194 terms

AP Statistics

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How do you check if there is outliers?
calculate IQR; anything above Q3+1.5(IQR) or below Q1-1.5(IQR) is an outlier
If a graph is skewed, should we calculate the median or the mean? Why?
median; it is resistant to skews and outliers
If a graph is roughly symmetrical, should we calculate the median or the mean? Why?
mean; generally is more accurate if the data has no outliers
What is in the five number summary?
Minimum, Q1, Median, Q3, Maximum
Relationship between variance and standard deviation?
variance=(standard deviation)^2
variance definition
the variance is roughly the average of the squared differences between each observation and the mean
standard deviation
the standard deviation is the square root of the variance
What should we use to measure spread if the median was calculated?
IQR
What should we use to measure spread if the mean was calculated?
standard deviation
What is the IQR? How much of the data does it represent?
Q3-Q1; 50%
How do you calculate standard deviation?
1. Type data into L1
2. Find mean with 1 Variable Stats
3. Turn L2 into (L1-mean)
4. Turn L3 into (L2)^2
5. Go to 2nd STAT over to MATH, select sum(
6. Type in L3
7. multiply it by (1/n-1)
8. Square root it
What is the formula for standard deviation?
Categorical variables vs. Quantitative Variables
Categorical: individuals can be assigned to one of several groups or categories
Quantitative: takes numberical values
If a possible outlier is on the fence, is it an outlier?
No
Things to include when describing a distribution
Center (Mean or Median), Unusual Gaps or Outliers, Spread (Standard Deviation or IQR), Shape (Roughly Symmetric, slightly/heavily skewed left or right, bimodal, range)
Explain how to standardize a variable. What is the purpose of standardizing a variable?
Subtract the distribution mean and then divide by standard deviation. Tells us how many standard deviations from the mean an observation falls, and in what direction.
What effect does standardizing the values have on the distribution?
shape would be the same as the original distribution, the mean would become 0, the standard deviation would become 1
What is a density curve?
a curve that (a) is on or above the horizontal axis, and (b) has exactly an area of 1
Inverse Norm
when you want to find the percentile: invNorm (area, mean, standard deviation)
z
(x-mean)/standard deviation
pth percentile
the value with p percent observations less than is
cumulative relative frequency graph
can be used to describe the position of an individual within a distribution or to locate a specified percentile of the distribution
How to find and interpret the correlation coefficient r for a scatterplot
STAT plot, scatter, L1 and L2 (Plot 1: ON); STAT --> CALC --> 8:LinReg(a+bx)
No r? --> 2nd 0 (Catalog) down to Diagnostic ON
r
tells us the strength of a LINEAR association. -1 to 1. Not resistant to outliers
r^2
the proportion (percent) of the variation in the values of y that can be accounted for by the least squares regression line
residual plot
a scatterplot of the residuals against the explanatory variable. Residual plots help us assess how well a regression line fits the data. It should have NO PATTERN
regression line
a line that describes how a response variable y changes as an explanatory variable x changes. We often use a regression line to predict the value of y for a given value of x.
residual formula
residual=y-y(hat) aka observed y - predicted y
What method do you use to check if a distribution or probability is binomial?
BINS:
1. Binary: There only two outcomes (success and failure)
2. Independent: The events independent of one another?
3. Number: There is a fixed number of trials
4. Success: The probability of success equal in each trial
What method do you use to check if a distribution or probability is geometric?
BITS:
1. Binary: There only two outcomes (success and failure)
2. Independent: The events independent of one another
3. Trials: There is not a fixed number of trials
4. Success: The probability of success equal in each trial
n
number of trials
p
probability of success
k
number of successes
Binomial Formula for P(X=k)
(n choose k) p^k (1-p)^(n-k)
Binomial Calculator Function to find P(X=k)
binompdf(n,p,k)
Binomial Calculator Function for P(X≤k)
binomcdf(n,p,k)
Binomial Calculator Function for P(X≥k)
1-binomcdf(n,p,k-1)
mean of a binomial distribution
np
standard deviation of a binomial distribution
√(np(1-p))
Geometric Formula for P(X=k)
(1-p)^(k-1) x p
Geometric Calculator Function to find P(X=k)
geometpdf(p,k)
Geometric Calculator Function for P(X≤k)
geometcdf(p,k)
Geometric Calculator Function for P(X≥k)
1-geometcdf(p,k-1)
Mean of a geometric distribution
1/p=expected number of trials until success
Standard deviation of a geometric distribution
√((1-p)/(p²))
What do you do if the binomial probability is for a range, rather than a specific number?
Take binomcdf(n,p,maximum) - binomcdf(n,p,minimum-1)
how do you enter n choose k into the calculator?
type "n" on home screen, go to MATH --> PRB --> 3: ncr, type "k"
μ(x+y)
μx+μy
μ(x-y)
μx-μy
σ(x+y)
√(σ²x+σ²y)
What does adding or subtracting a constant effect?
Measures of center (median and mean).
Does NOT affect measures of spread (IQR and Standard Deviation) or shape.
What does multiplying or dividing a constant effect?
Both measures of center (median and mean) and measures of spread (IQR and standard deviation).
Shape is not effected.
For variance, multiply by a² (if y=ax+b).
σ(x-y)
√(σ²x+σ²y) --> you add to get the difference because variance is distance from mean and you cannot have a negative distance
calculate μx by hand
X1P1+X2P2+.... XKPK (SigmaXKPK)
calculate var(x) by hand
(X1-μx)²p(1)+(X2-μx)²p(2)+.... (Sigma(Xk-μx)²p(k))
Standard deviation
square root of variance
discrete random variables
a fixed set of possible x values (whole numbers)
continuous random variables
-x takes all values in an interval of numbers
-can be represented by a density curve (area of 1, on or above the horizontal axis)
What is the variance of the sum of 2 random variables X and Y?
(σx)²+(σy)², but ONLY if x and y are independent.
mutually exclusive
no outcomes in common
addition rule for mutually exclusive events
P (A U B)
P(A)+P(B)
complement rule
P(A^C)
1-P(A)
general addition rule (not mutually exclusive)
P(A U B)
P(A)+P(B)-P(A n B)
intersection
P(A n B)
both A and B will occur
conditional probability
P (A | B)
P(A n B) / P(B)
independent events (how to check independence)
P(A) = P(A|B)
P(B)= P(B|A)
multiplication rule for independent events
P(A n B)
P(A) x P(B)
general multiplication rule (non-independent events)
P(A n B)
P(A) x P(B|A)
sample space
a list of possible outcomes
probability model
a description of some chance process that consists of 2 parts: a sample space S and a probability for each outcome
event
any collection of outcomes from some chance process, designated by a capital letter (an event is a subset of the sample space)
What is the P(A) if all outcomes in the sample space are equally likely?
P(A) = (number of outcomes corresponding to event A)/(total number of outcomes in sample space)
Complement
probability that an event does not occur
What is the sum of the probabilities of all possible outcomes?
1
What is the probability of two mutually exclusive events?
P(A U B)= P(A)+P(B)
five basic probability rules
1. for event A, 0≤P(A)≤1
2. P(S)=1
3. If all outcomes in the sample space are equally likely, P(A)=number of outcomes corresponding to event A / total number of outcomes in sample space
4. P(A^C) = 1-P(A)
5. If A and B are mutually exclusive, P(A n B)=P(A)+P(B)
When is a two-way table helpful
displays the sample space for probabilities involving two events more clearly
In statistics, what is meant by the word "or"?
could have either event or both
When can a Venn Diagram be helpful?
visually represents the probabilities of not mutually exclusive events
What is the general addition rule for two events?
If A and B are any two events resulting from some chance process, then the probability of A or B (or both) is P(A U B)= P(A)+P(B)-P(A n B)
What does the intersection of two or more events mean?
both event A and event B occur
What does the union of two or more events mean?
either event A or event B (or both) occurs
What is the law of large numbers?
If we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value, which we can call the probability of that outcome
the probability of any outcome...
is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions
How do you interpret a probability?
We interpret probability to represent the most accurate results if we did an infinite amount of trials
What are the two myths about randomness?
1. Short-run regularity --> the idea that probability is predictable in the short run
2. Law of Averages --> people except the alternative outcome to follow a different outcome
simulation
the imitation of chance behavior, based on a model that accurately reflects the situation
Name and describe the four steps in performing a simulation
1. State: What is the question of interest about some chance process
2. Plan: Describe how to use a chance device to imitate one repetition of process; clearly identify outcomes and measured variables
3. Do: Perform many repetitions of the simulation
4. Conclude: results to answer question of interest
What are some common errors when using a table of random digits?
not providing a clear description of the simulation process for the reader to replicate the simulation
What does the intersection of two or more events mean?
both event A and event B occur
sample
The part of the population from which we actually collect information. We use information from a sample to draw conclusions about the entire population
population
In a statistical study, this is the entire group of individuals about which we want information
sample survey
A study that uses an organized plan to choose a sample that represents some specific population. We base conclusions about the population on data from the sample.
convenience sample
A sample selected by taking the members of the population that are easiest to reach; particularly prone to large bias.
bias
The design of a statistical study shows ______ if it systematically favors certain outcomes.
voluntary response sample
People decide whether to join a sample based on an open invitation; particularly prone to large bias.
random sampling
The use of chance to select a sample; is the central principle of statistical sampling.
simple random sample (SRS)
every set of n individuals has an equal chance to be the sample actually selected
strata
Groups of individuals in a population that are similar in some way that might affect their responses.
stratified random sample
To select this type of sample, first classify the population into groups of similar individuals, called strata. Then choose a separate SRS from each stratum to form the full sample.
cluster sample
To take this type of sample, first divide the population into smaller groups. Ideally, these groups should mirror the characteristics of the population. Then choose an SRS of the groups. All individuals in the chosen groups are included in the sample.
inference
Drawing conclusions that go beyond the data at hand.
margin of error
Tells how close the estimate tends to be to the unknown parameter in repeated random sampling.
sampling frame
The list from which a sample is actually chosen.
undercoverage
Occurs when some members of the population are left out of the sampling frame; a type of sampling error.
nonresponse
Occurs when a selected individual cannot be contacted or refuses to cooperate; an example of a nonsampling error.
wording of questions
The most important influence on the answers given to a survey. Confusing or leading questions can introduce strong bias, and changes in wording can greatly change a survey's outcome. Even the order in which questions are asked matters.
observational study
Observes individuals and measures variables of interest but does not attempt to influence the responses.
experiment
Deliberately imposes some treatment on individuals to measure their responses.
explanatory variable
A variable that helps explain or influences changes in a response variable.
response variable
A variable that measures an outcome of a study.
lurking variable
a variable that is not among the explanatory or response variables in a study but that may influence the response variable.
treatment
A specific condition applied to the individuals in an experiment. If an experiment has several explanatory variables, a treatment is a combination of specific values of these variables.
experimental unit
the smallest collection of individuals to which treatments are applied.
subjects
Experimental units that are human beings.
factors
the explanatory variables in an experiment are often called this
random assignment
An important experimental design principle. Use some chance process to assign experimental units to treatments. This helps create roughly equivalent groups of experimental units by balancing the effects of lurking variables that aren't controlled on the treatment groups.
replication
An important experimental design principle. Use enough experimental units in each group so that any differences in the effects of the treatments can be distinguished from chance differences between the groups.
double-blind
An experiment in which neither the subjects nor those who interact with them and measure the response variable know which treatment a subject received.
single-blind
An experiment in which either the subjects or those who interact with them and measure the response variable, but not both, know which treatment a subject received.
placebo
an inactive (fake) treatment
placebo effect
Describes the fact that some subjects respond favorably to any treatment, even an inactive one
block
A group of experimental units that are known before the experiment to be similar in some way that is expected to affect the response to the treatments.
inference about the population
Using information from a sample to draw conclusions about the larger population. Requires that the individuals taking part in a study be randomly selected from the population of interest.
inference about cause and effect
Using the results of an experiment to conclude that the treatments caused the difference in responses. Requires a well-designed experiment in which the treatments are randomly assigned to the experimental units.
lack of realism
When the treatments, the subjects, or the environment of an experiment are not realistic. Lack of realism can limit researchers' ability to apply the conclusions of an experiment to the settings of greatest interest.
institutional review board
A basic principle of data ethics. All planned studies must be approved in advance and monitored by _____________ charged with protecting the safety and well-being of the participants.
informed consent
A basic principle of data ethics. Individuals must be informed in advance about the nature of a study and any risk of harm it may bring. Participating individuals must then consent in writing.
simulation
a model of random events
census
a sample that includes the entire population
population parameter
a number that measures a characteristic of a population
systematic sample
every fifth individual, for example, is chosen
multistage sample
a sampling design where several sampling methods are combined
sampling variability
the naturally occurring variability found in samples
levels
the values that the experimenter used for a factor
the four principles of experimental design
control, randomization, replication, and blocking
completely randomized design
a design where all experimental units have an equal chance of receiving any treatment
interpreting p value
if the true mean/proportion of the population is (null), the probability of getting a sample mean/proportion of _____ is (p-value).
p̂1-p̂2 center, shape, and spread
center: p1-p2
shape: n1p1, n1(1-p1), n2p2, and n2(1-p2) ≥ 10
spread (if 10% condition checks): √((p1(1-p1)/n1)+(p2(1-p2)/n2)
probability of getting a certain p̂1-p̂2 (ex. less than .1)
plug in center and spread into bell curve, find probability
Confidence intervals for difference in proportions formula
(p̂1-p̂2) plus or minus z*(√((p1(1-p1)/n1)+(p2(1-p2)/n2))
When do you use t and z test/intervals?
t for mean
z for proportions
Significance test for difference in proportions
...
What is a null hypothesis?
What is being claimed. Statistical test designed to assess strength of evidence against null hypothesis. Abbreviated by Ho.
What is an alternative hypothesis?
the claim about the population that we are trying to find evidence FOR, abbreviated by Ha
When is the alternative hypothesis one-sided?
Ha less than or greater than
When is the alternative hypothesis two-sided?
Ha is not equal to
What is a significance level?
fixed value that we compare with the P-value, matter of judgement to determine if something is "statistically significant".
What is the default significance level?
α=.05
Interpreting the p-value
if the true mean/proportion of the population is (null), the probability of getting a sample mean/proportion of _____ is (p-value).
p value ≤ α
We reject our null hypothesis. There is sufficient evidence to say that (Ha) is true.
p value ≥ α
We fail to reject our null hypothesis. There is insufficient evidence to say that (Ho) is not true.
reject Ho when it is actually true
Type I Error
fail to reject Ho when it is actually false
Type II Error
Power definition
probability of rejecting Ho when it is false
probability of Type I Error
α
probability of Type II Error
1-power
two ways to increase power
increase sample size/significance level α
5 step process: z/t test
State --> Ho/Ha, define parameter
Plan --> one sample, z test
Check --> random/normal/independent
Do --> find p hat, find test statistic (z), use test statistic to find p-value
Conclude -->
p value ≤ α reject Ho
p value ≥ α fail to reject Ho
Formula for test statistic (μ)
Formula for test statistic (p̂) (where p represents the null)
(p̂-p)/(√((p)(1-p))/n)
probability of a Type II Error?
overlap normal distribution for null and true. Find rejection line. Use normalcdf
when do you use z tests?
for proportions
when do you use t tests?
for mean (population standard deviation unknown)
finding p value for t tests
tcdf(min, max, df)
Sample paired t test
state--> Ho: μ1-μ2=0 (if its difference)
plan --> one sample, paired t test
check --> random, normal, independent
do --> find test statistic and p value
conclude --> normal conclusion
What does statistically significant mean in context of a problem?
The sample mean/proportion is far enough away from the true mean/proportion that it couldn't have happened by chance
When doing a paired t-test, to check normality, what do you do?
check the differences histogram (μ1-μ2)
How to interpret a C% Confidence Level
In C% of all possible samples of size n, we will construct an interval that captures the true parameter (in context).
How to interpret a C% Confidence Interval
We are C% confident that the interval (_,_) will capture the true parameter (in context).
What conditions must be checked before constructing a confidence interval?
random, normal, independent
C% confidence intervals of sample proportions, 5 step process
State: Construct a C% confidence interval to estimate...
Plan: one sample z-interval for proportions
Check: Random, Normal, Independent
Do: Find the standard error and z, then p hat +/- z
Conclude: We are C% confident that the interval (_,_) will capture the true parameter (in context).
What's the z interval standard error formula?
How do you find z*?
InvNorm(#)
How do you find the point estimate of a sample?
subtract the max and min confidence interval, divide it by two (aka find the mean of the interval ends)
How do you find the margin of error, given the confidence interval?
Ask, "What am I adding or subtracting from the point estimate?"
So find the point estimate, then find the difference between the point estimate and the interval ends
Finding sample size proportions: When p hat is unknown, or you want to guarantee a margin of error less than or equal to:
use p hat=.5
Finding the confidence interval when the standard deviation of the population is known
x bar +/- z*(σ/√n)
Checking normal condition for z* (population standard deviation known)
starts normal or CLT
Finding the confidence interval when the standard deviation of the population is unknown (which is almost always true)
x bar +/- t*(Sx/√n)
degrees of freedom
n-1
How do you find t*?
InvT(area to the left, df)
What is the standard error?
same as standard deviation, but we call it "standard error" because we plugged in p hat for p (we are estimating)
a point estimator is a statistic that...
provides an estimate of a population parameter.
Explain the two conditions when the margin of error gets smaller.
Confidence level C decreases, sample size n increases
Does the confidence level tell us the chance that a particular confidence interval captures the population parameter?
NO; the confidence interval gives us a set of plausible values for the parameter
Sx and σx: which is which?
Sx is for a sample, σx is for a population
How do we know when do use a t* interval instead of a z interval?
you are not given the population standard deviation
Checking normal condition for t* (population standard deviation unknown)
Normal for sample size...
-n


-n<15: if the data appears closely normal (roughly symmetric, single peak, no outliers)
How to check if a distribution is normal for t*, population n<15
plug data into List 1, look at histogram. Conclude with "The histogram looks roughly symmetric, so we should be safe to use the t distribution)
t* confidence interval, 5 step process
State: Construct a __% confidence interval to estimate...
Plan: one sample t interval for a population mean
Check: Random, Normal, Independent
(for Normal, look at sample size and go from there)
Do: Find the standard error (Sx/√n) and t, then do x bar +/- tt*(standard error)
Conclude: We are __% confident that the interval (_,_) will capture the true parameter (in context).
margin of error formula
z or t (standard error)
When calculating t interval, what is it and where do you find the data?
x bar plus or minus t* (Sx/√n)
-get x bar and Sx using 1 Var Stats
-t*=Invt(area to the left, df)
-population (n) will be given
What is it looking for if it asks for the appropriate critical value?
z/t* interval