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AP Statistics Practice Quiz: Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 14 questions to check your progress.

Question 1 of 14

A 95% confidence interval for the difference in the proportion of students who prefer online classes versus in-person classes (p_online - p_in-person) is (-0.08, 0.12). Based on this interval, what is the most appropriate conclusion?

All Questions (14)

A 95% confidence interval for the difference in the proportion of students who prefer online classes versus in-person classes (p_online - p_in-person) is (-0.08, 0.12). Based on this interval, what is the most appropriate conclusion?

A) There is a significant difference between the two proportions because the interval is wide.

B) The proportion of students who prefer online classes is significantly higher.

C) The proportion of students who prefer in-person classes is significantly higher.

D) Since zero is in the interval, there is not convincing evidence of a difference between the two population proportions.

Correct Answer: D

A confidence interval provides an interval of plausible values for the true difference in population proportions. Because the value 0 is included in the interval (-0.08, 0.12), it is a plausible value for the difference. Therefore, we cannot conclude there is a significant difference between the proportion of all students who prefer online classes and the proportion of all students who prefer in-person classes.

A researcher constructed a 90% confidence interval for the difference in proportions of adults who use social media platform A versus platform B (p_A - p_B). The resulting interval is (0.02, 0.09). Which claim is justified by this interval?

A) There is no difference in the proportion of adults who use platform A and platform B.

B) A greater proportion of adults use platform A than platform B.

C) A greater proportion of adults use platform B than platform A.

D) The difference in the sample proportions is not statistically significant.

Correct Answer: B

The confidence interval provides an interval of plausible values for the true difference in population proportions. Since the entire interval (0.02, 0.09) is above zero, all plausible values for the difference (p_A - p_B) are positive. This provides convincing evidence that the proportion of all adults who use platform A is greater than the proportion of all adults who use platform B.

Which of the following is a correct interpretation of a 99% confidence interval of (-0.15, -0.03) for the difference in the proportion of residents in City X who own a pet and the proportion of residents in City Y who own a pet (p_X - p_Y)?

A) We are 99% confident that the difference in the sample proportions of pet owners from City X and City Y is between -0.15 and -0.03.

B) There is a 99% probability that the true difference in the population proportions of pet owners from City X and City Y is between -0.15 and -0.03.

C) We are 99% confident that the true difference in the population proportions of pet owners (City X - City Y) is between -0.15 and -0.03.

D) In repeated sampling, 99% of the sample differences will fall between -0.15 and -0.03.

Correct Answer: C

A correct interpretation of a confidence interval should reference the confidence level, the parameter being estimated (the true difference in population proportions), and the interval of plausible values. It must reference the populations, not the samples from which the interval was calculated.

A political analyst wants to compare the proportion of voters in District 1 who support a policy with the proportion in District 2. A 95% confidence interval for the difference (p_1 - p_2) is calculated. Which statement best describes the meaning of the 95% confidence level?

A) There is a 95% probability that the true difference in proportions of voters who support the policy is captured in the calculated interval.

B) If this process of sampling and constructing intervals were repeated many times, approximately 95% of the resulting intervals would capture the true difference in population proportions.

C) 95% of the voters in the two districts have a difference in opinion that falls within the calculated interval.

D) The calculated interval contains 95% of all possible sample differences.

Correct Answer: B

The confidence level refers to the long-run success rate of the method. In repeated sampling, we expect that approximately C% (in this case, 95%) of the confidence intervals created from those samples will successfully capture the true difference in population proportions. It is not a probability statement about a single, already-calculated interval.

A 95% confidence interval for the difference in population proportions (p_1 - p_2) is (-0.05, 0.09). Based on this interval, which of the following is a plausible value for the true difference in proportions?

A) 0.10

B) 0.00

C) -0.06

D) 0.12

Correct Answer: B

A confidence interval provides an interval of plausible values for the parameter. Any value within the interval [-0.05, 0.09] is considered a plausible value. Of the options provided, only 0.00 falls within this range.

A pharmaceutical company tests a new drug. They calculate a 98% confidence interval for the difference in the proportion of patients experiencing relief with the new drug versus a placebo (p_new - p_placebo). The interval is (0.04, 0.11). Does this interval support the claim that the new drug is more effective than the placebo?

A) No, because the interval does not contain zero, we cannot make a conclusion.

B) No, because the difference could be as small as 0.04, which is not a large effect.

C) Yes, because all plausible values for the difference in proportions are positive, suggesting the new drug has a higher relief rate.

D) Yes, because the sample proportion for the new drug was higher than for the placebo.

Correct Answer: C

To justify a claim, we examine the interval of plausible values. Since the entire interval (0.04, 0.11) is positive, zero is not a plausible value for the difference. This provides strong evidence that the true proportion of patients experiencing relief is higher for the new drug than for the placebo, thus supporting the claim.

A sociologist claims there is no difference in the proportion of urban residents who volunteer and the proportion of suburban residents who volunteer. A 95% confidence interval for the difference (p_urban - p_suburban) is found to be (-0.09, -0.01). What does this interval suggest about the sociologist's claim?

A) The claim is supported because the interval is close to zero.

B) The claim is not supported because the interval does not contain zero.

C) The claim is supported because the sample difference was small.

D) The claim cannot be evaluated without knowing the sample proportions.

Correct Answer: B

The sociologist's claim is that the true difference is zero. The confidence interval provides an interval of plausible values for this difference. Since zero is not contained in the interval (-0.09, -0.01), it is not a plausible value. Therefore, the data does not support the claim of no difference.

Two different manufacturing processes, A and B, are used to create a product. A 99% confidence interval for the difference in the proportion of defective items (p_A - p_B) is (-0.04, 0.05). Which statement provides the best interpretation and conclusion?

A) We are 99% confident that the true difference in the proportion of defective items is between -0.04 and 0.05. Since 0 is a plausible value, there is no convincing evidence of a difference in the defect rates of the two processes.

B) There is a 99% chance the true difference in defect rates is between -0.04 and 0.05. Therefore, process A is better.

C) We are 99% confident that the difference in the sample proportions of defective items is between -0.04 and 0.05. We cannot conclude that one process is better than the other.

D) In 99% of samples, the difference in defect rates will be between -0.04 and 0.05. Since the interval is mostly positive, process B is likely better.

Correct Answer: A

This option correctly interprets the confidence interval by referencing the confidence level and the true difference in population proportions. It also correctly justifies the claim by noting that since zero is a plausible value within the interval, there is not sufficient evidence to conclude a difference exists between the two processes.

When justifying a claim about the difference between two population proportions, why is the value of zero so important in the confidence interval?

A) If zero is in the interval, it means the sample sizes were too small.

B) If zero is in the interval, it means the two sample proportions were exactly equal.

C) Zero represents the point of maximum uncertainty in the interval.

D) Zero represents a difference of none, so if it is a plausible value, we cannot claim a true difference exists.

Correct Answer: D

A confidence interval provides a range of plausible values for the true difference in population proportions. A difference of zero means the two population proportions are equal. If zero is contained within the interval, it is considered a plausible outcome, and therefore we cannot justify a claim that a difference exists between the two populations.

An interpretation of a confidence interval for a difference in proportions must reference both the samples and the populations. Which part of the interpretation 'We are 95% confident that the interval from 0.03 to 0.11 captures the true difference in the proportion of all teenagers who use App A and all teenagers who use App B' correctly references the populations?

A) We are 95% confident

B) the interval from 0.03 to 0.11

C) the proportion of all teenagers who use App A and all teenagers who use App B

D) This interpretation does not reference the samples.

Correct Answer: C

The phrase 'the proportion of all teenagers...' correctly specifies that the interval is an estimate for a parameter of the populations, not a statistic from the samples. A complete interpretation should be based on data from samples to make an inference about populations.

The statement 'A confidence interval for a difference in proportions provides an interval of plausible values to support a claim' is a foundational concept. How does this concept of 'plausible values' allow statisticians to make a judgment about a claim?

A) It provides a list of all possible values for the sample statistic, allowing a direct comparison to the claim.

B) It determines the probability that a claim is true.

C) It creates a range of believable values for the population parameter; if the claimed value falls outside this range, the claim is not supported by the data.

D) It guarantees that the true population parameter is within the interval, making any claim outside the interval definitively false.

Correct Answer: C

The confidence interval gives a range of values for the true difference in population proportions that are consistent with the observed sample data. If a claim posits a specific value for this difference (e.g., zero for a claim of 'no difference'), we check if that value is within our interval of plausible values. If it is not, we have evidence to reject the claim.

A 90% confidence interval for the difference in the proportion of satisfied customers for Company A and Company B (p_A - p_B) is (-0.20, -0.05). Based on this interval, which claim is supported?

A) There is no significant difference in customer satisfaction between the two companies.

B) Company A has a significantly higher proportion of satisfied customers than Company B.

C) Company B has a significantly higher proportion of satisfied customers than Company A.

D) The difference in the sample proportions was exactly -0.125.

Correct Answer: C

The interval of plausible values for the difference p_A - p_B is entirely negative. This means that p_A - p_B < 0, which implies p_A < p_B. Therefore, the evidence supports the claim that Company B has a higher proportion of satisfied customers than Company A.

A student interprets a 95% confidence level by saying, 'This means that if we took 100 random samples from the same populations and calculated 100 intervals, exactly 95 of them would capture the true difference in population proportions.' What is the primary flaw in this student's reasoning?

A) The confidence level applies to the population data, not to sampling.

B) The number of intervals that capture the true difference is approximately 95, not exactly 95.

C) The confidence level should be interpreted as a 95% probability that the true difference is in our one interval.

D) The student should have specified that the samples must be of the same size.

Correct Answer: B

The confidence level describes the long-run relative frequency of the method producing an interval that captures the true parameter. In repeated sampling, we expect *approximately* C% of the intervals to be successful. It is a probabilistic concept, not an exact count for any finite set of 100 samples.

A study on remote work found that a 95% confidence interval for the difference in the proportion of employees reporting high job satisfaction (p_remote - p_office) was (0.01, 0.15). Which of the following claims is NOT supported by this interval?

A) There is evidence of a difference in job satisfaction between the two groups.

B) The proportion of remote employees with high job satisfaction is greater than that of office employees.

C) A difference of 0 is not a plausible value.

D) The true difference in the proportion of employees with high job satisfaction is 0.08.

Correct Answer: D

A confidence interval provides a range of plausible values, not a single point estimate for the true difference. While 0.08 (the midpoint) is a plausible value and our point estimate, the interval does not claim it is the true difference. The interval supports claims A, B, and C because it does not contain zero and is entirely positive.