AP Statistics Practice Quiz: Carrying Out a Test for the Difference of Two Population Means
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 difference between the two sample means is 2.45.
B) The observed difference between the sample means is 2.45 standard deviations above the hypothesized difference of zero.
C) The probability of observing a difference as extreme as the one found is 2.45%.
D) The p-value for the test is 2.45.
Correct Answer: B
Based on Content 1, the test statistic for a difference of two means measures how far the observed difference in sample means is from the hypothesized difference (usually 0), in terms of standard error. It is not the difference itself, a probability, or the p-value.
A) There is a 3% chance that the two population means are actually equal.
B) There is a 3% chance that the alternative hypothesis is true.
C) Assuming the two population means are equal, there is a 3% chance of observing a difference in sample means as extreme or more extreme than the one observed.
D) Assuming the two population means are different, there is a 3% chance of observing a difference in sample means as extreme or more extreme than the one observed.
Correct Answer: C
Content 2 and 5 state that interpreting a p-value involves calculating the probability of observing a result as or more extreme than the sample result, under the assumption that the null hypothesis is true. Option C correctly frames this conditional probability.
A) We have convincing evidence that there is a difference in the mean battery life between the two brands.
B) We have convincing evidence that there is no difference in the mean battery life between the two brands.
C) We do not have convincing evidence that there is a difference in the mean battery life between the two brands.
D) The mean battery life is the same for both brands.
Correct Answer: C
Based on Content 3 and 6, a formal decision is made by comparing the p-value to alpha. Since the p-value (0.09) is greater than alpha (0.05), we fail to reject the null hypothesis. This means we lack convincing evidence to support the alternative hypothesis (that there is a difference).
A) It is exactly a t-distribution with degrees of freedom equal to the smaller sample size minus one.
B) It is an approximate t-distribution, with the degrees of freedom typically calculated using technology.
C) It is exactly a Normal distribution, as long as both sample sizes are greater than 30.
D) It is an approximate F-distribution with two different degrees of freedom.
Correct Answer: B
Content 4 explicitly states that the sampling distribution of the t-statistic for a difference of two means is an approximate t-distribution, with degrees of freedom found by technology. The conservative 'n-1' method is an approximation, but the more accurate method requires a complex formula typically handled by software.
A) The alternative hypothesis is true.
B) The null hypothesis is true.
C) Both population means are greater than zero.
D) The sample means are equal to the population means.
Correct Answer: B
Content 5 is clear that interpreting the p-value for a test for a difference of means assumes the null hypothesis is true. The p-value is the probability of getting our observed result (or more extreme) *given that* the null hypothesis is the true state of the populations.
A) Reject the null hypothesis because 0.021 < 0.05.
B) Fail to reject the null hypothesis because 0.021 < 0.05.
C) Reject the null hypothesis because 0.021 is a small number.
D) Fail to reject the null hypothesis because 0.05 > 0.021.
Correct Answer: A
Content 6 outlines the formal decision rule: compare the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, we reject the null hypothesis. Here, 0.021 is less than 0.05, so the correct decision is to reject H₀.
A) It proves with certainty that the new drug is better.
B) It provides statistical reasoning to support the claim that the new drug is more effective.
C) It shows that the difference observed in the sample will be the exact same difference in the population.
D) It confirms that no other factors could have influenced the results.
Correct Answer: B
According to Content 7, the results of a significance test provide statistical reasoning for a research question. A significant result does not provide absolute proof but offers evidence to support a claim, which in this case is that the new drug is more effective.
A) Because the p-value is large, we have strong evidence that the mean commute times in the two cities are the same.
B) Because the p-value is large, we fail to reject the null hypothesis and cannot conclude there is a difference in mean commute times.
C) Because the p-value is large, we accept the null hypothesis and conclude the mean commute times are equal.
D) Because the p-value is large, the test is invalid and no conclusion can be made about the population.
Correct Answer: B
Based on Content 3, we must justify a claim. A large p-value (e.g., 0.45 > 0.05) leads to a failure to reject the null hypothesis. This does not prove H₀ is true (so A and C are incorrect), but rather that we lack sufficient evidence to conclude H₀ is false. Option B uses the correct language of 'failing to reject' and 'cannot conclude' a difference.
A) The difference between the two sample means (x̄₁ - x̄₂).
B) The hypothesized difference between the population means (μ₁ - μ₂), usually 0.
C) The standard deviations and sizes of both samples.
D) The population standard deviations (σ₁ and σ₂).
Correct Answer: D
Based on Content 1, calculating the appropriate test statistic for a difference of two means involves a t-statistic, not a z-statistic. The t-statistic is used precisely because the population standard deviations (σ₁ and σ₂) are unknown and must be estimated from the sample standard deviations (s₁ and s₂). The other options are all components of the t-statistic calculation.
A) The p-value is greater than 0.01.
B) The p-value is less than 0.01.
C) The null hypothesis is true.
D) The test statistic was less than 1.
Correct Answer: A
Content 6 describes the formal decision process. If the decision is to 'fail to reject' the null hypothesis, it means the p-value was not small enough to provide convincing evidence. Specifically, the p-value must be greater than the significance level, alpha. Therefore, p-value > 0.01.
A) Assuming the medications have the same mean effectiveness, the probability of observing a difference as large as we did is 4%.
B) The probability that the null hypothesis is true is 4%.
C) The result is statistically significant at the α = 0.05 level.
D) We have evidence to suggest a difference in mean effectiveness between the two medications.
Correct Answer: B
Based on Content 2 and 5, the p-value is a conditional probability, calculated *assuming* the null hypothesis is true. It does not represent the probability of the null hypothesis itself being true. This is a very common and critical misinterpretation. Option A is the correct interpretation.
A) Prove that the sample means are different.
B) Determine if the observed difference in sample means is likely due to random sampling variation or a true difference in the populations.
C) Calculate the exact difference between the two population means.
D) Confirm that the data was collected without bias.
Correct Answer: B
Content 7 states that a significance test provides statistical reasoning. The core of this reasoning is to assess whether an observed difference is statistically significant (unlikely to be chance) or if it could be plausibly explained by the natural variability that occurs when taking random samples.
A) By adding the two sample sizes and subtracting two (n₁ + n₂ - 2).
B) By using the smaller of n₁ - 1 or n₂ - 1.
C) By using a complex formula (Satterthwaite approximation) that is typically calculated by statistical software.
D) By taking the average of the two sample sizes and subtracting one.
Correct Answer: C
Content 4 specifies that the degrees of freedom are 'found by technology'. This refers to the Satterthwaite approximation, a complex formula that provides a more accurate value for the degrees of freedom than the simpler, more conservative methods mentioned in options A and B.
A) We have very strong evidence to conclude that μ₁ is not equal to μ₂.
B) We have proven that μ₁ is different from μ₂.
C) We can be 99.2% confident that μ₁ is not equal to μ₂.
D) We have very strong evidence that the sample means, x̄₁ and x̄₂, are different.
Correct Answer: A
Based on Content 3, we must justify a claim about the population. A very small p-value (0.008) provides strong evidence against the null hypothesis and in favor of the alternative. Option A uses the appropriate language of 'strong evidence' and makes a conclusion about the population means (μ). Option B is too strong ('proven'), C misinterprets the p-value as a confidence level, and D makes a claim about the samples, which we already know are different.
A) The sample means and the population means.
B) The test statistic and the critical value.
C) The p-value and the significance level (α).
D) The sample size and the degrees of freedom.
Correct Answer: C
Content 6 explicitly states that a formal decision compares the p-value to alpha to reject or fail to reject the null hypothesis. This is the standard procedure for the p-value approach to hypothesis testing.
A) Assuming the methods are equally effective, there is a 7% chance of observing a difference in sample means as large as we did. We have convincing evidence the new method is better.
B) Assuming the methods are equally effective, there is a 7% chance of observing a difference in sample means as large as we did. We do not have convincing evidence the new method is better.
C) There is a 7% chance the methods are equally effective. We do not have convincing evidence the new method is better.
D) The new method is 1.50 points better on average than the standard method, but this is not enough evidence to make a conclusion.
Correct Answer: B
This question combines Content 2, 3, 5, and 6. The first sentence of option B is the correct interpretation of the p-value (Content 2 & 5). The second sentence is the correct conclusion because the p-value (0.07) is greater than alpha (0.05), so we fail to reject H₀ (Content 3 & 6). Option A draws the wrong conclusion. Option C misinterprets the p-value. Option D misinterprets the t-statistic.