Chapter 9: Testing a Claim

AP Statistics | 2024-25

What is a significance test?

a formal procedure for using observed data to decide between two competing claims (hypotheses)

What is the null hypothesis (H0)?

the claim that we weigh evidence against in a significance test; usually a statement of “no difference”

What is the alternative hypothesis (Ha)?

the claim that we are trying to find evidence for

When is the alternative hypothesis one-sided?

if it states that a parameter is greater or less than the null value

When is the alternative hypothesis two-sided?

if it states that the parameter is different from the null value

What is the p-value of a test?

the probability of getting evidence for the alternative hypothesis Ha as strong or stringer than the observed evidence when the null hypothesis (H0)

True or false: hypotheses always refer to a population, not a sample

true

What conclusion do you make if the p-value is large?

“Because [p-value] > α, we fail to reject H0. We do not have convincing evidence for [Ha in context]”

What conclusion do you make if the p-value is small?

“Because [p-value] < α, we reject H0. We do not have convincing evidence for [Ha in context]

What is significance level α?

the value that we use as a boundary for deciding whether an observed result is unlikely to happen by chance alone when the null hypothesis is true

When does a Type I error occur?

if we reject H0 when Ha is true; the data give convincing evidence that Ha is true when it really isn’t

When does a Type II error occur?

if we fail to reject H0 when Ha is true; the data do not give convincing evidence that Ha is true when it really is

What are the most common significance level α?

0.05, 0.01, and 0.10

What is the probability of a Type I error?

significance level α

How can you reduce the probability of a Type I error?

by using a smaller significance level

What is the drawback of using a smaller significance level?

Type II error increases

What does a standardized test statistic measure?

how far a sample statistic is from what we would expect if the null hypothesis H0 were true, in standard deviation units

What is the formula for a standardized test statistic?

(insert)

What is the State step of a significance test?

state the hypotheses you want to test and the significance level, and define any parameters you use

What is the Plan step of a significance test?

identify the appropriate inference method and check conditions

What is the Do step of a significance test?

if the conditions are met, perform calculations/give the sample statistic(s)/calculate the standardized test statistic/find the p-value

What is the Conclude step of a significance test?

make a conclusion about the hypotheses in the context of the problem

What is the power of a test?

the probability that the test will find convincing evidence for Ha when a specific alternative value of the parameter is true

What is the interpretation for p-value?

“Assuming H0 is true (p = ___ ), there is a [p-value] probability of getting a p̂ of ___ or [greater/less

What is the interpretation for power of a test?

“If the true [parameter in context] is […], there is a [power of the test value] probability that [individual/organization] will find convincing evidence for Ha”

What is the power of the test in relation to the probability of Type II error?

power = 1 – P(Type II error)

What is the probability of Type II error in relation to the power of the test?

P(Type II error) = 1 – Power

When will the power of the test be larger?

when:/sample size n is larger/the significance level α is larger/the null and alternative parameter values are further apart

What is the general form of the null hypothesis in a test for comparing two proportions?

H0: p1 – p2 = hypothesized value

What is the formula for p̂C for a test for comparing two proportions?

(insert)

What is the formula for a standardized test statistic in a test for comparing two proportions?

(insert)

How do you test the hypothesis H₀: p1 – p2 = 0?

first find the pooled proportion p̂_C of success in both samples combined, then compute the standardized test statistic