AP Statistics Practice Quiz: Carrying Out a Test for the Slope of a Regression Model
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) A t-distribution with 23 degrees of freedom.
B) A t-distribution with 24 degrees of freedom.
C) A Normal distribution.
D) A chi-square distribution with 23 degrees of freedom.
Correct Answer: A
The sampling distribution for the slope of a regression model is a t-distribution with df = n-2 degrees of freedom. In this case, n=25, so the degrees of freedom are 25 - 2 = 23.
A) 0.10
B) 0.40
C) 2.50
D) 5.00
Correct Answer: C
An appropriate test statistic for the slope is calculated as t = (b - beta) / SEb. In this case, t = (0.5 - 0) / 0.2 = 2.50.
A) There is a 3% chance that the true slope of the population regression line is 0.
B) Assuming the true slope of the population regression line is 0, there is a 3% chance of observing a sample slope as extreme or more extreme than the one obtained.
C) There is a 3% chance that the alternative hypothesis is true.
D) The probability of making a Type I error is 0.03.
Correct Answer: B
Interpreting the p-value for a test for slope assumes the null hypothesis is true. The p-value is the probability of observing a test statistic (and corresponding sample slope) as extreme or more extreme than the one calculated, given that the null hypothesis (beta = 0) is true.
A) Reject the null hypothesis because the p-value is greater than alpha.
B) Reject the null hypothesis because the p-value is not zero.
C) Fail to reject the null hypothesis because the p-value is greater than alpha.
D) Fail to reject the null hypothesis because the p-value is less than 0.10.
Correct Answer: C
A formal decision compares the p-value to alpha. If the p-value is greater than alpha, we fail to reject the null hypothesis. Here, 0.085 > 0.05, so we fail to reject H0.
A) There is convincing evidence of a linear relationship between study hours and exam score.
B) There is convincing evidence that there is no linear relationship between study hours and exam score.
C) There is not convincing evidence of a linear relationship between study hours and exam score.
D) Study hours have no effect on exam scores.
Correct Answer: C
Failing to reject the null hypothesis means we lack sufficient evidence to support the alternative hypothesis. This allows us to justify a claim that there is not convincing evidence of a linear relationship. We cannot prove the null hypothesis is true.
A) n - 1, where n is the sample size.
B) n - 2, where n is the sample size.
C) The smaller of (n1 - 1) or (n2 - 1).
D) n, where n is the sample size.
Correct Answer: B
The sampling distribution of t = (b - beta) / SEb has a t-distribution with df = n - 2, where n is the number of data points in the sample.
A) The slope of the sample regression line.
B) The standard error of the sample slope.
C) The hypothesized value of the population slope from the null hypothesis.
D) The y-intercept of the population regression line.
Correct Answer: C
The term 'beta' in the test statistic formula represents the true population slope. For a significance test, its value is taken from the null hypothesis, which is typically H0: beta = 0.
A) It suggests that the sample data is not representative of the population.
B) It provides strong evidence to support the claim of a linear relationship between education and income in the population.
C) It proves that more education causes a higher income.
D) It indicates that there is no linear relationship between the variables.
Correct Answer: B
The results of a significance test for slope provide statistical reasoning for a research question. A small p-value leads to rejecting the null hypothesis of no relationship, thus providing evidence for a linear relationship in the population.
A) The binomial distribution.
B) The chi-square distribution.
C) The F-distribution.
D) The t-distribution.
Correct Answer: D
The null distribution for the slope of a regression model is a t-distribution. This distribution is used to determine the probability (p-value) of observing the calculated test statistic if the null hypothesis were true.
A) We have strong evidence that the price of a car is related to its age.
B) We have strong evidence that the price of a car is not related to its age.
C) We do not have convincing evidence of a linear relationship between the price of a car and its age.
D) We can be 95% confident that the true slope is 0.34.
Correct Answer: C
Since the p-value (0.34) is greater than alpha (0.05), we fail to reject the null hypothesis. This allows us to justify the claim that we lack convincing statistical evidence of a linear relationship between the two variables.
A) The alternative hypothesis is true.
B) The sample data is perfectly linear.
C) The null hypothesis is true.
D) The sample size is large.
Correct Answer: C
Interpreting the p-value for a test for slope assumes the null hypothesis is true. The p-value is the probability of obtaining a sample result as or more extreme than the one observed, under the condition that there is no linear relationship in the population (H0: beta = 0).
A) t = (1.5 - 0) / 0.4 = 3.75
B) t = 1.5 / (0.4 / sqrt(30))
C) z = (1.5 - 0) / 0.4 = 3.75
D) t = (1.5) / (0.4 * sqrt(28))
Correct Answer: A
To calculate an appropriate test statistic for the slope, use the formula t = (b - beta) / SEb. The sample slope b is 1.5, the standard error SEb is 0.4, and the null hypothesis value for beta is 0. The correct distribution is a t-distribution, not a z-distribution.
A) Because the p-value of 0.002 is greater than 0, we fail to reject H0. There is no evidence of a linear relationship.
B) Because the p-value of 0.002 is less than the alpha of 0.01, we reject H0. This provides strong statistical evidence of a linear relationship between hours of sleep and reaction time in the population.
C) Because the sample slope was not zero, we can conclude there is a linear relationship in the population.
D) Because the p-value of 0.002 is very small, it proves that lack of sleep causes slower reaction times.
Correct Answer: B
To justify a claim, we make a formal decision by comparing the p-value to alpha. Since 0.002 < 0.01, we reject the null hypothesis. This result provides the statistical reasoning to claim there is evidence of a linear relationship in the population. The test does not prove causation.
A) The sample slope (b) and the population slope (beta).
B) The p-value and the significance level (alpha).
C) The test statistic (t) and the sample size (n).
D) The p-value and the sample slope (b).
Correct Answer: B
A formal decision to reject or fail to reject the null hypothesis is made by comparing the calculated p-value to the pre-determined significance level, alpha.
A) Fail to reject H0 because the test statistic is negative.
B) Fail to reject H0 because the degrees of freedom (10) are too small.
C) Reject H0 because the absolute value of the test statistic |-2.50| is greater than the critical value 2.23.
D) Reject H0 because the sample size of 12 is greater than the test statistic of -2.50.
Correct Answer: C
The degrees of freedom are n-2 = 12-2 = 10. A formal decision can be made by comparing the test statistic to the critical value. Since the absolute value of our test statistic (2.50) is greater than the critical value (2.23), the result is statistically significant. This is equivalent to the p-value being less than alpha, leading us to reject the null hypothesis and justify a claim of a linear relationship.
A) The linear relationship observed in the sample data is strong.
B) The linear relationship observed in the sample data is not due to random chance alone and likely exists in the broader population.
C) The data points in the population form a perfect straight line.
D) The sample slope is exactly equal to the population slope.
Correct Answer: B
The purpose of a significance test is to justify a claim about the population based on sample data. Rejecting the null hypothesis provides statistical reasoning that the observed relationship is unlikely to be due to sampling variability and that a linear relationship exists in the population.