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Elementary Statistics Test 2
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Terms in this set (106)
The rare event rule
If, under a given assumption, the probability of a particula observed event is extremely small, we conclude that the assumption is probably not correct
Event
Any collection of results or outcomes of a procedure (example: rolling a dice six times)
Simple event
An outcome or an event that cannot be further broken down into simpler components (example: having three children can be broken down; having two girls and one boy is broken down the most)
Sample Space
For a procedure consists of all possible simple events; that is the _______________ consists of all possible outcomes that cannot be broken down any further.
probability
The likelihood of an event occurring
P
Denotes a probability
A, B, C
Denotes specific events
P(A)
Denotes the probability of event A occurring
0 and 1
The probability of an event is between what numbers inclusive?
An impossible event
If P(A)=0, then A is
A certain event
If P(A)=1, then A is
Relative frequency approximation
Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on actual results.
Number of times A occurred/number of times procedure was repeated
Relative frequency approximation P(A) is calculated how?
Classical approach
This requires equally likely outcomes. Assume that a given procedure has n (number of different simple events) different simple events and that each of those simple events have an equal chance of occurring.
S/n meaning number of ways A can occur/number of different simple events
Classical approach P(A) is calculated how?
Subjective probabilities
The probability of event A is estimated by using knowledge of the relevant circumstances. (Example: Estimating the probability of your teacher walking in with a feather hat)
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 (classical)
Incorrectly assuming that outcomes are equally likely
What is one common mistake with probability?
Complementary event
It consist of all outcomes in which the event does not occur
1-P(A)
What is the formula to find the complementary event?
Three significant digits
When the probability is a decimal, then round it to...
When it is not a simple fraction
When should probability not be expressed as a fraction?
Unlikely
When an event has a small probability, it is
0.05 or less
For an event to be unlikely, it's probability is very small such as...
Unusual
When an event has an extreme result meaning that the number of outcomes of a particular type is far below or far above, then it is...
P(Ā)/P(A)
What is the formula for ACTUAL odds against event A?
a:b
How is ACTUAL odds against event A expressed in ratio form?
P(A)/P(Ā)
What is the formula for ACTUAL odds in favor of event A?
b:a
How is ACTUAL odds in favor of event A expressed in ratio form?
(Net profit):(amount bet)
What is the formula for PAYOFF odds against event A?
Compound event
Any event combining 2 or more simple events.
Event A occurs or event B occurs it they both occur
What does P(A or B) mean?
Counting an outcome twice
When dealing with a compound event and counting the number of ways event A and B can occur, what do you have to be careful of?
P(A or B)=P(A)+P(B)-P(A and B)
What is the formal addition rule formula?
Counting event A and B without counting an outcome more than once, then dividing by the total number of outcomes
What is the Intuitive Addition rule?
Disjoint
When events A and B cannot occur at the same time, then they are ...
Occur at the same time
If two events overlap then they can ...
No
Is it possible for a complement of an event to occur at the same time and overlap?
1
P(A)+P(Ā)=
1-P(A)
P(Ā)=
1-P(Ā)
P(A)=
Addition
"Or" implies ___________
Multiplication
"And" implies __________
Independent events
If the occurrence of one event does not affect the probability of the occurrence of the other event.
P(A and B)=P(A) times P(B|A)
What is the formal multiplication rule?
You are taking into account that event A has occurred. Thus, the probability of B given event A.
What does P(B|A) mean?
Intuitive Multiplication rule
When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of B given that event A already happened.
P(A and B)=P(A) times P(B)
What is the formula for the Multiplication rule for Independent Events?
Independent or dependent
When applying the multiplication rule, always consider whether the events are ...
Independent
Is sampling with replacement a dependent or independent event?
Dependent
Is sampling without replacement a dependent or independent event?
Random Variable
A variable (typically represented by x) that has a single numerical value, determined by chance, for each outcome of a procedure
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
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 that they result from a counting process
Continuous random variable
Has infinitely many values, and the collection of values is non-countable; those values can be associated with measurements on a continuous scale without gaps or interruptions
X should be a numerical random variable, and its value should be associated with corresponding probabilities, the sum of the probabilities should equal one, and the values for x should be between 0 and 1.
What are the requirements for probability distribution?
Probability histogram
Histogram that is very similar to a relative frequency histogram, but the vertical scale shows probabilities. The horizontal scale shows the variables of x.
μ=Σ[x•P(x)]
What is the formula for the mean of a probability distribution?
σ^2=Σ[x^2•P(x)]-μ^2
What is the formula for the variance of a probability distribution?
σ= Square root of (Σ[x^2•P(x)]-μ^2)
What is the formula for the standard deviation of a probability distribution?
Carry one more decimal place than the number of decimal places used for the random variable x.
How do you round for μ, σ, and σ^2 if they are not integers?
To one decimal place
How do you round for μ, σ, and σ^2 if they are integers?
Expected value
The ________________ of a discrete random variable; it represents the mean value of the outcomes. It is denoted by E. It is obtained by finding the value of Σ[x•P(x)].
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 and minimum values.
μ+2σ
Maximum usual value:
μ-2σ
Minimum usual value:
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.
Unusually high
Part of the rare event rule. X successes among n trials is an unusually high number of successes if P(x or more) less than or equal to 0.05.
Unusually low
Part of the rare event rule. X successes among n trials is an unusually low number of successes if P(x or fewer) is less than or equal to 0.05.
The procedure has a fixed number of trials, the trials are independent, each trial must have all outcomes classified into two categories like successes and failures, and the probability of a success remains the same in all trials.
What are the requirements for a binomial probability distribution?
S
Success is denoted by
F
Failure is denoted by
Probability of success
P(S)=p
Probability of failure
P(F)=1-p=q
Probability of success
P represents
Probability of failure
Q represents
1-p
How do you find probability of failure?
n
Denotes the number of trials
x
Denotes a specific number of successes in n trials, so it can be any whole number between 0 and n (number of trials), inclusive.
p
Denotes the probability of success in one of the n trials.
q
Denotes the probability of failure in one of the n trials.
P(x)
Denotes the probability of getting exactly x successes among the n trials.
Success
X and p refer to the same category which is called
n<0.05N
When sampling is without replacement, they can be considered independent if ...
(n!/(n-x)!x!)•p^x•q^(n-x)
To find the P(x), what is the formula used?
It is the event we are trying to determine
What does it mean when something is categorized as a success?
μ=n•p
What is the Binomial distribution mean formula?
σ^2=n•p•q
What is the Binomial distribution variance formula?
σ=square root of n•p•q
What is the Binomial distribution standard deviation formula?
Uniform distribution
A continuous random variable has an even distribution of its values are spread evenly over the range of probabilities. It's graph results in the shape of a rectangle.
Density Curve
The graph of a continuous probability distribution. It must satisfy the following properties: area under curve equals 1 and every point on curve must have a vertical height that is 0 or greater.
The area under the curve equals one and the curve cannot fall below the x-axis or be negative
What are the density curve requirements?
Probability
Because the total area under the density curve is equal to 1, there is a correspondence between area and ______________.
Standard normal distribution
It is a normal distribution with μ=0 and σ=1. The total area under its density curve is equal to 1.
Standard normal distribution
What is Table A-2 designed for?
A positive and negative
Table A-2 is on two pages which each consist of what kind of page?
Left
The Table A-2 is cumulative to the ________.
Z score
Distance along the horizontal scale of the standard normal distribution.
Area
Region under the curve
P(a<z<b)
Denotes the probability that the z score is between a and b.
P(z>a)
Denotes the probability that the z score is greater than a.
P(z<a)
Denotes the probability that the z score is less than a.
It is a z score that separates unlikely values from those that are likely to occur.
What is a critical value?
The z score with an area of α to its right
The expression z subscript α denotes
(x-μ)/σ
What is the formula for a z score?
2
How many decimal places is the z score rounded to?
;