AP Statistics Practice Quiz: Justifying a Claim About the Slope of a Regression Model Based on a Confidence Interval
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
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A) We are 95% confident that the interval from 2.1 to 4.5 captures the true population slope.
B) There is a 95% probability that the true population slope is between 2.1 and 4.5.
C) 95% of the sample data points will have a slope between 2.1 and 4.5.
D) The slope of the sample regression line is 95% likely to be between 2.1 and 4.5.
Correct Answer: A
This option correctly states the confidence in the interval's ability to capture the true population parameter. A confidence interval interpretation should reference the population parameter (true slope) and the confidence level.
A) There is strong evidence of a positive linear relationship.
B) There is strong evidence of a negative linear relationship.
C) Since the interval contains 0, we cannot conclude there is a linear relationship because a slope of 0 is a plausible value.
D) The sample size was too small to make any conclusion.
Correct Answer: C
A confidence interval provides plausible values for the slope. Since 0 is included in the interval, it is a plausible value for the true population slope. A slope of 0 indicates no linear relationship, so we cannot justify the claim that a relationship exists.
A) The width of the interval would tend to increase.
B) The width of the interval would tend to decrease.
C) The width of the interval would remain unchanged.
D) The effect on the width cannot be determined without knowing the data.
Correct Answer: B
The content states that the width of the confidence interval for the slope tends to decrease as the sample size increases. A larger sample provides a more precise estimate.
A) There is a 99% probability that the true population slope is contained within our specific calculated interval.
B) In repeated sampling, approximately 99% of the confidence intervals constructed will capture the true population regression slope.
C) 99% of the data from the population will fall within the range of our confidence interval.
D) We are 99% confident that the sample slope is equal to the population slope.
Correct Answer: B
The confidence level (C%) refers to the long-run success rate of the method. In repeated sampling, we expect approximately C% of the intervals created from many different samples to capture the true population parameter.
A) No, because the interval is too wide to be certain.
B) No, because the interval does not contain 0.
C) Yes, because all plausible values for the slope in the interval are positive.
D) Yes, but only if the sample slope is also positive.
Correct Answer: C
The confidence interval provides a range of plausible values for the true population slope. Since all values in the interval (3.2, 7.8) are positive, 0 is not a plausible value. This supports the claim of a positive linear relationship.
A) For every one-year increase in a car's age, the price of a car from our sample decreased by an amount between $900 and $1500.
B) Based on this sample, we are 90% confident that the interval from -1500 to -900 captures the true population slope of the linear relationship between a car's age and its price.
C) In 90% of all possible samples, the calculated sample slope will be between -1500 and -900.
D) There is a 90% probability that the true population slope is between -1500 and -900.
Correct Answer: B
A correct interpretation must reference both the sample (as the basis for the calculation) and the population parameter (the true slope) it is estimating. It should also use the standard confidence interval phrasing.
A) It would likely be narrower.
B) It would likely be wider.
C) It would likely be the same width.
D) The change in width is unpredictable.
Correct Answer: B
The width of the confidence interval for the slope tends to decrease as sample size increases. Therefore, a smaller sample size (decreasing from 100 to 25) will tend to produce a wider, less precise confidence interval.
A) The exact value of the true population slope.
B) A set of plausible values that can be used to support a claim about the population slope.
C) The value of the slope calculated from the specific sample data.
D) A prediction for the response variable for every value of the explanatory variable.
Correct Answer: B
The primary purpose of a confidence interval is to provide a range of plausible or believable values for an unknown population parameter, in this case, the slope. This range can then be used to justify claims.
A) Yes, because the interval is not centered at 2.
B) Yes, because the interval is too wide to be useful.
C) No, because 2 is a plausible value for the slope contained within the interval.
D) No, because the interval does not contain 0.
Correct Answer: C
The confidence interval (1.5, 2.8) represents the range of plausible values for the true population slope. Since the manufacturer's claimed value of 2 is inside this interval, the data does not provide sufficient evidence to reject the claim; the claim is plausible.
A) The conclusion is correct; this is the definition of a confidence interval.
B) The true population slope is a fixed value; the 95% refers to the success rate of the method in repeated sampling, not a probability about the parameter.
C) The probability should be calculated based on the sample slope, not the interval.
D) The confidence level only applies to the sample, not the population.
Correct Answer: B
The true population slope is a fixed constant, not a random variable. Therefore, it does not make sense to assign a probability to it. The 95% confidence level refers to the long-run proportion of intervals constructed this way that would capture the true slope.
A) As sample size increases, the width of the confidence interval tends to increase.
B) As sample size increases, the width of the confidence interval tends to decrease.
C) Sample size has no effect on the width of the confidence interval.
D) The effect of sample size depends on the confidence level.
Correct Answer: B
The provided content explicitly states that the width of the confidence interval for the slope tends to decrease as sample size increases. This reflects the increased precision of the estimate from a larger sample.
A) There is no evidence of a linear relationship between the variables.
B) There is evidence of a positive linear relationship between the variables.
C) There is evidence of a negative linear relationship between the variables.
D) The true population slope is exactly -2.7, the midpoint of the interval.
Correct Answer: C
The interval provides a range of plausible values for the true population slope. Since all values in the interval (-4.1, -1.3) are negative and the interval does not contain 0, it provides strong evidence to support the claim of a negative linear relationship.
A) We are 95% confident that the interval from 10 to 30 captures the true population slope.
B) The interval (10, 30) provides a range of plausible values for the true population slope.
C) Based on our sample, we are 95% confident that the slope of our sample regression line is between 10 and 30.
D) If we were to take many samples, about 95% of the resulting confidence intervals would capture the true population slope.
Correct Answer: C
A confidence interval is used to estimate an unknown population parameter, not a sample statistic. The slope of the sample regression line is a single point estimate that is known exactly from the data; we do not need an interval to estimate it.
A) It proves that there is no linear relationship.
B) It indicates that 0 is a plausible value for the true population slope, so we cannot conclude a linear relationship exists.
C) It means the sample data was collected incorrectly.
D) It proves that there is a strong linear relationship.
Correct Answer: B
A slope of 0 means there is no linear relationship. If 0 is included in the confidence interval, it is considered a plausible value for the true population slope. Therefore, we do not have sufficient evidence to justify a claim that a linear relationship exists.
A) A less reliable estimate of the population slope.
B) A more precise estimate of the population slope.
C) A guarantee that the interval will capture the true slope.
D) A higher confidence level for the same interval width.
Correct Answer: B
Larger samples reduce the variability of the sampling distribution of the slope. This leads to a smaller standard error for the slope, which in turn results in a narrower, more precise confidence interval, as the estimate is more concentrated around the true population value.
A) All 100 of them.
B) Exactly 95 of them.
C) Approximately 95 of them.
D) It is impossible to say.
Correct Answer: C
The 95% confidence level describes the long-run capture rate of the method. In repeated sampling, we expect about 95% of the constructed intervals to contain the true population parameter. Due to random sampling variability, it is unlikely to be exactly 95, but it will be approximately 95.