AP Statistics Practice Quiz: Carrying Out a Chi-Square Test for Homogeneity or Independence
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Test your understanding with short quizzes. This quiz has 16 questions to check your progress.
Question 1 of 16
All Questions (16)
A) The sum of (Observed - Expected)^2 / Expected
B) The sum of (Observed - Expected) / Expected
C) The product of (Observed - Expected)^2 / Expected
D) The sum of (Observed - Expected)^2 / Observed
Correct Answer: A
Based on the provided content, 'The chi-square test statistic is the sum of (Observed - Expected)^2 / Expected'.
A) (rows + 1)(cols + 1)
B) (rows - 1)(cols - 1)
C) (rows * cols) - 1
D) (rows - 1) + (cols - 1)
Correct Answer: B
The content states that for the chi-square test statistic, df = (rows - 1)(cols - 1).
A) 20
B) 19
C) 12
D) 8
Correct Answer: C
The degrees of freedom are calculated as (rows - 1)(cols - 1). For a 4x5 table, this is (4 - 1)(5 - 1) = 3 * 4 = 12.
A) The probability of obtaining a test statistic as or more extreme than the observed value, assuming the null hypothesis (H0) is true.
B) The probability that the null hypothesis (H0) is true.
C) The proportion of the chi-square distribution smaller than the calculated test statistic.
D) The probability that the alternative hypothesis (Ha) is false.
Correct Answer: A
The content explicitly states, 'The p-value is the probability of obtaining a test statistic as or more extreme than the observed value, assuming H0 is true.'
A) By comparing the test statistic to the degrees of freedom.
B) By comparing the p-value to the significance level, alpha.
C) By comparing the observed counts to the expected counts directly.
D) By comparing the sample size to the number of cells in the table.
Correct Answer: B
The provided content states, 'A decision to reject or fail to reject H0 is based on comparing the p-value to alpha.'
A) Reject H0 because 0.027 > 0.05.
B) Fail to reject H0 because 0.027 > 0.05.
C) Reject H0 because 0.027 < 0.05.
D) Fail to reject H0 because 0.027 < 0.05.
Correct Answer: C
The decision rule is to reject H0 if the p-value is less than or equal to alpha. Since 0.027 is less than 0.05, the null hypothesis should be rejected.
A) There is not convincing evidence of an association between the two variables in the population.
B) There is convincing evidence of an association between the two variables in the population.
C) The p-value was greater than 0.01, providing convincing evidence of an association.
D) The test statistic was small, providing convincing evidence that the variables are independent.
Correct Answer: B
Rejecting the null hypothesis (which states no association/independence) means there is sufficient evidence to support the alternative hypothesis (which states there is an association). Therefore, there is convincing evidence of an association.
A) We have convincing evidence that the distribution of the categorical variable is the same for all populations.
B) We have convincing evidence that the distribution of the categorical variable is different across the populations.
C) We do not have convincing evidence that the distribution of the categorical variable is different across the populations.
D) We do not have convincing evidence that the distribution of the categorical variable is the same for all populations.
Correct Answer: C
Failing to reject the null hypothesis (H0: distributions are the same) means we lack sufficient evidence to conclude the alternative hypothesis (Ha: distributions are different) is true. We cannot prove H0 is true, only that we failed to find evidence against it.
A) The proportion of the distribution equal to or smaller than the test statistic.
B) The proportion of the distribution between 0 and the test statistic.
C) The proportion of the distribution equal to or larger than the test statistic.
D) Twice the proportion of the distribution larger than the test statistic.
Correct Answer: C
The content specifies that 'For a two-way table test, the p-value is the proportion of the chi-square distribution equal to or larger than the test statistic.'
A) There is a 4% chance that the two variables are independent.
B) There is a 4% chance of obtaining a test statistic as or more extreme than the one observed.
C) There is a 96% chance that the two variables are associated.
D) There is a 4% chance that the conclusion to reject the null hypothesis is wrong.
Correct Answer: B
The p-value is the probability of getting a result (test statistic) at least as extreme as the one observed, given that the null hypothesis is true. Option B is the correct interpretation based on the provided content.
A) The probability of observing a specific cell count.
B) The squared difference between observed and expected counts for one cell, scaled by the expected count.
C) The total difference between all observed and expected counts.
D) The degrees of freedom associated with a single cell.
Correct Answer: B
Each term, (Observed - Expected)^2 / Expected, is calculated for a single cell in the table. It measures the contribution of that cell's discrepancy to the overall test statistic.
A) Only by manual calculation using the probability density function.
B) By using a z-table or a t-table.
C) By using a chi-square distribution table or technology.
D) Only by using advanced statistical simulation software.
Correct Answer: C
The content states, 'The p-value for a chi-square test for independence or homogeneity is found using a table or technology.' In this context, the appropriate table is a chi-square distribution table.
A) Reject H0; there is convincing evidence that the distributions are different.
B) Reject H0; there is not convincing evidence that the distributions are different.
C) Fail to reject H0; there is convincing evidence that the distributions are the same.
D) Fail to reject H0; there is not convincing evidence that the distributions are different.
Correct Answer: D
Since the p-value (0.15) is greater than alpha (0.05), we fail to reject the null hypothesis. This means we do not have sufficient evidence to conclude the alternative hypothesis (that the distributions are different) is true.
A) The p-value is likely to be large, suggesting the data are consistent with the null hypothesis.
B) The p-value is likely to be small, suggesting the data are not consistent with the null hypothesis.
C) The p-value cannot be inferred without knowing the sample size.
D) The p-value will be approximately equal to the significance level, alpha.
Correct Answer: B
The p-value is the area in the tail of the chi-square distribution to the right of the test statistic. A very large test statistic will be far out in the right tail, resulting in a very small area (p-value). This indicates an extreme result under the null hypothesis.
A) The p-value was large, and the decision was to fail to reject H0.
B) The calculated chi-square statistic was smaller than the critical value.
C) The p-value was less than or equal to the chosen significance level, alpha.
D) The observed counts in the table were exactly equal to the expected counts.
Correct Answer: C
A conclusion of 'convincing statistical evidence of an association' implies that the null hypothesis of independence was rejected. According to the decision rule provided, H0 is rejected when the p-value is less than or equal to alpha.
A) There is a 45% chance that H0 is true.
B) If H0 were true, the probability of observing a test statistic as or more extreme than the one found is 0.45.
C) The result is statistically significant at the α = 0.05 level.
D) There is a 45% chance that the distributions for the populations are different.
Correct Answer: B
The p-value is the probability of obtaining the observed result (or a more extreme one) if the null hypothesis is true. A p-value of 0.45 means that such a result is not unusual under the assumption that H0 is true. This is the direct interpretation based on the provided content.