping groups. A simple random sample is then selected from each of the groups
symmetric
a distribution is this if the two halves on either side of the center look approximately like mirror images of each other
Type I Error
The error that is committed when a true null hypothesis is rejected erroneously. The probability of a Type I Error is abbreviated with the lowercase Greek letter alpha.
Type II Error
the error of failing to reject a null hypothesis when in fact it is false (also called a "false negative"). the probability of a Type II error is commonly denoted β and depends on the effect size.
unbiased estimator
a statistic whose sampling distribution is centered over the population parameter
undercoverage
occurs when some groups in the population are left out of the process of choosing the sample
variance
standard deviation squared, a measure of spread
voluntary response
Individuals with strong feelings about a subject are more likely than others to respond. Such a study is interesting but not reflective of the population.
y-intercept
predicted value when the x variable is zero
z score
a measure of how many standard deviations you are away from the norm (average or mean)
Addition Rule
If A and B are disjoint events: P(A or B)=P(A) + P(B)
Conditional Distribution
the distribution of a variable restricting the who to consider only a smaller group of individuals
Conditional Probability
A probability that takes into avvount a given condition.
Disjoint Events
mutually exclusive, events that have no outcomes in common
General Addition Rule
For any two events (meaning disjoint or not disjoint), A and B, the probability of A or B is:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
General Multiplication Rule
If A and B are any two events, then
P(A & B) = P(A) x P(B|A)
Independence (Casually)
Two events are indpendent if knowing whether one event occurs does not alter the probability that the other event occurs
Independence (Formally)
P(BlA) = P(B) when A and B are independent.
Sample Space
The collection of all possible outcomes
Tree Diagram
a diagram used to show the total number of possible outcomes in a probability experiment
Addition and Subtraction
-Mean, Median, and Mode are affected
-Cannot subtract SD; Only square, add, and square root
-Range is NOT affected
Continuous Random Variable
-Random variable that assumes values associated with one or more intervals on the number line
Discrete Random Variable
-Random variable with a countable number of outcomes
Event
-Subset of the sample space
Independent Events
-If the knowledge of one event having occurred does not change the probability that the other event occurs
Law of Large Numbers
-States that the proportion of successes in the simulation should become, over time, close to the true proportion in population
Multiplication and Division
-Mean, Median, Mode, Range, and SD are affected
Mutually Exclusive Event
-If they have no outcomes in common
-One cannot happen with the other
Probability Distribution for a Discrete Random Variable
-Possible values of the discrete random variable together with their respective probabilities
Probability Distribution for a Random Variable
-Possible values of the random variable X together with the probabilities corresponding to those values
Random Phenomenon
-An activity whose outcome we can observe or measure but we do not know how it will turn out on any single trial
complementary
two events that cannot occur together, but one must happen
dependent
events do impact each other's probability
Independent
events do not impact each other's probability
Mutually exclusive/disjoint
two events that cannot occur simultaneously
Bayes's Theorem
Suppose that A₁, A₂, ... Ak are disjoint events whose probabilities are not 0 and add to exactly 1, i.e. any outcome must be exactly one of those events. The, if B is any other event whose probability is not 0 or 1,
P(Ai|B) = P(B|Ai)P(Ai) / P(B|A₁)P(A₁) + ... P(B|Ak)P(Ak)
Conditional Probability
The probability of some event given that some other event occurs.
Disjoint ≠ Independent
If two events are disjoint, then the occurrence of one would mean the non-occurrence of the other. If events are independent, then non/occurrence is moot.
General Addition Rule for Any Two Events
For any two events A and B,
P(A or B) = P(A) + P(B) - P(A and B)
General Multiplication Rule for Any Two Events
The probability that both of two events A and B happen together can be found by
P(A and B) = P(A)P(B|A)
Independence Definition
When the outcome of one event cannot influence the outcome of a second event.
Outcomes for a diagnostic test
There are four possible outcomes:
- true positive
- true negative
- false positive
- false negative
Positive Predictive Value
PPV = # of true positives / total # positives
Prevalence
# of diseased individuals / total # of individuals
Sensitivity
P(positive|diseased)
Want to be as high as possible, to diagnose.
Specificity
P(negative|non-diseased)
Want to be as high as possible to avoid false positives.
The Multiplication Rule for Independent Events
P(A and B) = P(A)P(B)
Tree Diagrams
Diagrams that will show P(A) as independent branches, then P(B|A) as branches coming off those branches, etc. until a final event is reached. The probability of any one event occurring can be calculated by multiplying the probabilities of each branch along the way.
Venn Diagram
A diagram showing a sample space S and events as areas within S. Overlaps indicate non-disjoint events.
When P(A)>0, the conditional probability of event B occurring given A occurs is
P(B|A) = P(A and B) / P(A)
"At least one"
1 - P(E)
"None"
P(E)
Addition Rule 1
When two events A and B are mutually exclusive, the probability that A or B will occur is
P(A or B) = P(A) + P(B)
Addition Rule 2
If A and B are NOT mutually exclusive, then
P(A or B) = P(A) + P(B) - P(A and B)
Classical Probability
P(E) = # of ways the trial can occur
total # of outcomes
Whenever you are finding probability where the sample space is the same.
Combinations Rule
Used when selecting a smaller number from a larger number but the order is NOT important.
nCr= n!
r! (n-r)!
n=sample size, r=smaller objects selecting
On calculator: enter amount(n), math, PRB, 3, enter amount(r), enter
Complement Rule
P(E)
Is the set of outcomes in the sample space that are not included in the outcomes of E
Conditional Probability
The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. The formula is
P(B!A) = P(A and B)
P(A)
Dependent Events
When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed.
*without replacement = dependent events
Empirical Probability
P(E) = frequency for the class = f
total frequencies in the distribution n
Relies on actual experience to determine the likelihood of outcomes.
Factorial Rule
Use this when you have "n" objects and you want to know how many different ways they can be arranged.
n!
On calculator: enter amount, math, arrow left to PRB, 4, enter
Fundamental Counting Rule
Use this when you have different positions and you want to know how many options there are within those positions.
___*___*___*___*___*___*___________*___*___*___*___*___________*___*___*___*___*___*___*___= 2 =512
Independent Events
Two events A and B are independent events if the fact that A occurs does NOT affect the probability of B occurring.
*with replacement = independent events
Multiple Combinations Rule
When you are taking more than one combination in a problem.
nCr * nCr
Multiplication Rule 1
When two events are independent, the probability of both occurring is
P(A and B) = P(A) * P(B)
Multiplication Rule 2
When two events are dependent, the probability of both occurring is
P(A and B) = P(A) * P(B!A)
Mutually Exclusive Events
Two events that cannot occur at the same time (i.e., they have no outcomes in common).
Permutations Rule
Used when selecting a smaller group from a larger group and you put them in a specific order.
*ORDER IS IMPORTANT
nPr= n!
(n-r)!
n=sample size, r=smaller objects selecting
On calculator: enter amount(n), math, PRB, 2, enter amount(r), enter
Subjective Probability
Uses a probability value based on an educated guess or estimate, employing opinions and inexact information.
These are counting Rules.....
NOT trying to find probability!
bar graph
quickly compares data in column form, the heights can also show percents
box plots
made based off of the 5 number summary
modified - shows outliers
cases
When the objects are people in a set of data
catagorical variable
an individual into one of two or more groups or categories
density curve
the overall pattern of a distribution, areas underneath give proportions of observations for the distribution
distribution
a variable tells us what values it takes and how often it takes these values
of categorical - gives us either the count of the percent of individuals that fall in each category
examining terms for distribution
overall pattern, deviations, shape, center, spread, outlier
exploratory data analysis
examination of data and describe its main features
five number summary
median, quartials and the min and max number
histogram
breaks the the range of values of a variable into classes and displayus only the count or percent of the observations that fall into each class, no space inbetween each bar
individuals
the objects described in a set of data.
interquartile range
difference between quartiles
use 1.5 x IQR to solve for any outliers
linear transformations
changes the original variable x into the new variable x(new) given by the euation ***
mean
the arithmetic average
median
midpoint of the data
modes
unimodal - one peak
bimodal - two peaks
multimodal- multiple peaks
normal distributions
bell curve, symmetric, unimodal density curves
normal quartile plot
a pattern on such a plot that deviates substantially from a staight line indicates that the data are not normal
pie chart
shows us the percents or count in relationship to a whole
quantitative variable
numerical values for which arithmetic operations such as adding and averaging make sense
quartiles
describes the distribution further
Q1 : 1/4 of the data
Q3 : 3/4 of data
resistance measure
any aspect of a distribution is relatively unaffected by changes in the numerical value of a small proportion of the total number of oberservations no matter how large these changes are
splitting stem/ trim
terms to slim down the size of your stem plot. helpful when you have large sets of data
standard deviation
is zero when there is no spread and gets larger as the increase spreads
stem plot
gives a quick picture of the shape of a distribution while including the actual numerical values in the graph
time plot
a variable plots each observation against the time at which it was measured
time series
measurements of a variable taken at regular intervals over time
trend
persistent, long term rise or fall
variable
any characteristic of an individual
varience
commons measure of spread about the mean as center
z score
how many standard deviations x lies from the distribution mean
block
a group of individuals sharing some common features that might affect the treatment
census
in which measurements or observations from the entire population are used
cluster sampling
divide population into pre-existing segments; select random clusters; include every member of each selected cluster
completely randomized experiment
one in which a random process is used to assign each individual to one of the treatments
confounded
when the effects of one of the two variables canot be distinguished from the effects of the other
control group
receives a dummy treatment, enabling the researchers to control for the placebo effect; used to account for the influence of other known or unknown variables that might be an underlying cause of a change in response in the experimental group
convenience sampling
create a sample by using data from population members that are readily available
Descriptive statistics
involves methods of organizing, picturing, and summarizing information from samples or populations
experiment
a treatment is deliberatrly imposed on the individuals in order to observe a possible change in the response or variable being measured
Individuals
the people or objects included in the study
inferential statistics
involves methods of using information from a sample to draw conclusions regarding the population
interval level of measurement
applies to data that can be arranged in order; differences are meaningfull
lurking variable
one for which no data have been collected but that nevertheless has influence on other variables in the study
multistage sampling
use a variety of sampling methods to create successively smaller groups at each stage. The final sample consists of clusters
nominal level of measurement
applies to data that consist of names, labels, or categories; cannot be ordered
nonsampling error
the result of poor sample design, sloppy data collection, faulty measuring instruments, bias in questionnaires, and so on.
observational study
observations and measurements of individuals are conducted in a way that doesn't change the response or the variable being measured
ordinal level of measurement
applies to data that can be arranged in order; differences between data are meaningless
parameter
numerical measure that describes an aspect of a population
placebo effect
occurs when a subject receives no treatment but (incorrectly) believes he is receiving treatment and responds favorably
population data
the data are from every individual of interest
qualitative variable
describes an individual by placing the individual into a category or group
quantitative variable
has a value or numerical measurement
randomization
used to assign individuals to the two treatment groups; helps prevent bias in selecting group members
randomized block experiment
individuals are first sorted into blocks, and then a random process is used to assign each individual in the block to one of the treatments
ratio level of measurement
applies to data that can be arranged in order; differences are meaningfull; true zero
replication
reduces the possibility that the differences in pain relief for the two groups occurred by chance alone
sample data
the data are form only some of the individuals of interest
sample
in which measurements or observations from part of the population are used
sampling error
the difference between measurements from a sample and corresponding measurements from the respective population; caused by the fact that the sample does not perfectly represent the population
sampling frame
a list of individuals form which a sample is actually selected
simple random sample
a subset of the population selected in a manner such that every sample of size n from the population has an equal chance of being selected
simulation
a numerical facsimile or representation of a real-world phenomenon
statistics
the study of how to collect, organize, analyze, and interpret numerical information from data
statistic
numerical measure that describes an aspect of a sample
stratified sampling
divide the entire population into distinct subgroups called strata. The strata are based on a specific characteristic. All members of a stratum share the specific charactersitic. Draw random samples from each stratum
systematic sampling
number all members of the population then from a random starting point, select every kth member
undercoverage
results when population members are omitted from the sample frame
variable
a characteristic of the individual to be measured or observed
Bell-Shaped Distribution
Has a single peak, tapers odd at either end; and is approximately symmetric
Bimodal
Distribution has tow peaks of about the same height
Categorical Frequency Distribution
Used for data that can be placed into specific categories, such as nominal or ordinal level data
Class Boundaries
Used to separate the classes so that there are no gaps in the frequency distribution
Class Width
Found by subtracting the lower class limit one from the lower class limit of the next class; can also be used with upper limits
Class
Each raw data value is placed into a quantitative or qualitative category
Frequency Distribution
The organization of raw data in table form; consists of classes and frequencies
Frequency Polygon
Graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes
Frequency
The number of data values contained in a specific class
Grouped Frequency Distribution
When the data is large and the data must be grouped into classes that are more than one unit in width.
Histogram
A graph that displays the data by using adjacent vertical bars of various heights to represent the frequencies of the classes
J-Shaped Distribution
A few data values on the left that increases as one moves to the right
Lower Class Limit
Represents the smallest data value that can be included in the class
Negatively Skewed
When the data values are clustered to the right and taper off to the left
Positively Skewed
When the peak of the distribution is to the left and the data values taper off to the right
Raw Data
When data are in their original form; little information can be obtained from looking at this
Relative Frequency Graphs
When the frequencies can be converted into proportions
Reverse J-Shaped Distribution
Opposite if the J-shaped distribution
Uniform Distribution
Basically flat or rectangular
Unimodal
Distribution with one peak
Upper Class Limit
Represents the largest data value that can be included in the class
à (Complement)
Outcomes that Do Not occur in A
A-B or A\B
Outcomes that are in A but not in B
A∩B∩C = Ø
Disjoint or Mutually Exclusive Events
A∪B∪C = Ω
Exhaustive
C(n, k) = (n k) = n! / k!(n-k)!
Combination Without Replacement
Cr(n, k) = k+n-1! / k!(n-1)!
Combination With Replacement
Distinguishable
Different order yields a different outcome
Events
A Set of Outcomes and a Set of the Sample Space.
i.e., gender of a person, card is black
Exhaustive
AT LEAST ONE WILL OCCUR for sure
Indistinguishable
Order is not important
Mutually Exclusive
Will NEVER occur at the same time
P(A|B) = Conditional Probability
Probability that A occurs given B is known to occur
P(A|B) = P(A)
Independent
P(A|B) = P(A∩B) / P(B)
Conditional Probability
P(A∩B) = P(A) • P(B)
Independent Events
P(A∩B) = P(A|B) • P(B)
Intersection (Not necessarily independent)
P(A∪B) = P(A) + P(B) - P(A∩B)
Not Mutually Exclusive
P(A∪B) = P(A) + P(B)
IF Mutually Exclusive
P(B|A) = P(A|B) P(B) / P(A)
Bayes Rule Single Event
P(B|A) = P(A|B) P(B) / P(A|B) P(B) + P(A|B⁻) P(B⁻)
Bayes Rule Two Events
P(E) = # favorable outcomes / total outcomes
Probability of an Event
P(E)
Probability of an Event
P(n, k) = n! / (n-k)!
Permutation Without Replacement (Distinguishable)
P(Ã) = 1-P(A)
Complement
Pr(n, k) = n^k
Permutation With Replacement (Distinguishable)
Probability
A finite measure with any value from 0 to 1
With Replacement
Each time same probability as the probability of the initial set
Without Replacement
Set of possibilities reduces by 1 after each selection
Ω (Sample Space)
A Collection of all Outcomes of an experiment.
i.e., Heads, Tails
∩ (Intersection)
Outcome that is in one "AND" the other
∪ (Union)
Outcome that is in one "OR" the other
4.1 randomness
...
4.2 probability models
...
4.3 random variables
-random variable is a variable that assigns a number to each outcome of an experiment. This is not to be confused with an algebraic variable.
-the probability distribution of a random variable is a listing of each possible outcome of a random variable together with that outcomes probability
-X: X1, X2, X3...
-P(X): P1,P2,P3...
example: toss a coin 3 times. Let X=the number of heads
-X:0,1,2,3
-P(X): 1/8,3/8,3/8,1/8
4.4 properties of random variables
definitions:
expected value (or mean) of a random variable: this is denoted E(X)
Variance of a random variable: this is denoted V(X)
at a hospital, the probability of a patient having surgery is 12%, and obstetric treatment 16% and the probability of both is 2%. What is the probability that a patient will have neither treatment?
.74
benford's law, also called the first-digit law
states that for certain kinds of data, the first digit in each data value has a curious frequency
-this can be used to access the legitimacy of certain date
-for appropriate data, first digits have the following distribution )with the last value missing
-first digit: 1 2 3 4 5 6 7 8 9
-Probability: .301 .176 .125 .097 .079 .067 .058 .051 ?
-1. what is the probability that the first digit is 9? .046
-2. What is the probability that the first digit is at least 2? (pp says .699 but I don't understand that)
Better example
-woman visits her doc and gets tested for rare disease
-doc indicates that the test is 99% accurate (false positive=1%)
-woman tests positive, she concludes there is a 99% chance she has the disease
-this is a rare disease, suppose the incidence in the population is 1 in 50,000.
-if 50,000 people are tested, we would expect 500 to test positive even though only one person has the disease
-thus, even after testing positive, she only has a 1 in 500 chance of having the disease
Calculating probability: Roll a die twice, what is the probability that the sum of the faces will be 8?
P(Sum=8)=5/36
Caution
a random variable does not share the same properties as an algebraic variable
-for an algebraic variable X: X+X+X=3X
-for a random variable, each X may turn out differently, so X+X+X doesnotequal 3X
-this distinction matter when calculating variance.
-X+X+X should really be written X1+X2+X3
Class Problem: A card is drawn from a deck of 52 cards.
-what is the probability that it is neither a diamond nor an ace?
-What is the probability that it is either not a diamond or it is not an ace?
-13 cards are diamonds and 3 more are aces, that leaves 36 cards, so 36/52= .6923
-there is only one card that doesn't fit either category-the ace of diamonds, so 51/52= .9808
Class problem: employee bonuses are awarded at the end of the year. Thomas realizes it is possible for him to get a $5000 bonus, but it is unlikely. He is twice as likely to get a $2000 bonus, seven times as likely to get a $1000 bonus, and ten times as likely to get a $500 bonus.
-construct the probability distribution for Thomas's bonus (first call the probability of getting a $5000 bonus p)
Bonus: 5000 2000 1000 500
probability: p 2p 7p 10p
-sum of probabilities = 1 > 20p=1 > p=0.05
...
bonus: 5000 2000 1000 500
Probability: .05 .10 .35 .50
...
E(X)=(5000)(.05)+(2000)(.10)+(1000)(.35)+(500)(.5)=1050
V(X)=(5000-1050)^2(.05)+(2000-1050)^2(.10)+(1000-1050)^2(.35)+(500-1050)^2(.5)=(powerpoint says 1,022,500 but I got 378,996,250)
...
CLASS PROBLEM: In real estate ads it is found that 64% of homes have garages, 9% have pools, and 28% have a finished basement. 5% have a garage and a pool, 19% have a garage and a basement, 4% have a basement and a pool, and 2% have all three. What percentage of homes do not have any of these three?
G=64-2=62-3-17=42
P=9-2=7-3-2=2
B=28-2=26-17-2=7
G&P=5-2=3
G&B=19-2=17
B&P=4-2=2
All=2
100-42-2-7-3-17-2-2=25
Class problem: John is suing his landlord. If he wins. he will be awarded $6000 and will not have to pay any court costs. If he loses, he will have to pay court fees totaling $200.
-john has found a lawyer that will represent him for $1200. If he hires this lawyer, there is an 80% chance he will win, and if he represents himself there is only a 60% chance that he will win.
-should john hire this lawyer? (calculate his expected net winnings using the lawyer and his expected net winnings not using the lawyer)
With lawyer: 4800 -1400
P(X): .8 .2
-E(X)= (4800)(.8)+(-1400)(.2)=3560
Without lawyer: 6000 -200
P(X): .6 .4
-E(X)=(6000)(.6)+(-200)(.4)=3520
Class problem: K, A, and M have completed several relay triathlons. K-swimming, A-bikes, M-runs. Their respective completion times (in hours) have means .77, 1.33, and .9, and their respective standard deviations are .05, .08, and .06.
a) what is their expected team finish time?
b) what is the standard deviation of the team finish time?
c) assume their team finish times are normally distributed. What is the probability that they finish the triathlon 15 minutes earlier than usual?
a)E(K+A+M)=E(K)+E(A)+E(M)=.77+1.33+.9=3
b)V(K+A+M)=V(K)+V(A)+V(M)=.0025+.0064+.0036=.0125
oK+A+M=Square root of .0125=.1118
c) T N(3, .1118) > P(T<2.75)=P(Z<2.236)=0.0127
Class Problem: Out of 125 students surveyed, 12 were accounting majors, 24 were business majors, and 34 were either an accounting major or business major (or both). Draw and label a Venn Diagram
Acc 10 Both 2 Bus 22
Class problem: Roll a die twice, what is the probability that the number on the second cast is greater than the one on the first cast?
P(2nd>1st) = 15/36=5/12
CLASS PROBLEM: The probability of encountering heavy traffic on a Monday is 0.8, and the probability of encountering heavy traffic on a Tuesday is 0.6
1. someone claims the probability of heavy traffic occurring both days is .3, why is this impossible?
2. the person retracts their claim, but insists that Monday and Tuesday are independent of each other. What is the probability of encountering heavy traffic on Monday or Tuesday?
3. What is the probability of encountering heavy traffic at least one Tuesday in a Month? (successive Tuesdays are independent)
1. 0.8+0.6-0.3=1.1
2. P(M or T)= P(M)+P(T)-P(M and T)= 0.8+0.6-(0.8)(0.6)=0.92
3. P(equal to or greater than 1)=1-P(none)=1-(0.4)^4=0.9744
CLASS PROBLEM: Toss a coin, if it lands heads, roll a die once. If it lands tails, flip the coin one more time. What is the sample space, and what is the size of the sample space?
S={(H,1), (H,2),...(H,6), (T,H), (T,T)} lsl=8
Events
-An event is some set of outcomes from the sample space. events are denoted by capital letters A,B,C....
-The complement of an event A is the event that A doesn't happen
...
-It is denoted A^c and may be thought of as the event "not A"
...
-Two events are Independent if the probability of one occurring is not influenced by the other occurring
...
Events
-The event A and B is the set of outcomes that belong to both sets (their overlap)
-The event A or B is both sets taken together.
...
-the Empty set, denoted ø is the set containing no elements at all
...
-two events are disjoint if they cannot both occur
...
-disjoint events are sometimes called mutually exclusive, since the occurrence of one excludes the possibility of the other occurring
...
Example:
S = {0,1,2,3,4,5,6,7,8}
A = {2,3,6,7} B = {0,3,6,8}
A and B = {3,6}
A or B = {0,2,3,6,7,8}
A^c and B = {0,8}
A^c or B^c = {0,1,2,4,5,7,8}
(A and B)^c = {0,1,2,4,5,7,8}
(A or B)^c = {1,4,5}
A and A^c = {ø}
Example: the american vet ass. claims that the annual cost of medical care for dogs averages $100 with a standard deviation of 30$, and the annual cost of medical care for cats averages $130 with a standard deviation of $35
a) what's the expected difference in cost between cats and dogs?
b) what's the standard deviation of the difference between cats and dogs?
c) if the differences in costs is normally distributed, what's the probability that the medical expenses for a woman's dog is greater than that for her ca?
a) E(C-D)=E(C)-E(D)=120-100=$20
b) V(C-D)=V(C)+V(D)=1225+900=2125 > O c-d=$46.1
c) we are told the difference is normal, and we already found the center and spread. Difference N(20,46.1)
P(difference<0)=P(Z<(0-20/46.1)=P(Z<-.4338)=.3322
Example:
suppose X and Y are independent, and E(X)=120 ox=12 E(Y)=300 ox=16
Find the mean and standard deviation of 2X-5Y
E(2X-5Y)=2E(X)-5E(Y)=2(120)-5(300)=-1260
V(2X-5Y)=V(2X)+V(5Y)=4V(X)+25V(Y)=4(144)+25(256)=6976 > o2x-5y=square root of 6976=83.522
Examples:
calculate the mean and standard deviation of the following random variable:
-X: -2 3 7
-P(X): .3 .1 .6
-E(X)= (-2)(.3)+(3)(.1)+7(.6)=3.9
-V(X)=(-2-3.9)^2(.3)+(3-3.9)^2(.1)+(7-3.9)^2(.6)=16.29
Examples:
In a game, a die is thrown. Alan pays Sally $1 if the die falls 1,2, or 3, and $3 if the die falls 4 or 5. If the die falls 6, Sally has to pay Alan $8. What is the expected value and standard deviation of the amount Sally wins?
Winnings X: 1 3 -8
P(X): 0.5 0.333 0.1667
-E(X)=(1)(0.5)+(3)(0.333)+(-8)(0.1667)=(power point got $0.1667 but my calculations were $0.1654)
-PwPtV(X)=(1-.1667)^2(0.5)+(3-.1667)^2(.333)+(-8-.1667)^2(.1667)= 14.13
-myV(X)= (1-.1654)^2(0.5)+(3-.1654)^2(.333)+(-8-.1654)^2(.1667)=14.13
Gambler's fallacy, or "law of averages"
psychological prejudice that assumes observations will behave as expected much sooner than necessary.
In other words, thinking an event is "due" or "not due"
-playing a different lottery number than last week's winning number because the chances it would come up twice in a row are so small.
-building your home in the exact spot that a meteor struck reasoning it would almost impossible for a meteor to strike in the same place twice.
-a man brings a bomb on a plane. he reasons "the chances of there being a bomb on a plane are so small, so the chances of there being another one are almost zero"
INDEPENDENCE:
two events A and B are independent if P(A and B)=P(A)*P(B)
Example: P(A)= .3 P(B)= .5 P(A and B)= .10
.15 does not equal .10 so A and B are not independent
Example: P(A)=.2 P(B)= .6 P(A or B)= .68
are A and B independent? (first use addition rule)
By the addition rule. P(A and B)=.12 and (.2)(.6)=.12, so A and B are independent.
...
-do not confuse independence with disjoin. Independence cannot be illustrated on a Venn diagram.
...
Law of Large Numbers
states that as an experiment is repeated over and over, the observed frequency of an outcome gets closer to its expected frequency.
probability
the probability of an outcome is the proportion of times that it would occur over many repetitions.
-often, people expect the outcomes to settle into some regularity much sooner than they actually do.
Properties of Mean and Variance
E(c)=c V(c)=0 E(X+/-Y)=E(X)+/-E(Y)
E(cX)=cE(X) V(cX)=c^2V(X)
if X and Y are independent: V(X+/-Y)=V(X)+V(Y)
Prosecutor's fallacy
-a man is on trial for a crime, and forensic evidence is found at the scene which implicates him.
-a prosecutor has an expert witness testify that the probability of finding this forensic evidence is 1 in 20,000 if the person is innocent
-by itself, this argument is misleading...
-the defense counters that there are 1,000,000 ppl in this city and so there are 50 people who could have left this evidence.
-thus there is still only a 1 in 50 chance that the defendant is the one that left this evidence
-the prosecutor would have to make an argument that significantly narrows down this pool of 40 people, like additional evidence.
-this is tantamount to someone winning the lottery, and the prosecutor charging them of cheating because the odds of winning were so low.
Random
a phenomenon is random if any individual outcome is unpredictable, but the distribution of outcomes over many repetitions is known
example: toss a coin. no flip is predictable, but many flips will result in approximately half heads and half tails
-remember that random does not mean that each outcome is equally likely, it only means that a particular outcome cannot be predicted with certainty
record the number of people that walk into a post office each day.
a) what is the sample space?
b) How do you think the outcomes will be distributed (what shape)
a) S={0,1,2,3,....) lsl= infinity
b) skewed-right
Rules of Thumb (1)
1) "and" means multiply when the events are independent
-toss a coin three times. what is the probability of all three being tails?
-that is, tails first AND tails second AND tails third
-since coin flips are independent we multiple, .5x.5x.5=.125
Rules of Thumb (2)
2) "or" means add when the events are disjoint
-roll two dice. What is the probability that the sum of the faces is 5 or 11?
-since the sum cannot be 5 and 11 at the same time, these are disjoint outcomes, so we add: P(sum=5)+P(sum=11)=4/36+2/36=1/6
Rules of Thumb (3 continued)
-if there are 23 people in a room, what is the probability that at least two of them have the same b-day?
-P(at least 2) = 1-P(all different)= # different bdays for 23 people/# possible bdays for 23 people = 1- (365364363...343/365365365...*365)=1-.4927 = 50.73%
Rules of Thumb (3)
3) for any probability question, first decide whether it is easier to calculate it directly, or easier to calculate the opposite and subtract from 1.
-a coin is tossed 7 times, what is the probability of tails occurring at least once?
-easier to answer the opposite: P(tails at least once)=1-P(no tails)
-"no tails" means "heads first AND heads second AND..."
P(no tails)=.5 x .5 x .5 x .5 x .5 x .5 x .5= .0078
P(at least once)=1-P(no tails)=1-0.0078=.9922
sample space
the sample space is the set of all possible outcomes, denoted S
example: toss a coin three times. The sample space is ... S={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
-the size of S is denoted lSl.
-example: toss a die twice. The sample space is... S={(1,1), (1,2),...(1,6), (2,1), (2,2),...(2,6),...(6,6)}
example: pull two cards from a well-shuffled deck. How many elements are in the sample space?
The Addition Rule:
P(A or B)=P(A) + P(B)-P(A and B)
Overlap counted twice, subtract out once.
In an office building of 80 people, 28 work on Saturday, 11 work on Sunday, and 3 people work on both Sunday and Saturday. What is the probability that a person in this office works at least one of these days?
P(Sat or Sun)= P(Sat) + P(Sun) - P(Both) = 28/80+11/80-3/80=.45
Basic Rules for
Computing Probability (Rule 1) -
Relative Frequency Approximation of Probability
P(A) = # of times A occurred / # of times procedure was repeated
Basic Rules for
Computing Probability (Rule 2) -
Classical Approach to Probability
(Requires Equally Likely Outcomes)
P(A) = number of ways A can occur / number of different simple events
...
Basic Rules for
Computing Probability (Rule 3) - Subjective Probabilities
P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.
Combinations Rule Formula
nCr = n! / ( (n - r)! * r!)
Combinations Rule
Requirements:
There are n different items available.
We select r of the n items (without replacement).
We consider rearrangements of the same items to be the same. (The combination of ABC is the same as CBA.)
Complementary Events
The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur
Complements: The Probability of "At Least One"
"At least one" is equivalent to "one or more."
The complement of getting at least one item of a particular type is that you get no items of that type.
...
Compound Event
P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)
Conditional probability
Find the probability of an event when we have additional information that some other event has already occurred.
Confusion of the Inverse
To incorrectly believe that P(A|B) and P(B|A) are the same, or to incorrectly use one value for the other, is often called confusion of the inverse.
Dependent and Independent
Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are similarly independent if the occurrence of any does not affect the probabilities of the occurrence of the others.) If A and B are not independent, they are said to be dependent.
Disjoint or Mutually Exclusive
Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.)
Event
any collection of results or outcomes of a procedure
Factorial Rule
A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects the fact that the first item may be selected in n different ways, the second item may be selected in n - 1 ways, and so on.)
Finding the Probability of "At Least One"
P(at least one) = 1 - P(none)
Formal Multiplication Rule
P(A and B) = P(A) • P(B | A)
Note that if A and B are independent events, P(B A) is really the same as P(B).
...
Formal Addition Rule
When the event-A and event-B are NOT mutually exclusive, use
P(A or B) = P(A) + P(B) - P(A and B)
...
where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure.
...
Fundamental Counting Rule
For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.
General Rule for a
Compound Event
When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find that total in such a way that no outcome is counted more than once
Intuitive Approach to Conditional Probability
The conditional probability of B given A can be found by assuming that event A has occurred, and then calculating the probability that event B will occur.
Law of Large Numbers
As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.
Notation for
Probabilities
P - denotes a probability.
A, B, and C - denote specific events.
P(A) - denotes the probability of event A occurring.
Notation for Conditional Probability
P(B|A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B|A as "B given A.")
Permutations Rule
(when items are all different)
Requirements:
There are n different items available. (This rule does not apply if some of the items are identical to others.)
We select r of the n items (without replacement).
...
We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.)
...
Permutations Rule
(when some items are identical to others)
Requirements:
There are n items available, and some items are identical to others.
We select all of the n items (without replacement).
...
We consider rearrangements of distinct items to be different sequences.
...
Permutations Rule Formula
(when items are all different)
nPr = n! / (n - r)!
Permutations Rule Formula
n! / (n(1)! n(2)! ... n(k))
Permutations versus Combinations
When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem.
Probability Limits
Always express a probability as a fraction or decimal number between 0 and 1.
The probability of an impossible event is 0.
...
The probability of an event that is certain tooccur is 1.
...
For any event A, the probability of A is between 0 and 1 inclusive.
That is, 0 <= P(A) <= 1
...
Probability of "at least one"
Find the probability that among several trials, we get at least one of some specified event.
Rare Event Rule for Inferential Statistics
If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.
Rounding Off Probabilities
When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)
Rule of Complementary Events
P(A) + P(Abar) = 1
P(Abar) = 1 - P(A)
P(A) = 1 - P(Abar)
Sample Space
for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further
Simple Event
an outcome or an event that cannot be further broken down into simpler components
The 5% Guideline for Cumbersome Calculations
If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).
complement
E (event) does not occur
event
specified result that may or may not occur when an experiment is performed
exhaustive events
If one includes all the possible outcomes, then A and A' are exhaustive because one of them must happen.
experiment
action whose outcome cannot be determined with certainty
frequentist interpretation of probability
probability of an event proportional to number of times event occurs in a large number of repetitions of the experiment
inferential statistics
used to interpret data and draw conclusions of whole based on sample
joint probabilities
probabilities that correspond to the events represented in the cells of the contingency table
marginal probabilities
probabilities that correspond to the events represented in the margin of the contingency table.
mutually exclusive events
Two events that cannot occur at the same time; no common outcomes
posterior probability
the probability that a hypothesis is true after consideration of the evidence
prior probability
initial probability of a state of nature before sample information is used with Bayes Theorem
probability model
a mathematical description of a random phenomenon consisting of a sample space and a way of assigning probabilities to events
probability
a measure of how likely it is that some event will occur; science of uncertainty
building blocks of probability
experiment, outcome, sample space, event
classical method of assigning probabilities
used when an experiment has equally likely outcomes
equally likely outcomes
outcomes that have the same probability of occurring
experiment
any activity for which the outcome is uncertain
outcome
the result of a singe performance of an experiment; a set of outcomes is denoted with braces {}
probability model
a table or listing of all the possible outcome of an experiment, together with the probability of each outcome; must follow the Rules of Probability
probability
of an outcome is defined as the long-term proportion of times the outcome occurs; a number that indicates how likely the particular outcome is
sample space
the collection of all possible outcomes; denoted with an S
simulation
uses methods such as rolling dice or computer generation of random numbers to generate results from an experiment.
what does the notion P(A) stand for?
the probability that outcome A occurred
addition rule for disjoint events
If E and F are disjoint events, then P(E or F) = P(E) +P(F)
Benford's Law
Mathematical algorithm that accurately predicts that, for many data sets, the first digit of each group of numbers in a random sample will begin with 1 more than a 2, a 2 more than a 3, a 3 more than a 4, and so on. Predicts the percentage of time each digit will appear in a sequence of numbers.
certainty
an event with a probability of 1
combination
a collection, without regard to order, of n distinct objects without repetition.
complement of an event
the probability that an event does not occur; all outcomes in a sample space that are not outcomes in the event
complement rule
P(Ec) = 1 - P(E)
conditional probability
the probability that an event occurs, given that another event has occurred
contingency table
A table that relates two categories of data; two-way table. Variables are placed in rows and columns; each intersection of variables is a cell in the table.
dependent events
events where the probability of one affects the probability of the other
disjoint events
two events that have no outcomes in common; mutually exclusive events
equation for approximating probabilities using the empirical approach
P(E) ≈ relative frequency of E =
(frequency of E)/(number of trials of experiment)
equation for computing probability using the classical method
P(E) = (number of ways that E can occur)/ (number of possible outcomes) = m/n
event
any collection of outcomes from a probability experiment, consisting of one or more outcomes
experiment
any process with uncertain results that can be repeated
E
event
factorial symbol (n!)
if n ≥ 0 is an integer, the factorial symbol, n!, is defined as follows:
n! = n(n-1)∗⋅⋅⋅∗3∗2∗1
fair die
a die where each possible outcome is equally likely
general addition rule
P(E or F) = P(E) + P(F) - P(E and F)
general multiplication rule
P(E and F) = P(E) ∗ P(F|E)
impossible
an event with a probability of 0
independent events
events whose probability do not affect each other
multiplication rule for independent events
P(E and F) = P(E) ∗ P(F)
multiplication rule for n independent events
P(E and F and G and ...) = P(E) ∗ P(F) ∗P(G)
multiplication rule of counting
If as task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, etc., then the task of making these selections can be done in
p∗q∗r∗⋅⋅⋅ ways
mutually exclusive events
two events that have no outcomes in common; disjoint events
m
the number of ways that an event E can occur
nCr
combination of n objects taken r at a time
nPr
permutation of n objects taken r at a time
number of combinations of n distinct objects taken r at a time
The number of different arrangements of n objects using r ≤ n of them, in which
1. the n objects are distinct
2. repetition of objects is not allowed
3. order is not important
nCr = n! / [r!(n-r)!]
...
number of permutations of distinct objects in groups
The number of arrangements of r objects chosen from n objects in which
1. the n objects are distinct
2. repetition of objects is not allowed
3. order is important
nPr = n!/(n-r)!
...
n
number of equally likely outcomes
permutation
an arrangement in which r objects are chosen from n distinct objects, repetition is not allowed, and order is important.
probability model
lists the possible outcomes of a probability experiment and each outcome's probability
probability of an outcome
the long-term proportion with which a certain outcome is observed
probability
a measure of the likelihood of a random phenomenon or chance behavior
Rules of probability
1. The probability of any event must be between 0 and 1, inclusive. 0 ≤ P(E) ≤ 1.
2. The sum of the probabilities of all outcomes must equal 1.
3. If E and F are disjoint events, then P(E or F) = P(E) + P(F). If E and F are not disjoint events, then P(E or F) = P(E) + P(F) - P(E and F)
4. If E represents any event and Ec represents the complement of E, then P(Ec) = 1 - P(E)
5. If E and F are independent events, then P(E and F) = P(E)∗P(F)
sample space
the collection of all possible outcomes
simple event
an event with only one outcome
subjective probability
a probability obtained on the basis of personal judgment
S
sample space
the law of large numbers
as the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome
tree diagram
a diagram to determine a sample space that lists the equally likely outcomes of an experiment
unusual event
an event that has a low probability of occurring, typically less than 5%
Venn diagram
A diagram that uses circles contained within a rectangle to display elements of different sets. The rectangle represents the sample space, and circles represent events.
simulation
the imitation of change behavior, based on a model that accurately reflect the phenomenon under consideration
independent
when the results of one variable don't affect another
randInt(min,max,num)
randomly selects num integers from min to max
random
when individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions
probability
the long-term relative frequency
sample space (S)
the set of all possible outcomes
event
and outcome of a random phenomenon, a subset of the sample space
probability model
a mathematical description of a random phenomenon consisting of a sample space and a way of assigning probabilities to events
tree diagram
enumerates each outcome in the sample space
Multiplication Principle
If you can do one task in n1 ways and a second task in n2 ways, then both tasks can be done in n1*n2 ways
sampling with replacement
when the second draw is exactly like the first
sampling without replacement
when you don't replace the things you select
disjoint
have no outcomes in common and cannot occur simultaneously
complement
the probability that something doesn't happen
union
the or of two sets
joint event
the simultaneous occurence of two events
conditional probability
the probability of one event given another
intersection
the and of some events
Area and Probability
Because the total area under the density curve is equal to 1, there is a correspondence between area and probability.
Binomial Probability Distribution
1.The procedure must have a fixed number of trials.
2. The trials must be independent.
3. Each trial must have all outcomes classified into two categories (commonly, success and failure).
4.The probability of success remains the same in all trials
Central Limit Theorem - continued
Conclusions:
1. The distribution of sample x will, as the sample size increases, approach a normal distribution.
2. The mean of the sample means is the population mean µ.
3. The standard deviation of all sample means is σ / (n)^(1/2)
Central Limit Theorem Description
for a population with any distribution, the distribution of the sample means approaches a normal distribution as the sample size increases.
Central Limit Theorem Requirements
Given:
1. The random variable x has a distribution (which may or may not be normal) with mean µ and standard deviation σ
2. Simple random samples all of size n are selected from the population. (The samples are selected so that all possible samples of the same size n have the same chance of being selected.)
continuity correction
When we use the normal distribution (which is a continuous probability distribution) as an approximation to the binomial distribution (which is discrete), a continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by the interval from
x - 0.5 to x + 0.5
(that is, adding and subtracting 0.5).
Conversion Formula
z = (x - μ) / σ
Density Curve
A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
1. The total area under the curve must equal 1.
2. Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)
Helpful Hints
1. Don't confuse z scores and areas. z scores are points along the horizontal scale, but areas are regions under the normal curve.
2. Choose the correct (right/left) side of the graph.
3. A z score must be negative whenever it is located in the left half of the normal distribution.
4. Areas (or probabilities) are positive or zero values, but they are never negative.
mean of the sample means formula
µ(xbar) = µ
Practical Rules Commonly Used
1. For samples of size n larger than 30, the distribution of the sample means can be approximated reasonably well by a normal distribution. The approximation gets closer to a normal distribution as the sample size n becomes larger.
2. If the original population is normally distributed, then for any sample size n, the sample means will be normally distributed (not just the values of n larger than 30).
standard deviation of sample mean or standard error of the mean~σ(x) = σ / (n)^(1/2)
Standard Normal Distribution
The standard normal distribution is a normal probability distribution with μ = 0 and σ = 1. The total area under its density curve is equal to 1.
Uniform Distribution
A continuous random variable has a uniform distribution if its values are spread evenly over the range of probabilities. The graph of a uniform distribution results in a rectangular shape.
using a normal distribution as an approximation to the binomial probability distribution.
If the conditions of np ≥ 5 and nq ≥ 5 are both satisfied, then probabilities from a binomial probability distribution can be approximated well by using a normal distribution with mean μ = np and standard deviation σ = (n p * p * q) ^ (1/2)
Chi-Square Distribution formula
X^2 = ( [ n -1 ] * s^2) / σ^2
Chi-Square Distribution
In a normally distributed population with variance σ^2 assume that we randomly select independent samples of size n and, for each sample, compute the sample variance s2 (which is the square of the sample standard deviation s). The sample statistic x^2 (pronounced chi-square) has a sampling distribution called the chi-square distribution.
Choosing the Appropriate Distribution
Use the normal (z) distribution
If σ known and normally distributed population or σ known and n > 30
Use t distribution
if σ not known and normally distributed population or σ not known and n > 30
...
Use a nonparametric method or bootstrapping
If Population is not normally distributed and n ≤ 30
...
confidence interval (or interval estimate)
range (or an interval) of values used to estimate the true value of a population parameter. A confidence interval is sometimes abbreviated as CI.
Confidence Interval for Estimating a Population Mean (with σ Known)
1. The sample is a simple random sample. (All samples of the same size have an equal chance of being selected.)
2. The value of the population standard deviation σ is known.
3. Either or both of these conditions is satisfied: The population is normally distributed or n > 30.
Confidence Interval for Estimating a Population Mean (with σ Known)
xbar - E < μ < xbar + E
or
xbar +/- E
or
(xbar - E, xbar + e)
where E = z(α/2) * ( σ / (n)^(1/2) )
Confidence Interval for Estimating a Population Proportion p notation
p̂ - E < p̂ < p̂ + E
p̂ +/- E
(p̂ - E, p̂ + E)
Confidence Interval for Estimating a Population Proportion p
p̂ - E < p̂ < p̂ + E
where
E = z(α/2) ( [p̂ * q̂] / n ) ^(1/2)
Confidence Interval for Estimating a Population Standard Deviation or Variance
( [n - 1] s^2 ) / X(r)^2 < σ^2 < ( [ n - 1 ] * s^2) / X(L)^2
confidence interval limits
xbar - E, xbar + E
Confidence Intervals for Comparing Data Caution
Confidence intervals can be used informally to compare the variation in different data sets, but the overlapping of confidence intervals should not be used for making formal and final conclusions about equality of variances or standard deviations.
confidence level, degree of confidence, or the confidence coefficient.
is the probability 1 - α (often expressed as the equivalent percentage value) that the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times.
Critical Value
A critical value is the number on the borderline separating sample statistics that are likely to occur from those that are unlikely to occur.
degrees of freedom
The number of degrees of freedom for a collection of sample data is the number of sample values that can vary after certain restrictions have been imposed on all data values. The degree of freedom is often abbreviated df.
degrees of freedom = n - 1 in this section
Finding a Sample Size for Estimating a Population Mean
n = ( [z(α/2) * σ] / E)^2
Finding the Point Estimate and E from a Confidence Interval
Point estimate of µ:
xbar = (upper confidence limit + lower confidence limit) / 2
Margin of Error:
E = (upper confidence limit - lower confidence limit) / 2
...
Finding the Point Estimate and E from a Confidence Interval
Point estimate of
p̂ = (upper confidence limit + lower confidence limit) / 2
Margin of error E = (upper confidence limit - lower confidence limit) / 2
Important Properties of the Student t Distribution
1. The Student t distribution is different for different sample sizes (see the following slide, for the cases n = 3 and n = 12).
2. The Student t distribution has the same general symmetric bell shape as the standard normal distribution but it reflects the greater variability (with wider distributions) that is expected with small samples.
3. The Student t distribution has a mean of t = 0 (just as the standard normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the sample size and is greater than 1 (unlike the standard normal distribution, which has a σ = 1).
5. As the sample size n gets larger, the Student t distribution gets closer to the normal distribution.
Margin of Error for Proportions
E = z(α/2) ( [p̂ * q̂] / n ) ^(1/2)
Point Estimate of the Population Mean
The sample mean xbar is the best point estimate of the population mean µ.
point estimate
single value (or point) used to approximate a population parameter.
Procedure for Constructing a Confidence Interval for p
1.Verify that the required assumptions are satisfied. (The sample is a simple random sample, the conditions for the binomial distribution are satisfied, and the normal distribution can be used to approximate the distribution of sample proportions because np >= 5, and nq >= 5 are both satisfied.)
2. Refer to Table A-2 and find the critical value z(α/2) that corresponds to the desired confidence level.
3. Evaluate the margin of error
4. Using the value of the calculated margin of error, E and the value of the sample proportion, p, find the values of p - E and p + E. Substitute those values in the general format for the confidence interval: p̂ - E < p̂ < p̂ + E
5. Round the resulting confidence interval limits to three significant digits.
Procedure for Constructing a Confidence Interval for µ (With σ Unknown)
1. Verify that the requirements are satisfied.
2. Using n - 1 degrees of freedom, refer to Table A-3 or use technology to find the critical value t(α/2) that corresponds to the desired confidence level.
3. Evaluate the margin of error E = t(α/2) • [ s / n^(1/2 ] .
4. Find the values of xbar - E and xbar + E. Substitute those values in the general format for the confidence interval: xbar - E < μ < xbar + E
5. Round the resulting confidence interval limits
Procedure for Constructing a Confidence Interval for σ or σ^2
1. Verify that the required assumptions are satisfied.
2. Using n - 1 degrees of freedom, refer to Table A-4 or use technology to find the critical values X(r)^2 and X(L)^2 that correspond to the desired confidence level/
3. Evaluate the upper and lower confidence interval limits using this format of the confidence interval:
( [n - 1] s^2 ) / X(r)^2 < σ^2 < ( [ n - 1 ] * s^2) / X(L)^2
4. If a confidence interval estimate of is desired, take the square root of the upper and lower confidence interval limits and change σ^2 to σ.
5. Round the resulting confidence level limits. If using the original set of data to construct a confidence interval, round the confidence interval limits to one more decimal place than is used for the original set of data. If using the sample standard deviation or variance, round the confidence interval limits to the same number of decimals places.
Procedure for Constructing a Confidence Interval for µ (with Known σ)
1. Verify that the requirements are satisfied.
2. Refer to Table A-2 or use technology to find the critical value z(α/2) that corresponds to the desired confidence level
3. Evaluate the margin of error E = z(α/2) * ( σ / (n)^(1/2) )
4. Find the values of xbar - E and xbar + E. Substitute those values in the general format of the confidence interval
5. Round using the confidence intervals round-off rules.
Properties of the Distribution of the Chi-Square Statistic
1. The chi-square distribution is not symmetric, unlike the normal and Student t distributions.
As the number of degrees of freedom increases, the distribution becomes more symmetric.
2. The values of chi-square can be zero or positive, but they cannot be negative.
3. The chi-square distribution is different for each number of degrees of freedom, which is df = n - 1. As the number of degrees of freedom increases, the chi-square distribution approaches a normal distribution.
In Table A-4, each critical value of X^2 corresponds to an area given in the top row of the table, and that area represents the cumulative area located to the right of the critical value.
Round-Off Rule for Confidence Intervals Used to Estimate µ
When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data.
When the original set of data is unknown and only the summary statistics (n, x, s) are used, round the confidence interval limits to the same number of decimal places used for the sample mean.
Round-Off Rule for Determining Sample Size
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.
Round-Off Rule for Sample Size n
If the computed sample size n is not a whole number, round the value of n up to the next larger whole number.
Sample Mean
1. For all populations, the sample mean x is an unbiased estimator of the population mean xbar, meaning that the distribution of sample means tends to center about the value of the population mean μ.
2. For many populations, the distribution of sample means x tends to be more consistent (with less variation) than the distributions of other sample statistics.
Sample Mean
The sample mean is the best point estimate of the population mean.
sample proportion
The sample proportion is the best point estimate of the population proportion.
Sample Size for Estimating Proportion p
When an estimate of p̂ is known
n = (z(α/2)^2 * q̂) / E^2
When no estimate of p is known:
n = (z(α/2)^2 * 0.25) / E^2
Student t Distribution
If the distribution of a population is essentially normal, then the distribution of
t = (xbar - μ) / [ s / n^(1/2) ]
0 ≤ P ≤ 1
Any probability is a number between 0 and 1.
Addition rule for disjoint events
P(A or B) = P(A) + P(B)
If two events are disjoint, the probability of getting one or the other is the sum of their individual probabilities.
Continuous Probability Model
A probability model that assigns probabilities as areas under a density curve; the probability of any event is the area under the curve and above the values on the horizontal axis that make up the event.
Density Curve
A curve that is on or above the horizontal axis and has an area of exactly 1 underneath it. It describes the overall pattern of a distribution. Merely a model - no set of real data is exactly described by a density curve.
Discrete/Categotical Probability Model
A probability model with a sample space made up of a finite list of individual outcomes.
Disjoint
When two events have no outcomes in common and so can never occur together.
Empirical Probability
A probability calculated from our knowledge of numerous similar past events.
Event
An outcome or set of outcomes of a random phenom; a subset of the sample space.
Finite Sample Space
A sample space dealing with either discrete or categorical variables that can take on only certain values.
Intervals and Areas of Density Curves
For continuous probability models using a density curve, events are defined over intervals of values, and probability is computed as areas under the density curve.
Mean "mu" of a density curve
The point at which the density curve would be balanced, if it were physical.
Odds
The ratio of the probability of an outcome of a random phenom over the probability of that outcome not occurring.
P(A does not occur) = 1 - P(A)
The probability of an event not occurring is equal to 1 minus the probability of the event happening.
Personal/Subjective Probability
A number between 0 and 1 that expresses an individual's judgement of how likely the outcome is.
Probability Distribution
The distribution of a random variable X that tells us what values X can take and how to assign probabilities to those values.
Probability Model
A mathematical description of a random phenom consisting of two parts: a sample space S, and a way of assigning probabilities to events.
Probability
The proportion of times an outcome will occur in a very long series of repetitions.
Random Phenomenon
When the individual outcomes of a phenomenon are uncertain but there is nonetheless a regular distribution of outcomes in a large number of repetitions.
Random samples eliminate bias from the act of choosing a sample, but they can still be wrong because of...
... the variability that results when one chooses at random.
Random Variable
A variable whose value is a numerical outcome of a random phenom.
Risk
The probability of getting an undesired outcome.
Sample Space
S
The set of all possible outcomes of a random process.
Standard deviation "sigma" of a density curve
The equals-area point of a density curve.
The idea of probability
Chance behavior is unpredictable short-term, but has a regular and predictable pattern in the long run.
The outcome of a single individual outcome for a continuous probability model
All continuous probability models assign a probability of 0 to any individual outcome; only intervals of values can have positive probability.
Theoretical Probability
A probability calculated from understanding the phenom in the problem.
∑P = 1
The sum of all probability outcomes is equal to 1.
# of ways of choosing a subset without replacement. (no regard to order):
n! / (k! * (n - k)!)
# of ways of ordering n objects of different types:
n! /( n1! n2!.....nt!)
1 + 2a + 3a^2 + ........... =
1 / (1-a)^2
Basic Probability: A - B =
A & B'
Basic Probability: A =
(A & B) or (A & B')
Bayes's Theorem: P[A(giv)B] =
(P[B(giv)A] P[A]) / P[B(giv)A] P[A] + P[B(giv)A'] * P[A']
Binomial Distribution with Parameters n & p:
p(x) is the probability that there will be exactly x successes in the n trials
Binomial Distribution: E[X] =
np (mean of the binomial distribution)
Binomial Distribution: Mx(t) =
(1 - p + pe^t)^n
Binomial Distribution: p(x) =
(n choose x)(p^x)(q^n-x)
Binomial Distribution: Var [X] =
np(q)
Binomial Distrubution: p, q, x, n =
p = probability success, q = probability failure, x = successes, n = independent trials
Choosing ordered subset of size k withouth replacement from n objets:
n! / (n-k)! P(n,k)
Conditional Independence:
If P[A(giv)B] = P[A] or P[B(giv)A] = P[B]
Conditional Probability: P[B & A] =
P[B(giv)A] * P[A}
Conditional Probability: P[B] =
P[B(giv)A] P[A] + P[B(giv)A'] P[A']
Continuous Uniform Distribution: E[X] & VAR[X] =
(a + b) / 2 & (b-a)^2 / 12
Continuous Uniform Distribution: f(x) =
1 / (b - a)
Continuous Uniform Distribution: F(x) =
integral (a to x) f(x) dx = (x-a) / (b - a)
COV[X,X] =
VAR[X]
COV[X,Y] =
E[XY] - E[X] * E[Y]
Cumulative Distribution Function:
F(x) = P[X<x] (Probability to the left of, and including, the point x)
DeMorgan's Laws:
(A or B)' = A' & B' ; (A & B)' = A' or B'
Discrete Distribution:
Can only take on values from a finite infinite sequence.
Exhaustive Outcomes:
Combine to the entire probability space. Or, one of the outcomes must occur whenever the experiment is performed.
Exponential Distribution F(x) =
1 - e^(-lamn*x)
Exponential Distribution S(x) =
e^(-lamn*x)
Exponential Distribution with mean (1/lamn), f(x) =
lamn(e^(-lamn*x))
Exponential Distribution: E[X] & VAR[X] =
1/lamn & 1/(lamn^2)
Exponential Distribution: k-th moment E[X^k] =
k! / lamn^k
Exponential Distribution: Mx(t) =
lamn / (lamn - t) t<lamn
For a continuous random variable Mx(t) =
integral (-infinity to infinity): e^(tx) f(x) dx
For a discrete random variable Mx(t) =
Sum (e^(tx) * p(x)
For Continuour Random Variable E[X] =
integral (-infinity to infinity): x* f(x) dx
For Discrete Random Variable E[X] =
x1p(x1) + x2p(x2) + .......
For Discrete Uniform Distribution of N points: E[X] =
(N + 1) / 2
For Discrete Uniform Distribution of N points: p(x) =
1 / N
For Discrete Uniform Distribution of N points: Var[X] =
(N^2 - 1) / 12
For Uniform Joint Distribution, pdf =
1 / area of R
Geometric Distribution: E[X] =
(1-p) / p
Geometric Distribution: Mx(t) =
p / (1-qe^t)
Geometric Distribution: p(x) =
q^x *p
Geometric Distribution: VAR[X] =
(1-p) / p^2
Geometric Distribution: X represents:
the number of failures until the first success
Given n distinct objects, the number of ways in which the objects may be ordered:
n!
How do you Standardize Normal Distribution: P[ r < X < s)
Z = (X-u)/std, =P[(r-u)/std < (X-u)/std < (s-u)/std]
If X & Y are independent, then COV[X,Y] =
0
If X & Y are independent, then E[X*Y] =
E[X] * E[Y]
Joint Distribution of Random Variables:
The Probability of two or more random variables together as a joint distribution
Joint Distribution: If Independent, f(x,y) =
fx(x) * fy(y)
M''x(0) =
E[X^2]
M'x(0) =
E[X]
Marginal Distribution fx(x) =
integral of f(x,y) dy (continuous) sum(y's) f(x,y) (discrete)
Mutually Exclusive Outcomes:
Outcomes that cannot occur simultaneously. (disjoint)
Mx(0) =
1
Mx(t) =
E[e^(tx)]
Normal Distribution: mean, median, mode =
mean (u)
P[A or B (giv) C] =
P[A(giv)C] + P[B(giv)C] - P[A & B (giv) C]
P[A or B or C] =
P[A] + P[B] + P[C] - P[A & B] - P[A & C] - P[B & C] + P[A & B & C]
P[A or B] =
P[A] + P[B] - P[A & B]
P[A'(giv)B] =
1 - P[A(giv)B]
Poisson distribution used as a model for:
Counting the number of events of a certain type that occur in a certain period of time
Poisson Distribution: E[X] & Var[X] =
lamna
Poisson Distribution: Mx(t) =
e^(lamn(e^t-1))
Poisson Distribution: p(x) =
(lamna parameter) ((e^-lamn(lamn^x))/x!
Probability Mass:
The probability at a point of a discrete random variable.
Random Variable:
function on a probability space S
Standard Normal Distribution has mean and variance of:
mean: 0 and Variance: 1
Survival Function:
Complement of the Cumulative Distribution Function
Symmetric about the point c if:
f(c + t) = f(c- t)
The Mode of the Distribution is the point where:
the probability or density function is maximized.
the pdf of Normal Distribution f(x) =
(1 / (std sqr(2pi)) e^((-(x-mean)^2)/(2(std^2)))
The Varianc is a measure of the:
Dispersion of X about the mean. Will always be > or equal to 0
told: exponential distribution with mean of 3 (lamna =)
1/lamn = 3 lamna = 1 / theta
Uniform Probability:
Each sample point has the same probability of occurring
Var[aX + b] =
a^2Var[X]
VAR[X+Y] =
VAR[X] + VAR[Y] + 2*COV[X,Y]
VAR[X] =
E[X^2] - (E[x])^2
When considering combinations, the order of the elements in the set is:
irrelevant: {a,b} = {b,a}
When considering permutations, the order is:
important: {a,b} does not equal {b,a}
addition rule
If A and B are disjoint events: P(A or B)=P(A) + P(B)
boxplot
displays the 5-number summary as a central box with whiskers that extend to the non-outlying data values
combinations
number of ways to combine items in which order doesn't matter
condtional probrability
The conditional probablity of B given A, written P(B/ A) is the probabilty that event B will occur given that event A has occured
dependent event
If the occurrence of one event has no effect on the occurrence of the other
depentent event
Two events such that the occurrence of one event affect the occurrence of the other event
deviation
The difference of a data value and the mean of a data set
dotplot
graphs a dot for each case against a single axis
expected value
a number that makes a rational expression undefine
five- number summary
A list of numbers that lists the minimum, first quartile, median, third-quartile, and the maximum of a data set.
interquartile range
The difference of the upper in lower quartiles of a data set
mean absolute deviation
the average of the absolute deviations for all the data values in the sample
multiplication rule
if events A and B are independent, then P(A and B) = P(A)P(B)
mutually exclusive event
events that have no common outcome, two events that cannot occour at the same time
outliers
Numbers that are much greater or much less than the other numbers in the set
permutations
an arragement or listing in which an order or placement is important
quartile
when data in a set are arranged in order, quartiles are the numbers that split the data into quarters or fourths
random sample
A sample in which every member of the population has an equal chance of being selected
range
The range of a numerical data set is a measure of disperison
skew
Asymmetry in the distribution of the data values or how the is distributed across
symmetry
elements on both sides of a line that have the same shape, size and arrangement
animated
(Adj.) Full of life, lively, alive; (part.) moved to action.
available
(adj.) Ready for use, at hand.
brood
(n.) a family of young animals, especially birds; any group having the same nature or orgin; (v.) to think over in a worried, unhappy way.
cater
(v.) to satisfy the needs of, try to make things easy and pleasant; to supply food and service.
culminate
(v.) to reach a hight point of development; to end, climax.
Customary
(adj.) usual, espected, routine.
difference
the answer of a subtraction problem
dissuade
(v.) to persuade not to do something.
downright
(adv.) throughly; (adj.) absolute, complete; frank, blunt.
drone
(n.) a loafer, idler; a buzzing or humming sound; a remote-control device; a male bee. (v.) to make a buzzing sound; to spead in a dull tone of voice.
entrepreneur
(n.) a person who starts up and takes on the risk of a buisness.
expectancy
the average number of years a person can expect to live in a given country
firebrand
(n.) a piece of burning wood; a troublemaker; an extremely energetic or emotional person.
goad
(v.) to drive or urge on. (n.) something used to drive or urge on.
hdi
a ranking of countries, combining various statistics
homicide
(n.) the killing of one person by another
income
the mathematical average amount of money a typical family makes in US dollars
Indifference
(n.) a lack of interest or concern
Indignant
(adj.) filled with resentment or anger over something unjust, unworthy, or mean
Indispensable
(adj.) absolutely necessary, not to be neglected
indulge
(v.) to give in to a wish or desire, give oneself up to.
ingredient
(n.) one of the materials in a mixture, recipe, or formula
literacy
percent of adults in a given country who can read and write
literate
(adj.) able to read and write; showing an excellent educational background; having knowledge or training.
loom
(v.) to come into view; to appear in exaggerated form. (n.) a machine for weaving.
lubricate
(v.) to apply oil or grease; to make smooth, slippery, or easier to use
luster
(n.) the quality of giving off light, brightness, glitter, brilliance.
miscellaneous
(adj.) mixed, of different kinds
mortality
number of infants, for every 1000 born who die before the age of 1
mutual
(adj.) shared, felt, or shown equally by two or more
oration
(n.) a public speech for a formal occasion.
peevish
(adj) cross, complaining, irratable; contrary
plague
(n.) an easily spread disease causing a large number of deaths; a widespread evil; (v.) to annoy or bother
poised
(adj.part.) balanced, suspended; calm, controlled; ready for action
population
amount of people in a given area
probability
the chance or likleyhood of getting a certain number, word, or object
product
the answer of a multiplication problem
quotient
the answer of a division problem
regime
(n.) a goverment in power; a form or system of rule or management; a period of rule.
retard
(v.) To make slow, delay, hold back
seethe
(v.) to boil or foam; to be excited or disturbed.
singe
(v.) to burn slightly (n.) a burn at the ends or edges.
statistics
numbers which measure factors of life in a given country
sum
the answer of an addition problem
transparent
(adj.) allowing light to pass through; easily recongnized or understoof; easily seen through or detected
unique
(adj.) one of a kind unequaled; unusual; found only in a given, class, place or situation
unscathed
(adj.) wholly unharmed, not injured.
upright
(adj.) verticle, straight; good. honest; (adv) in a vertical position
verify
(v) to establish the truth or accuracy of, confirm.
yearn
(v.) to have a strong and earnest desire.
Bar Graphs
divides the data into bars, each bar is one category. The bars do not touch each other
Categorical Data
attribute data; puts individuals into a group or category
Census
a survey that attempts to gather data on the entire population
Completely Randomized
randomly place subjects of sample into all experimental groups
Continuous data
possible values are infinite; no gaps
Descriptive Statistics
data is summarized using numerical and graphical techniques in some useful way
Discrete data
number of possible values are finite or countable; gaps between possible values
Experiment
deliberately imposes some treatment on individuals in order to observe their responses. Used to study whether the treatment causes a change in the response
Individual
object described by a set of data. Individuals may be people, but they may also be animals or things
Inferential Statistics
data taken from only a sample is used to generalize to a larger population
Interval Level
like ordinal, but differences between values make sense; data does not have a natural zero or starting point
Line Graph
used to indicate a trend over time. Horizontal axis = time. Vertical axis = observed numerical data. Look for: overall pattern or trend, deviations, seasonal variations, pay specific attention to vertical scale
Nominal Level
data consists of names, labels, or categories only, no order
Observational Study
observes individuals and measures variables of interest but does not attempt to influence the responses. Purpose of study is to describe some group or situation
Ordinal level
data can be arranged in some order, but differences between values are meaningless
Parameter
a number describing or calculated from a population, usually the actual numerical value is unknown and we must describe the parameter in words
Pareto Chart
bar graph in which bars are organized from highest to lowest
Pictogram
uses pictures as part of the representation. Pictures are not often to scale, should not be used
Pie Chart
divides the data up into slices, where each slice represents one category. The size of each slice is determined by the relative frequency of each category. Used only when data represents parts of one whole
Population
entire group of individuals about which we want information
Quantitative Data
numerical variable for which it makes sense to do arithmetic operations; measurements
Randomized Block Design
one layer of classification not random
Ratio Level
like interval, but now there is a natural zero; ratios make sense
Representative Sample
matches the characteristics of the population well
sample survey
a survey done only on a sample of the population
Sample
the part of the population from which we actually collect information and is used to draw conclusions about the whole
Statistic
a number describing or calculated from a sample, usually the actual numerical value is known
Variable
a characteristic of an individual
Classical (or theorectical) Probability
is used when each outcome in a sample space is equally likely to occur
Complement of Event E
the set of all outcomes in a sample space that are not included in event E. the complement of event E is denoted by E & is read as "E Prime"
Empirical (or statistical) Probability
based on observations obtained from probability experiments
Event
consists of one or more outcomes and is a subset of the sample space
Law of Large Numbers
as you increase the # of times of probability experiment in repeated, the emirical probabiliy (relative frequency) of an event approaches the theoretical probability of the event
Outcome
the result of a single trial in a probability experiment
Probability Experiment
an action, or trail, through which specific results are obtained
Sample Space
the set of all possible outcomes of a probability experiment
Subjective Probability
result from intution educated guesses and estimates
"something has to happen rule"
the sum of the probabilities of all possible outcomes must be 1
addition rule
If A and B are disjoint events, then P(A∪B)=P(A)+P(B)
complement rule
the probability of one event occurring is 1 minus the probability that it does NOT occur.
P(A)= 1-P(A') A' read as A complement
Conditional Probability
P(B/A)= P(A∩B)/P(A)
Read as: "the probability of B given A."
disjoint (mutually exclusive)
Two events share NO outcomes in common. As a mater of fact, knowing that A occurs tells that B CANNOT occur.
event
a collection of outcomes, usually identified to attach probabilities to them; denoted by capital letters such as A,B, or C.
General Addition Rule
For any two events, A and B, the probability of A or B is P(A∪B)=P(A)+P(B)-P(A∩B)
General Multiplication Rule
For any two events, then the probability of A and B is P(A∩B)= P(A) X P(B/A)
independence (formally)
when P(B/A)=P(B)
independence (informally)
this happens between two events where the knowing whether or not one event occurs does NOT alter the probability that the other event occurs
law of large numbers
states that in the long-run relative frequency of repeated independent events settles down to the TRUE relative frequency as the number of trials increases.
legitimate probability assignment
each probability is between 0 and 1 (inclusive) and the sum of the probabilities is 1
multiplication rule
If A and B are independent events, then the probability of A and B is
P(A∩B)= P(A) X P(B)
outcome
an individual result of a component of a simulation; the value measured, observed, or reported or an individual instance of the trial
probability of an event
a number between 0 and 1 that reports the likelihood of the event's occurrence; can be derived from equally likely outcome, long-run relative frequency of the events occurence or from known probabilities. We write P(A) for the probability of an event
random event
an event where we know what outcome could happen, but not which particular values will happen
response variable
a record of the resulting values from each trial that corresponds to what we were interested in
sample space
the collection of all possible outcome values; has a probability of 1
simulation component
the most basic situation in which something happens at random
simulation
models random events by using random numbers to specify event outcomes with relative frequencies that correspond to the true real-world relative frequencies we are trying to model
tree diagram
a display of conditional events or probabilities that is helpful in thinking through conditioning.
trial
a single attempt or realization of a random event
trial
the sequence of several components representing events that we are pretending will take place
Array
an arrangement of data in ascending or descending order
Continuous Variable
any numerical value over an interval
-measured
Example: Height
Descriptive Statistics
Don't draw inferences
present data in a table format
Discrete Variable
Breaks between values
-counted
Example: # of books in a room
Experimental Study
the factors who effect to be assessed is manipulated appropriately by devising a suitable design
-conceptual
-data creates background
Inferential Statistics
drawing inferences
developing and using math tools to make forecasts
Observational Study
a survey of an existing population carried out by adopting a sample procedure (pre-existing/ in place)
Outlier
a value that differs abnormally from the other observations
Parameter
a quantitative measure that describes a characteristic of the Population
Population
a set of data that form the target of a study
example: student body of a school
QuaLitative Variable
can be identified by noting its presence describes observation as belonging to a set of categories
QuaNtitative Variable
values measured on a numerical side
Range
Larger - smallest
Max - Min
Raw Data
data collected in an investigation and not organized systematically
Reasons for Sampling 8
1. lower cost
2. less time
3. provides relevant information
4. population might be destroyed
5. population size might be infinite
6. population might not be available
7. Risk factor
8. Avoid administrative problems
Sample
a set of data values collected on some of the sampling units
example: a student
Sampling Unit
an item or object on which an observation can be recorded
example: grade level
Statistics
A science that deals with methods of collecting, organizing, and summarizing data in such a way that valid conclusions can be drawn from them
Statistic
a quantitative measured that describes a characteristic of the Sample
Variable
a characteristic observed on a sample unit
addition rule
if A and B are disjoint events, then the probability of A or B is _______
complement rule
the probability of an event occurring is 1 minus the probability that it doesn't occur
disjoint
two events that share no outcomes in common, mutually exclusive
event
a combination of outcomes usually for the purpose of attaching a probability to them
independent
the outcome of one trial doesn't influence or change the outcome of another
multiplication rule
if A and B are independent events, then the probability of A and B is _____
outcome
the value measured, observed, or reported for each trial
probability
the proportion of times the event occurs in many repeated trials of a random phenomenon (the long-term relative frequency of an event)
random phenomena
the rules and concepts of probability that give us a language to talk and think about ________
relative frequencies
a casual term for probability
sample space
the collection of all possible outcomes
something has to happen rule
the sum of the probabilities of all possible outcomes must be 1
the law of averages
assumes that the more something hasn't happened the more likely it becomes
the law of large numbers
the long run relative frequency of repeated independent events settles down to the true probability as the number of trials increases
trial
a single attempt or realization of a random phenomenon
1st Quartile
median of the portion of the entire data set that lies at or below the median of the entire data set
2nd Quartile
median of the entire data set
3rd Quartile
median of the portion of the entire data set that lies at or above the median of the entire data set
68.26% - 95.44% - 99.74% Rule
1) 68.26% of all observations lie within one standard deviation to either side of the mean
2) 95.44% of all observations lie within two standard deviations
3) 99.74% of all observations lie within three standard deviations
Bivariate Data
data for two variables from same population
Boxplot
A plot of data based on the five number summary. A line is drawn from the minimum observation to Q1; a box is drawn from Q1 to Q3 with a vertical line at the median and a line is drawn from Q3 to the maximum observation.
Census
obtain info on the entire population of interest
Combination
order doesn't matter
nCr = n! / r!(n-r)!
Continuous Variable
quantitative variable whose possible values form some interval of numbers
Descriptive Statistics
Consists of methods for organizing and summarizing info
Discrete Random Variable
random variable whose possible values from a finite or countably finite set of numbers
Discrete Variable
quantitative variable whose possible values form a finite (or countable infinite) set of numbers
Distribution of a Data Set
a table, graph, or formula that provides the values of the observations and how often they occur
Experimental Units
items or individuals on which the experiment is performed (subjects)
Factor
variable whose effect on response variable is of interest in the experiment
Five Number Summary
minimum, Q1, Q2, Q3, maximum
Independent Events
two events in which the outcome of one event does not affect the outcome of the other event.
Inferential Statistics
Consists of methods for drawing and measuring the reliability of conclusions about a population based on info obtained from a sample of the population
Interquartile Range (IQR)
difference between 1st and 3rd quartiles
IQR = Q3 - Q1
Levels
possible values of a factor
Mean
the sum of the observations divided by the number of observations
Median
measure not impacted by extremes
Mode
the number that occurs most often in a set of data
Mutually Exclusive Events
two or more events - if no two of them have outcomes in common
Normally Distributed Variable
its distribution has the shape of a normal curve
Observational Study compared to a Designed Experiment
Designed experiment - treatments are imposed and experiment is controlled
Observational study - experiment is only observed, no treatments imposed
Outliers
lower limit = Q1 - 1.5 x IQR
upper limit = Q3 + 1.5 x IQR
if data falls below lower limit or above upper limit, it is an outlier
Parameter
descriptive measure for a population
Percentiles
percentage of scores / data values that fall below a point
Permutation
order matters
nPr = n! / (n-r)!
Population
collection of all individuals or items under consideration in a statistical study
Qualitative Variable
non-numerically valued variable
Quantitative Variable
numerically valued variable
Range
the difference between the highest and lowest scores in a distribution
Representative Sample
sample that reflects as closely as possible the relevant characteristics of the population under consideration
Response Variable
characteristic of experimental outcome that is to be measured or observed
Sample
part of the population from which info is obtained
Simple Random Sampling
sampling procedure for which each possible sample of a given size is equally likely to be the one obtained
Skewed Distributions
reverse-j, j-shaped, right-skewed, left-skewed
Standard Deviation
descriptive measure of the spread of the data from the mean
Standard Normal Distribution
mean = 0, standard deviation = 1
Standardized Variable
z-value
Statistic
descriptive measure for a sample
Symmetric Distributions
bell-shaped, triangular, uniform (rectangle)
Three Conditions for Bernoulli Trials
1) each trial has two possible outcomes; p = success, q = 1-p
2) trials are independent
3) probability of a success remains the same from trial to trial
Three Principles of Experimental Design - Control
control effects due to factors other than ones of primary interest
Three Principles of Experimental Design - Randomization
divide into groups to avoid unintentional selection bias
Three Principles of Experimental Design - Replication
ensure randomization creates groups that resemble and increases chances of detecting differences among treatments
Three Standard Deviation Rule
for any data set almost all data values fall within three standard deviations of the mean
Treatments of an Experiment
each experimental condition
Unimodal, Bimodal, & Multimodal Distributions
unimodal - has one peak
bimodal - has two peaks
multimodal - has three or more peaks
Univariate Data
data for one variable from a population
Variance of a Data Set
to find, square the standard deviation
Z-Score
number of standard deviations an observation is away from the mean
(Mean of all values) µ
∑ x/N
(Mean) x bar
∑ x/n
10 - 90 Percentile Range
90th percentile - 10th percentile
5-Number Summary
For a set of data, the 5-number summary consists of the minimum value; the first quartile Q1; the median (or second quartile Q2); the third quartile, Q3; and the maximum value.
Arithmetic Mean (Mean)
the measure of center obtained by adding the values and dividing the total by the number of values
Bimodal
two data values occur with the same greatest frequency
Boxplot skeletal (or regular)
A boxplot (or box-and-whisker-diagram) is a graph of a data set that consists of a line extending from the minimum value to the maximum value, and a box with lines drawn at the first quartile, Q1; the median; and the third quartile, Q3.
Coefficient of Variation
The coefficient of variation (or CV) for a set of nonnegative sample or population data, expressed as a percent, describes the standard deviation relative to the mean.
Sample
CV = s/xbar * 100%
Population
CV = mu / µ * 100%
Comparing Variation in Different Samples
It's a good practice to compare two sample standard deviations only when the sample means are approximately the same.
When comparing variation in samples with very different means, it is better to use the coefficient of variation, which is defined later in this section.
Converting from the kth Percentile to the Corresponding Data Value flowchart
1) Sort the data
2) L = (k/100) * n
3) Is L a whole number?
Yes) The value of the kth perecntile is midway between the Lth value and the nest value in the sorted set of data. Find P(k) by adding the Lth value and the next value and dividing by two
No) Change L by rounding it up the next larger whole number
The value of P(k) is the Lth value, counting from the lowest.
Converting from the kth Percentile to the Corresponding Data Value
L = (k/100) * n
Description of mean
Advantages - Is relatively reliable, means of samples drawn from the same population don't vary as much as other measures of center. Takes every data value into account
Disadvantage - Is sensitive to every data value, one extreme value can affect it dramatically; is not a resistant measure of center
...
Description of Midrange
Sensitive to extremes , because it uses only the maximum and minimum values, so rarely used
Redeeming Features
(1)very easy to compute
(2)reinforces that there are several ways to define the center
(3)Avoids confusion with median
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Description of Ranged
It is very sensitive to extreme values; therefore not as useful as other measures of variation.
Empirical (or 68-95-99.7) Rule
For data sets having a distribution that is approximately bell shaped, the following properties apply:
About 68% of all values fall within 1 standard deviation of the mean.
...
About 95% of all values fall within 2 standard deviations of the mean.
...
About 99.7% of all values fall within 3 standard deviations of the mean.
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Finding the Median
First sort the values (arrange them in order), the follow one of these
1. If the number of data values is odd, the median is the number located in the exact middle of the list.
2. If the number of data values is even, the median is found by computing the mean of the two middle numbers.
Important Principles of Outliers
An outlier can have a dramatic effect on the mean.
An outlier can have a dramatic effect on the standard deviation.
An outlier can have a dramatic effect on the scale of the histogram so that the true nature of the distribution is totally obscured.
Interquartile Range (or IQR)
Quartile 3 - Quartile 1
Mean Absolute Deviation
(∑|x-xbar|)/n
Mean from a Frequency Distribution
xbar = (∑(f * x)) / ∑f
Measure of Center
the value at the center or middle of a data set
Midquartile
(Quartile 3 + Quartile 1) / 2
Midrange formula
(maximum value + minimum value) / 2
Midrange
the value midway between the maximum and minimum values in the original data set
Mode
the value that occurs with the greatest frequency
Data set can have one, more than one, or no mode
Modified Boxplot Construction
A modified boxplot is constructed with these specifications:
A special symbol (such as an asterisk) is used to identify outliers.
...
The solid horizontal line extends only as far as the minimum data value that is not an outlier and the maximum data value that is not an outlier.
...
Multimodal
more than two data values occur with the same greatest frequency
No Mode
no data value is repeated
Outliers
An outlier is a value that lies very far away from the vast majority of the other values in a data set.
Percentile formula
numbers of values less than x / total number of values * 100
Percentiles
are measures of location. There are 99 percentiles denoted P1, P2, . . . P99, which divide a set of data into 100 groups with about 1% of the values in each group.
Population Standard Deviation
∑ (((x - µ)^2)/N)^(1/2)
Properties of the Standard Deviation
Measures the variation among data values
Values close together have a small standard deviation, but values with much more variation have a larger standard deviation
...
Has the same units of measurement as the original data
...
For many data sets, a value is unusual if it differs from the mean by more than two standard deviations
...
Compare standard deviations of two different data sets only if the they use the same scale and units, and they have means that are approximately the same
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Q1 (First Quartile)
separates the bottom 25% of sorted values from the top 75%
Q2 (Second Quartile)
same as the median; separates the bottom 50% of sorted values from the top 50%
Q3 (Third Quartile)
separates the bottom 75% of sorted values from the top 25%.
Range Rule of Thumb for Estimating a Value of the Standard Deviation s
s approx = range/4
Range Rule of Thumb
is based on the principle that for many data sets, the vast majority (such as 95%) of sample values lie within two standard deviations of the mean.
Range
(maximum value) - (minimum value)
Rationale for using n - 1 versus n
There are only n - 1 independent values. With a given mean, only n - 1 values can be freely assigned any number before the last value is determined.
Dividing by n - 1 yields better results than dividing by n. It causes s2 to target 2 whereas division by n causes s2 to underestimate 2.
Round-off Rule for
Measures of Center
Carry one more decimal place than is present in the original set of values.
Round-Off Rule for Measures of Variation
When rounding the value of a measure of variation, carry one more decimal place than is present in the original set of data.
Round only the final answer, not values in the middle of a calculation.
...
Semi-interquartile Range
(Quartile 3 - Quartile 1) / 2
Skewed to the left
(also called negatively skewed) have a longer left tail, mean and median are to the left of the mode
Skewed to the right
(also called positively skewed) have a longer right tail, mean and median are to the right of the mode
Skewed
distribution of data is skewed if it is not symmetric and extends more to one side than the other
Standard Deviation -
Important Properties
The standard deviation is a measure of variation of all values from the mean.
The value of the standard deviation s is usually positive.
The value of the standard deviation s can increase dramatically with the inclusion of one or more outliers (data values far away from all others).
The units of the standard deviation s are the same as the units of the original data values.
standard deviation
The standard deviation of a set of sample values, denoted by s, is a measure of variation of values about the mean.
Also known as the Square root of variance
or ( { ∑(x-xbar)^2 } / (n-1) )^(1/2)
Symmetric
distribution of data is symmetric if the left half of its histogram is roughly a mirror image of its right half
Unbiased Estimator
The sample variance s2 is an unbiased estimator of the population variance 2, which means values of s2 tend to target the value of 2 instead of systematically tending to overestimate or underestimate 2.
Variance
The variance of a set of values is a measure of variation equal to the square of the standard deviation.
Sample variance: s2 - Square of the sample standard deviation s
...
Population variance: 2 - Square of the population standard deviation
...
Weighted Mean formula
x bar = ∑ (w • x) / ∑w
z Score (or standardized value)
the number of standard deviations that a given value x is above or below the mean
z score formula
z = (x - xbar)/standard devation
Census
Measurements or observations from the entire population are used.
Cluster Sampling
Divide the entire population into pre-existing segments or clusters. The clusters are often geographic. Make a random selection of clusters. Include every member of each selected cluster in the sample.
Completely Randomized Experiment
One in which a random process is used to assign each individual to one of the treatments.
Confounding Variable
When the effects of one [variable] cannot be distinguished from the effects of the other. [ ] variables may be part of the study, or they may be outside lurking variables.
Control Group
This group received a dummy treatment, enabling the researchers to control for the placebo effect. In general, a [ ] group is used to account for the influence of other known or unknown variables that might be an underlying cause of a change in response in the experimental group.
Convenience Sample
Select from a group of individuals that are easy to reach (severely biased!)
Descriptive Statistics
Organizing, picturing, and summarizing information from samples or populations
Double-Blind
Neither the individuals in the study nor the observers know which subjects are receiving the treatment.
Experiment
A treatment is deliberately imposed on the individuals in order to observe a possible change in the response or variable being measured.
Faulty Recall
Respondents may not accurately remember when or whether an event took place.
Hidden Bias
The question may be worded in such a way as to elicit a specific response. The order of questions might lead to biased responses. Also, the number of responses on a Likert scale may force responses that do not reflect the respondent's feelings or experience.
Individuals
People/objects included in a statistical study
Inferential Statistics
Using information from a sample to draw conclusions about the population
Interviewer Influence
Factors such as tone of voice, body language, dress, gender, authority, and ethnicity of the interviewer might influence responses.
Lurking Variable
One [variable] for which no data have been collected but that nevertheless had influence on other variables in the study.
Multistage Sampling
Use a variety of sampling methods to create successively smaller groups at each stage. The final sample consists of clusters.
Nonresponse
Individuals either cannot be contacted or refuse to participate. [ ] can result in significant undercoverage of a population.
Nonsampling Error
Result of poor sample design, sloppy data collection, bias, etc. (human error)
Observational Study
Observations and measurements of individuals are conducted in a way that doesn't change the response or the variable being measured.
Parameter
A numerical measure that describes an aspect of a population
Placebo Effect
Occurs when a subject receives no treatment but (incorrectly) believes he or she is, in fact, receiving treatment and responds favorably.
Population Data
The variable is from every relevant individual
Qualitative
Measures a non-numerical category (aka categorical)
Quantitative
Measures a numerical amount (discreet and continuous)
Random Sampling
Use a simple random sample from the entire population
Randomization
Used to assign the individuals to the two treatment groups. This helps prevent bias in selecting members for each group.
Replication
[ ] of the experiment on many patients reduces the possibility that the differences in pain relief for the two groups occurred by chance alone.
Sample Data
The variable is from some of the relevant individuals
Sampling Error
Difference between measurement from a sample and the population because the sample does not perfectly represent the population
Sampling Frame
A list of individuals from which a sample is actually selected.
Simple Random Samples
Take 'n' measurements from a population so that every sample of size 'n' has an equal chance of being selected and every individual has an equal chance of being included (use the random number table!)
Statistics
The study of how to collect, organize, analyze and interpret numerical information from data
Statistic
A numerical measure that describes an aspect of a sample
Stratified Sampling
Divide the entire population into distinct subgroups called strata. The strata are based on a specific characteristic such as age, income, education level, and so on. All members of a stratum share the specific characteristic. Draw random samples from each stratum.
Systematic Random Sample
Select every 'nth' subject (can be problematic if the subject is cyclical)
Systematic Sampling
Number all members of the population sequentially. Then, from a starting point selected at random, include every Nth member of the population in the sample.
Truthfulness of Response
Respondents may lie intentionally or inadvertently.
Undercoverage
Results when population members are omitted from the sample frame.
Vague Wording
Words such as "often", "seldom", and "occasionally" mean different things to different people.
Variables
Characteristics of the individual to be measured/observed
outlier
is a point lying far away from the other data points.
regression equation
The best-fitting straight line's equation
Regression Equation
yhat = b(0) + b(1) * x
regression line
The best-fitting straight line
residual plot
scatterplot of the (x, y) values after each of the y-coordinate values has been replaced by the residual value y - y (where y denotes the predicted value of y). That is, a residual plot is a graph of the points (x, y - y).
response variable or dependent variable
the y variable
Special Property
The regression line fits the sample points best.
Using the Regression Equation for Predictions cont
3. Use the regression line for predictions only if the data do not go much beyond the scope of the available sample data. (Predicting too far beyond the scope of the available sample data is called extrapolation, and it could result in bad predictions.)
4. If the regression equation does not appear to be useful for making predictions, the best predicted value of a variable is its point estimate, which is its sample mean.
...
Using the Regression Equation for Predictions
1.Use the regression equation for predictions only if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well.
2. Use the regression equation for predictions only if the linear correlation coefficient r indicates that there is a linear correlation between the two variables (as described in Section 10-2).
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Comparing Variation in Two Samples Requirements
1. The two populations are independent.
2. The two samples are simple random samples.
3. The two populations are each normally distributed. IT DOES NOT MATTER OF THE POPULATION IS > 30
Confidence Interval Estimate of p(1) - p(2)
E = z(𝞪/2) ( (p(1) q(1) / n(1)) (p(2) ) * q(2) / n(2)))
Confidence Interval Estimate of μ(1) - μ(2): Independent Samples
(x1 - x2) - E < (µ1 - µ2) < (x1 - x2) + E
where E = t(𝞪/2) * ( (s^2(1) / (n(1)) + (s^2(2) / n(2) ) ^ (1 / 2)
...
Confidence Interval: Independent Samples with σ1 and σ2 Both Known
See page 479
Confidence Intervals for Matched Pairs
see page 488
dependent
the sample values are paired
Hypothesis Test for Two Means: Independent Samples with σ(1) and σ(2) Both Known
see page 479
Hypothesis Test Statistic for Matched Pairs
...
Hypothesis Test Statistic for Two Means: Independent Samples
t = ( ( xbar(1) - xbar(2) ) - ( μ(1) - μ(2)) ) / ( (s^2(1) / n(1) ) + (s^2(2) / n(2)) )
independent
the sample values selected from one population are not related to or somehow paired or matched with the sample values from the other population.
Pooled Sample Proportion
pbar = ( x(1) + x(2) ) / ( n(1) + n(2) )
qbar = 1 - pbar
Properties of the F Distribution - continued
If the two populations do have equal variances, then F = s(1) / s(2) will be close to 1 because and are close in value.
If the two populations have radically different variances, then F will be a large number.
...
Test Statistic for Hypothesis Tests with Two Variances
SEE PAGE 498
Test Statistic for Two Proportions - cont
p(1) - p(2) = 0 (assumed in the null hypothesis)
phat(1) = x(1) / n(1)
phat(2) = x(2) / n(2)
Test Statistic for Two Proportions
z = ( phat(1) - phat(2) ) - ( p(1) - p(2) ) / ( (phat qhat / n(1) + (phat qhat / n(2) )
alternative hypothesis
The alternative hypothesis (denoted by H1 or Ha or HA) is the statement that the parameter has a value that somehow differs from the null hypothesis.
The symbolic form of the alternative hypothesis must use one of these symbols: , <, >.
Conclusions in Hypothesis Testing
We always test the null hypothesis. The initial conclusion will always be one of the following:
1. Reject the null hypothesis.
2. Fail to reject the null hypothesis.
critical region (or rejection region)
is the set of all values of the test statistic that cause us to reject the null hypothesis.
critical value
A critical value is any value that separates the critical region (where we reject the null hypothesis) from the values of the test statistic that do not lead to rejection of the null hypothesis. The critical values depend on the nature of the null hypothesis, the sampling distribution that applies, and the significance level 𝞪
Decision Criterion
P-value method:
Using the significance level :
If P-value <= 𝞪 , reject H0.
If P-value > 𝞪 , fail to reject H0.
hypothesis test (or test of significance)
is a standard procedure for testing a claim about a property of a population.
hypothesis
claim or statement about a property of a population
null hypothesis (denoted by H0)
The null hypothesis (denoted by H0) is a statement that the value of a population parameter (such as proportion, mean, or standard deviation) is equal to some claimed value.
We test the null hypothesis directly.
Either reject H0 or fail to reject H0.
Obtaining P
phat sometimes is given directly
phat sometimes must be calculated:
phat = x/n
P-Value note
The null hypothesis is rejected if the P-value is very small, such as 0.05 or less.
P-Value
The P-value (or p-value or probability value) is the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true.
Requirements for Testing Claims About σ or σ^2
X^2 = (n - 1) * s^2 / σ^2
Requirements for Testing Claims About a Population Mean (with σ Not Known)
1) The sample is a simple random sample.
2) The value of the population standard deviation σ is not known.
3) Either or both of these conditions is satisfied: The population is normally distributed or n > 30.
Requirements for Testing Claims About a Population Mean (with σ Known)
1) The sample is a simple random sample.
2) The value of the population standard deviation σ is known.
3) Either or both of these conditions is satisfied: The population is normally distributed or n > 30.
Requirements for Testing Claims About a Population Proportion p
1) The sample observations are a simple random sample.
2) The conditions for a binomial distribution are satisfied.
...
3) The conditions np >= 5 and nq >= 5 are both satisfied, so the binomial distribution of sample proportions can be approximated by a normal distribution with µ = np and σ = (npq)^(1/2) . Note: p is the assumed proportion not the sample proportion.
...
significance level (denoted by 𝞪)
is the probability that the test statistic will fall in the critical region when the null hypothesis is actually true. This is the same 𝞪 introduced in Section 7-2. Common choices for 𝞪 are 0.05, 0.01, and 0.10.
Tails
Two tailed test: <> used
Left-Tailed Test < used
Right-Tailed test > used
Test statistic for mean
z = (xbar - μ) / (σ / (n)^1/2 ) or t = (xbar - μ) / (s / (n)^(1/2) )
Test statistic for proportion
z = (phat - p) / (p * q / n)^(1/2)
Test statistic for standard deviation
X^2 = (n - 1) * s^2 / σ^2
Test Statistic for Testing a Claim About a Mean (with σ Not Known)
t = (xbar - μ(xbar) ) / (s / (n)^(1/2) )
Test Statistic for Testing a Claim About a Mean (with σ Known)
z = (xbar - μ(xbar) / (σ / (n)^(1/2) )
μ(xbar) = population mean of all sample means from samples of size n
...
test statistic
The test statistic is a value used in making a decision about the null hypothesis, and is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.
Type I Error
A Type I error is the mistake of rejecting the null hypothesis when it is actually true.
The symbol 𝞪(alpha) is used to represent the probability of a type I error.
...
Type II Error
A Type II error is the mistake of failing to reject the null hypothesis when it is actually false.
The symbol β (beta) is used to represent the probability of a type II error.
...
Binomial Distribution: Mean
µ = n • p
Binomial Distribution: Standard Deviation
σ = (n • p • q)^(1/2)
Binomial Distribution: Variance
σ^2 = = n • p • q
binomial probability distribution
1. The procedure has a fixed number of trials
2. The trials must be independent. (The outcome of any individual trial doesn't affect the probabilities in the other trials.)
3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).
4. The probability of a success remains the same in all trials.
Continuous random variable
infinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions
Discrete random variable
either a finite number of values or countable number of values, where "countable" refers to the fact that there might be infinitely many values, but they result from a counting process
expected value formula
E = ∑ [x • P(x)]
expected value
The expected value of a discrete random variable is denoted by E, and it represents the mean value of the outcomes. It is obtained by finding the value of [x • P(x)].
Identifying Unusual Results Range Rule of Thumb
According to the range rule of thumb, most values should lie within 2 standard deviations of the mean.
We can therefore identify "unusual" values by determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ - 2σ
Mean of a Probability Distribution
µ = ∑[x • P(x)]
Methods for Finding Probabilities - Method 1: Using the Binomial
Probability Formula
P(x) = (n! / ((n - x)! x!))) p^x * q^(n-x)
Notation for Binomial Probability Distributions
P(S) = p (p = probability of success)
P(F) = 1 - p = q (q = probability of failure)
...
Probability Distributions
describe what will probably happen instead of what actually did happen, and they are often given in the format of a graph, table, or formula.
Probability distribution
a description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula
Random variable
a variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure
Rare Event Rule for Inferential Statistics
If, under a given assumption (such as the assumption that a coin is fair), the probability of a particular observed event (such as 992 heads in 1000 tosses of a coin) is extremely small, we conclude that the assumption is probably not correct.
Requirements for Probability Distribution
P(x) = 1
where x assumes all possible values.
0 <= P(x) = 1
for every individual value of x.
Roundoff Rule for μ, σ, σ^2
Round results by carrying one more decimal place than the number of decimal places used for the random variable x.
If the values of x are integers, round µ, σ, and σ^2 and 2 to one decimal place.
Standard deviation (probability distribution)
([∑x^2 • P(x) ] - µ^2)^(1/2)
Standard Deviation of a Probability Distribution
σ = (∑[x^2 • P(x)] - µ^2)^(1/2)
Variance (probability distribution)
[∑x^2 • P(x) ] - µ^2
Variance (shortcut) of a Probability Distribution
σ^2 = ∑ [x^2 • P(x)] - µ^2
Variance of a Probability Distribution
σ^2 = ∑[(x - µ)^2 • P(x)]
association
Although there may be a strong ________________ between variables this does not necessarily imply there is causation.
block design
the random assignment of units to treatments is carried out separately within each section.
causation
Although there may be a strong association between variables this does not necessarily imply there is this
clinical trials
experiments that student the effectiveness of medical treatment on actual patients.
coefficient of determination
r2
completely randomized design
when all experimental units are allocated at random among all treatments
confidential
all individual data on subject must be this
confounders
two variables whose effect cannot be distinguished from one another
correlation
measures the direction and strength of the linear relationship between two quantitative variables.
density curve
the overall pattern of a distribution can be described by this
distribution
tells us what values it takes and how often it takes these values
double-blind
Neither the test subject, nor the administrator knows the treatment given.
explanatory variable
x variable. explains or causes changes in the y variables
extrapolation
the use of a regression line to make predictions for the values outside the range of x.
influential point
changes the regression line if removed from the data.
informed consent
All individuals who are subjects in a student must give this before data is collected
institutional review board
protects the rights and welfare of humans subjects participating in research activities
interquartile range
the difference between quartiles
lurking variables
may explain the relationship between explanatory and response variables.
mean
the arithmetic average of the observations
median
the midpoint of a distribution
normal distribution
a family of symmetrical bell-shaped density curves.
normal quartile plot
a pattern on such a plot that deviates substantially from a straight line indicates that the data are not Normal.
outliers
observations that lie outside the overall patter of the distribution
parameter
a number that describes the population
placebo effect
an improvement in health not due to any treatment, but only to the patient's belief that he or she will improve.
placebo
a control, does nothing to the text subject
Principles of experimental design
control, randomize, and repeat
regression line
a line that describes how the response variable changes as the explanatory variable changes.
residual
the difference between an observed value of the response variable and the value predicted by the regression line
response variable
y variable. measures an outcome of a study
standard deviation
describes the variation around the mean
statistic
a number that can be computed from the data.
symmetric
same on both sides
transformation
if the data does not appear linearly distributed it may be necessary to do this on the data to change the distribution to a linear distribution
treatment
used in experimental studies, its what is given to a text subject.
variable
any characteristic of an individual
voluntary response sampling
may lead to a large amount of bias
Z score
measures the number of standard deviations a data value is from the mean
Voluntary Response
Individuals with strong feelings about a subject are more likely than others to respond. Such a study is interesting but not reflective of the population.
0.5
What proportion of the area under a normal curve is to the right of a z-score of zero?
0.6915
The area under a normal curve to the left of z=0.5 is __________.
0.7012
The area under the normal curve between z=-1 and z=1.08 is __________.
0.8869
The area under a normal curve to the right of z=-1.21 is __________.
0.9783
The area under the normal curve to the right of z=-2.02 is __________.
0
The z value of µ for a normal curve is always __________.
1
The total area under the normal curve is __________.
68.26%
P(µ-σ<x<µ+σ)=
74.5
A statistics student recieves a score of 85 on a statistics midterm. If the corresponding z-score equals 1.5 and the standard deviation equals 7, the average score on this exam is __________.
95.44%
P(µ-2σ<x<µ+2σ)=
99.74%
P(µ-3σ<x<µ+3σ)=
asymptotic
__________ means that the normal curve gets closer and closer to the x-axis but never actually touches it.
bell-shaped curve
The graph of a normal probability distribution curve is called a __________.
continuous probability distribution
What type of probability distribution is the normal distribution?
equal
In a normal distribution, the relationship between the mean, median and mode is __________.
left
A computed z for x values to the __________ of the mean is negative.
less than
A negative z-score indicates that the corresponding value in the original distribution is __________ the mean.
negative
For a normal distribution curve, the z value for an x value that is less than µ is always __________.
normal probability distribution
the most widely used continuous probability distribution, which plays a central role in statistical inference; can be used to describe almost all phenomena in real life situations
standard normal distribution
If we convert values of a normal distribution to a distribution that has a mean of 0 and a standard deviation of 1, this probability distribution is called __________.
standard normal distribution
Normal distribution with a mean µ=0 and a standard deviation σ=1 is known as __________.
x-µ/σ
z=
x:N(µ,σ)
parameters of a normal curve
z-score
A __________ is the distance between a selected value (x) and the population mean (µ) divided by the population standard deviation (σ).
z:N(0,1)
parameters of a standard normal curve
z=x-µ/σ
The formula to convert any normal distribution to the standard normal distribution is __________.
µ+zσ
x=
µ,σ
The parameters of the normal distribution are __________ and __________.
"not"
subtract from 1, 1 - P(example).
"Something has to happen rule"
The sum of the probabilities of all possible outcomes of a trial must be 1.
Addition Rule "or"
If A and B are disjoint events, then the probability of A or B is P(A U B) = P(A) + P(B).
Complement Rule or "at least one"
The probability of an event occurring is 1 minus the probability it doesn't occur. P(A) = 1 - P(Ac(it doesn't occur)).
Conditional Probability
P(BlA) = P(A and B)/P(A). P(BlA) is read " the probability of B given A."
Disjoint (mutually exclusive)
Two events are disjoint of they share no outcomes in common. If A and B are disjoint, then knowing that A occurs tells us that B cannot occur. Disjoint events are also called "mutually exclusive."
Disjoint Events
Two events are disjoint (or mutually exclusive) if they have no outcomes in common.
Event
A collection of outcomes. Usually, we identify events in order to attach probabilities to them. We denote events with bold capital letters such as A, B, or C.
General Addition Rule
For any two events, A and B, the probability of A or B is... P(A U B) = P(A) + P(B) - P(A and B).
General Multiplication Rule
For any two events, A and B, the probability of A and B is P(A and B) = P(A) x P(BlA).
Independence (informally)
Two events are independent if knowing whether one event occurs does not alter the probability of the other event occurring.
Independence (used casually)
Two events are independent if knowing whether one event occurs does not alter the probability that the other event occurs.
Independence (used formally)
P(BlA) = P(B) when A and B are independent.
Law of Large Numbers
States that long-run relative frequency of repeated independent events gets closer and closer to the true relative frequency as the number of trials increase.
Legitimate probability assignment
An assignment of probabilities to outcomes is legitimate if...
a) each probability if between 0 and 1 (inclusive).
b) the sum of the probabilities is 1.
Multiplication Rule, Upside down U
If A and B are disjoint events, then the probability of A and B is P(A and B) = P(A) x P(B).
Outcome
The outcome of a trial is the value measured, observed, or reported for an individual instance of that trial.
Outcomes are considered to be either
a) discrete if they have distinct values such as heads or tails (even if the values are numerals)
b) continuous if they take on numeric values in some rand of possible values
Probability
The probability of an event is a number between 0 and 1 that reports the likelihood of the event's occurrence. A probability can be derived from equally likely outcomes, from the long-run proportion of the event's occurrence, or from known proportions, We write P(A) for the probability of the event A.
Random Phenomenon
A phenomenon is random if we know what outcomes could happen, but not which particular values did or will happen.
Sample Space
The collection of all possible outcome values. The sample space has a probability of 1.
Tree Diagram
A display of conditional events or probabilities that is helpful in thinking through conditioning.
Trial
A single attempt or realization of a random phenomenon.
U, "or," Union
means to add
upside down U, "and," intersection
meant to multiply
binomial probability distribution function
The probability of obtaining x successes in n independent trials of a binomial experiment is given by
P(x) = nCx p^x(1-p)^(n-x)
x = 0,1, 2, ⋅⋅⋅,n
where p is the probability of success
...
binomial random variable
the number of successes in n trials of a binomial experiment
continuous random variable
Has infinitely many values. The values can be plotted on a line in an uninterrupted fashion.
criteria for a binomial probability experiment
1. The experiment is performed a fixed number of times. Each repetition is called a trial.
2. The trials are independent. The outcome of one trial will not affect the outcome of the other trials.
3. For each trial, there are two mutually exclusive (disjoint) outcomes: success or failure.
4. The probability of success is the same for each trial of the experiment.
cumulative distribution function
computes probabilities less than or equal to a specified value
discrete random variable
Has either a finite or countable number of values. The values can be plotted on a number line with space between each point.
expected value
the mean of the discrete random variable
interpretation of the mean of a discrete random variable
Suppose an experiment is repeated n independent times and the value of the random variable X is recorded. As the number of repetitions of the experiment increases, the mean value of the n trials will approach µx, the mean of the random variable X.
x̄ =( x₁ + x₂ + ⋅⋅⋅ + x-sub-n)/n
The difference between x̄ and µ-sub-x gets closer to 0 as n increases
mean and standard deviation of a binomial random variable
a binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the formulas
u-sub-x = np and σ-sub-x = √np(1-p)
mean of a discrete random variable
µ-sub-x = ∑ [x ∗ P(x)]
where x is the value of the random variable and P(x) is the probability of observing the variable x.
...
notation used in binomial probability distribution
1. There are n independent trials of the experiment.
2. P denotes the probability of success for each trial so that 1-p is the probability of failure for each trial.
3. X denotes the number of successes in n independent trials of the experiment. 0 ≤ x ≤ 1.
probability distribution
Provides the possible values of the random variable and their corresponding probabilities. Can be in the form of a table, graph, or mathematical formula.
probability histogram
a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of each value of the random variable
random variable
A numerical measure of the outcome of a probability experiment, so its value is determined by chance. Random variables are typically denoted using capital letters such as X.
rules for a discrete probability distribution
Let P(x) denote the probability that the random variable X equals x;
1. then ∑P(x) = 1
2. 0 ≤ P(x) ≤ 1
standard deviation of a discrete random variable
σ-sub-x = √σ²-sub-x
variance of a discrete random variable
σ²-sub-x = ∑ [x² P(x)] -µ²-sub-x
dimension
the length, width, or height of a shape
function
each input has exactly one output
negative slope
the line would decrease from left to right
positive slope
the line would increase from left to right
slope
the rise of a line divided by the run of the line
y-intercept
where the graph of a line crosses the y-axis
bcdf(n, p, k)
bcdf on calculator
gcdf(p,n)
gcdf on calculator
if P(B|A) = P(B)
Independent events
If S is the sample space in a probability model, then P(S)=1
Probability rule 2
if X and Y are random variables, meanx+y=meanx + meany
rule 2 for means
if X is a random variable and a and b are fixed numbers, mean of a+bX = a+bmeanX
rule 1 for means
If you can do one task n number of ways and a second m number of ways, then both tasks can be done in n*m ways.
Multiplication principle
mean = 1/p
Mean of geometric random variable
mean = np
mean of binomial
mean of X = multiply each possible value by its probability and then add it up
Mean of a discrete random variable
P(A and B) = P(A)*P(B)
Multiplication rule for independent events
P(A or B) = P(A)+P(B) - P(A and B)
General addition rule for unions of two events
P(B|A) = P(A and B)/P(A)
Conditional probability
P(X>n) = q^n
probability of more than n trials before success
q/p^2
std. dev of geometric random variable
random variable with either a finite (whole) number value or a countable number
Discrete random variable
Sx=sqrt(npq)
std. dev of binomial
takes all values in an interval of numbers, described by a density curve.
Continuous random variable
the complement of A is 1-P(A)
Probability rule 4
The P(A) of any event is between 0 and 1 (inclusive)
Probability rule 1
Two events A and B are disjoint if they have no outcomes in common. P(A or B)=P(A)+P(B)
Probability rule 3
variance of x = (x1-mean)^2p1...(xi-mean)^2pi
Variance of a discrete random variable
Disjoint/mutually exclusive
two events that have no outcomes in common (Cannot occur simultaneously)
Equation for conditional probability
P(B|A)= P(A and B)/P(A)
Event
any outcome or set of outcomes of a random phenomenon
Independent
knowing that one event occurs does not change the probability that the other occurs
P(A and B) (non-independent)
P(A)*P(B|A)
Probability Model
a mathematical description of a random phenomenon consisting of a sample space and way of assigning probability
Sample Space, S
the set of all possible outcomes of a random phenomenon
The Multiplication Principle
if event A has a possible outcomes and event B has b possible outcomes, then BOTH events considered together have (a*b) outcomes.
1 + 2r + 3r^2 + ........... =
1 / (1-r)^2
a +ar + ar^2 +............=
a / (1-r)
Binomial Distribution: E[X] =
np (mean of the binomial distribution)
Binomial Distribution: Mx(t) =
(1 - p + pe^t)^n
Binomial Distribution: p(x) =
(n choose x)(p^x)(q^n-x)
Binomial Distribution: Var [X] =
np(q)
Continuous Uniform Distribution: E[X] & VAR[X] =
(a + b) / 2 & (b-a)^2 / 12
Continuous Uniform Distribution: f(x) =
1 / (b - a)
Continuous Uniform Distribution: F(x) =
integral (a to x) f(x) dx = (x-a) / (b - a)
COV[X,X] =
VAR[X]
COV[X,Y] =
E[XY] - E[X] * E[Y]
E[a1 h1(x) + a2 h2(x) + b] =
a1 E[h1(x)] + a2 E[h2(x)] + b
E[aX + b] =
a E[x] + b
E[X + Y] =
E[X] + E[Y]
Exponential Distribution F(x) =
1 - e^(-lamn*x)
Exponential Distribution S(x) =
e^(-lamn*x)
Exponential Distribution with mean (1/lamn), f(x) =
lamn(e^(-lamn*x))
Exponential Distribution: E[X] & VAR[X] =
1/lamn & 1/(lamn^2)
Exponential Distribution: k-th moment E[X^k] =
k! / lamn^k
Exponential Distribution: Mx(t) =
lamn / (lamn - t) t<lamn
f(y(giv)x) * fx(x) =
f(x,y)
For a continuous random variable Mx(t) =
integral (-infinity to infinity): e^(tx) f(x) dx
For a discrete random variable Mx(t) =
Sum (e^(tx) * p(x)
For Continuour Random Variable E[X] =
integral (-infinity to infinity): x* f(x) dx
For Discrete Random Variable E[X] =
x1p(x1) + x2p(x2) + .......
For Discrete Uniform Distribution of N points: E[X] =
(N + 1) / 2
For Discrete Uniform Distribution of N points: p(x) =
1 / N
For Discrete Uniform Distribution of N points: Var[X] =
(N^2 - 1) / 12
For Uniform Joint Distribution, pdf =
1 / area of R
Geometric Distribution: E[X] =
(1-p) / p
Geometric Distribution: Mx(t) =
p / (1-qe^t)
Geometric Distribution: p(x) =
q^x *p
Geometric Distribution: VAR[X] =
(1-p) / p^2
How do you Standardize Normal Distribution: P[ r < X < s)
Z = (X-u)/std, =P[(r-u)/std < (X-u)/std < (s-u)/std]
If X & Y are independent, then COV[X,Y] =
0
If X & Y are independent, then E[X*Y] =
E[X] * E[Y]
If x & y are Independent, then f(y (giv) X=x) =
fy(y)
integral (0 to infinity) (x^n)(e^-cx)dx =
n! / c^n+1
integral of a^x =
a^x / ln a
Joint Conditional Distribution: If Independent, f(x,y) =
fx(x) * f(y(giv)X=x)
Joint Distribution Cumulative Distribution:
integral (-infinity to x) integral (-infinity to y) f(s,t) dt ds
Joint Distribution: If Independent, f(x,y) =
fx(x) * fy(y)
Mx(t) =
E[e^(tx)]
Negative Binomial Distribution Mx(t) =
(p / 1-qe^t)^r
Negative Binomial Distribution p(x):
(n + x - 1 choose x) p^r q^x
Negative Binomial Distribution VAR[X] =
r(1-p) / p^2
Negative Binomial Distribution:
Experiment performed repeatedly until the r-th success (X # of failures)
Negative Binomial Distrubution E[X] =
r(1-p) / p
Poisson Distribution: E[X] & Var[X] =
lamna
Poisson Distribution: Mx(t) =
e^(lamn(e^t-1))
Poisson Distribution: p(x) =
(lamna parameter) ((e^-lamn(lamn^x))/x!
the pdf of Normal Distribution f(x) =
(1 / (std sqr(2pi)) e^((-(x-mean)^2)/(2(std^2)))
Var[aX + b] =
a^2Var[X]
VAR[aX +bY + c] =
a^2VAR[X] + b^2VAR[Y] + 2abCOV[X,Y] (If X & Y are independent, COV[X,Y] = 0)
VAR[X+Y] =
VAR[X] + VAR[Y] + 2*COV[X,Y]
VAR[X] =
E[X^2] - (E[x])^2
blinding
is a technique where the subject does not know whether he or she is receiving a treatment or a placebo.(1.3)
blocks
groups of subjects with similar characteristics.(1.3)
census
is a count or measure of an entire population.(1.3)
class boundaries
are the numbers that separate classes without forming gaps between them.(2.1)
class width
is the distance between lower (or upper) limits of consecutive classes.(2.1)
cluster sample
divide the population into groups, called clusters, and select all of the members in one or more (but not all) of the clusters.(1.3)
completely randomized design
subjects are assigned to different treatment groups through random selection.(1.3)
confounding variable
occurs when an experimenter cannot tell the difference between the effects of different factors on a variable.(1.3)
cumulative frequency
is the sum of the frequency for that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n.(2.1)
data
consist of information coming from observations, counts, measurements, or responses.(1.1)
descriptive statistics
is the branch of statistics that involves the organization summarization, and display of data.(1.1)
double-blind experiment
neither the subject nor the experimenter knows if the subject is receiving a treatment or a placebo. The experimenter is informed after all the data have been collected. This type of experimental design is preferred by researchers.(1.3)
frequency distribution
is a table that shows classes or intervals of data entries with a count of the number of entries in each class.(2.1)
frequency f
is the number of data entries in the class.(2.1)
frequency histogram
is a bar graph that represents the frequency distribution of a data set.(2.1)
frequency polygon
is a line graph that emphasizes the continuous change in frequencies.(2.1)
inferential statistics
is the branch of statistics that involves using a sample to draw conclusions about a population. A basic tool in the study of inferential statistics is probability.(1.1)
interval level of measurement
data at this measurement can be ordered, and you can calculate meaningful differences between data entries. At the interval level, a zero entry simply represents a position on a scale; the entry is not an inherent zero.(1.2)
lower class limit
which is the least number that can belong to the class.(2.1)
matched-pairs design
where subjects are paired up according to a similarity. One subject in the pair is randomly selected to receive one treatment while the other subject receives a different treatment.(1.3)
midpoint
is the sum of the lower and upper limits of the class divided by two. Sometimes called the class mark and is calculated [(lower class limit)+(upper class limit)]/2. section (2.1)
nominal level of measurement
data at this level is qualitative only. Data at this level are categorized using names, labels, or qualities. No mathematical computations can be made at this level.(1.2)
observational study
a researcher observes and measures characteristics of interest of part of a population but does not change existing conditions.(1.3)
ordinal level of measurement
data at this level are qualitative or quantitative. Data at this level can be arranged in order, or ranked, but differences between data entries are not meaningful.(1.2)
parameter
is a numerical description of a population characteristic.(1.1)
placebo effect
occurs when a subject reacts favorably to a placebo when in fact, he or she has been given no medicated treatment at all.(1.3)
population
is the collection of all outcomes, responses, measurements, or counts that are of interest.(1.1)
qualitative data
consist of attributes, labels, or nonnumerical entries.(1.2)
quantitative data
consist of numerical measurements or counts.(1.2)
random sample
is one in which every member of the population has an equal chance of being selected.(1.3)
randomization
is a process of randomly assigning subjects to different treatment groups.(1.3)
randomized block design
divide subjects with similar characteristics into blocks, and then within each block, randomly assign subjects to treatment groups.(1.3)
ratio level of measurement
data at this measurement are similar to data at the interval level, with the added property that a zero entry is an inherent zero. A ratio of two data values can be formed so that one data value can be meaningfully expressed as a multiple of another.(1.2)
relative frequency
is the portion or percentage of the data that falls in that class. To find the relative frequency of a class, divide the frequency f by the sample size n.(2.1)
replication
is the repetition of an experiment using a large group of subjects.(1.3)
sample
is a subset of a population.(1.1)
sampling error
is the difference between the results of a sample and those of the population.(1.3)
sampling
is a count or measure of apart of a population, more commonly used in statistical studies.(1.3)
simple random sample
is a sample in which every possible sample of the same size has the same chance of being selected.(1.3)
simulation
is the use of a mathematical or physical model to reproduce the conditions of a situation or process.(1.3)
Statistics
is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions.(1.1)
statistic
is a numerical description of a sample characteristic.(1.1)
stratified sample
members of the population are divided into two or more subsets, called strata, that share a similar characteristic such as age, gender, ethnicity or even political preference. A sample is then randomly selected from each of the strata and ensures that each segment of the population is represented.(1.3)
survey
is an investigation of one or more characteristics of a population.(1.3)
systematic sample
is a sample in which each member of the population is assigned a number. The members of the population are ordered in some way, a starting number is randomly selected, and then sample members are selected at regular intervals from the starting number.(1.3)
upper class limit
which is the greatest number that can belong to the class.(2.1)
Addition Rule
If A and B are disjoint events: P(A or B)=P(A) + P(B)
Complement Rule
the probability of an event occurring is 1 minus the probability that it doesn't occur
P(E) = 1 - P(E`) or P(E`) = 1 - P(E)
...
Disjoint Events
events that have no outcomes in common
Independent Events
The outcome of one event does not affect the outcome of the second event
Law of Large Numbers
a statistical law stating that as sample size increases, the probability of an event outcome will more closely reflect the theoretical probability of the event.
Multiplication Rule
if events A and B are independent, then P(A and B) = P(A)P(B)
Probability
the likelihood that a possible future event will occur in any given instance of the event;
mathematical ratio:
what you want to happen
to total outcomes of what could happen
Sample Space
the set of all possible outcomes
Bar Graph
A type of graph which uses bars to show the differences or similarities in different sets of data.
Boxplot
displays the 5-number summary as a central box with whiskers that extend to the non-outlying data values
Center
The midpoint of the distribution of the graph.
Counts
Specific number for each category used for a graph or set of data.
Deviations
Pieces of data which do not follow the graphs overall pattern.
Distribution
tells what value the variable takes and how often it takes these values.
Dotplot
A type of graph where dots are used to represent data. The distribution of the dots can highlight similarities or differences in the data.
Five number summary
The smallest observation, the first quartile, the median, the third quartile, and the largest observation. Usually written from smallest to largest.
Histogram
A bar graph that shows the frequency of data within equal intervals.
Line Graph
Graph that shows change over time using lines and data from observations.
Mean
The average value from a set of observations.
Median
The number in a set of data which half the observations are small than and the other half of the observations are larger than.
Outlier
An individual observation that falls outside the overall pattern of the graph.
Overall pattern
The overall pattern of a graph is the basic trend the graph shows and can lead to a general explanation of the data presented.
Pictogram
A graph that uses pictures as representation instead of actual figures or dots.
Pie Chart
A type of graph which uses a 2 dimensional circle. Sections of the circle are filled out to show differences or similarities in data.
Quartiles
The medians of the two halves of a set of a data after the median of the entire set is found.
Rates
Percent or proportions used for each category in data or a graph.
Resistant
Not influenced by size of data, but by positioning of data when set in numerical order.
Roundoff error
Errors in rounding off a decimal. These add up over time.
Seasonal variation
a pattern that repeats itself at known regular intervals of time
Shape
The general shape of a graph which can be described in a few words.
Skewed to the left
When the left side of the graph extends much farther out than the ride side.
Skewed to the right
When the right side of the graph(containing the half of the observations with larger values) extends much farther out than the left side.
Spread
The difference between the highest and lowest data figures.
Standard Deviation
a measure of variability that describes an average distance of every score from the mean
Stemplot
Represents data by seperating each value into two parts, with the stem being the larger digit in the value and the leaf being the smaller digit.
Symmetric distribution
When the right and left sides of the graph are approximately mirror images of each other.
Trend
A noted tendency of change on a graph or set of data
Variance
The distance of each observation form the mean and square eache of these distances. Average the distances by dividing their sum by n-1. this average squared distance is called variance.
Complete Regression Analysis 3. Use a histogram and/or normal quantile plot to confirm that the values of the residuals have a distribution that is approximately normal.
4. Consider any effects of a pattern over time.
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Complete Regression Analysis
1. Construct a scatterplot and verify that the pattern of the points is approximately a straight-line pattern without outliers. (If there are outliers, consider their effects by comparing results that include the outliers to results that exclude the outliers.)
2. Construct a residual plot and verify that there is no pattern (other than a straight-line pattern) and also verify that the residual plot does not become thicker (or thinner).
correlation
A correlation exists between two variables when the values of one are somehow associated with the values of the other in some way.
explanatory variable, predictor variable or independent variable
the x variable
Hypothesis Test for Correlation Conclusion
If |r| > critical value from Table A-6, reject H0 and conclude that there is sufficient evidence to support the claim of a linear correlation.
If |r| ≤ critical value from Table A-6, fail to reject H0 and conclude that there is not sufficient evidence to support the claim of a linear correlation.
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Hypothesis Test for Correlation Requirements
1. The sample of paired (x, y) data is a simple random sample of quantitative data.
2. Visual examination of the scatterplot must confirm that the points approximate a straight-line pattern.
3. The outliers must be removed if they are known to be errors. The effects of any other outliers should be considered by calculating r with and without the outliers included.
influential points
points that strongly affect the graph of the regression line.
least-squares property
sum of the squares of the residuals is the smallest sum possible
linear correlation coefficient formula
see page 520
linear correlation coefficient r
numerical measure of the strength of the relationship between two variables representing quantitative data.
linear correlation coefficient
The linear correlation coefficient r measures the strength of the linear relationship between the paired quantitative x- and y-values in a sample.
marginal change
the amount that it changes when the other variable changes by exactly one unit
One-Tailed Tests
One-tailed tests can occur with a claim of a positive linear correlation or a claim of a negative linear correlation. In such cases, the hypotheses will be as shown here.
left tailed test: p < 0
right tailed test p > 0
For these one-tailed tests, the P-value method can be used as in earlier chapters.