AP Statistics 3.4: Proportion CI Claims
Use a proportion confidence interval to judge claims and explain how confidence level, sample size, and margin of error change its width.
- Syllabus
- Effective Fall 2025
- Course
- AP Statistics
Covered syllabus points
3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion question 1
[Maximum number: 1]
In a random sample of 500 American adults, 210 expressed the belief that journalists have "low" ethical standards. A 95\% confidence interval for the proportion of all American adults who believe that journalists have "low" ethical standards is given by (0.3767, 0.4633). Which of the following is a correct interpretation of the confidence level of 95\%?
A
There is a 0.95 probability that the true proportion of all American adults who believe that journalists have "low" ethical standards is between 0.3767 and 0.4633.
B
We are 95\% confident that the true proportion of all American adults who believe that journalists have "low" ethical standards is between 0.3767 and 0.4633 .
C
If the proportion of American adults who believe that journalists have "low" ethical standards is calculated for repeated random samples of 500 adults, 95\% of these sample proportions will fall between 0.3767 and 0.4633 .
D
If a confidence interval for the proportion of American adults who believe that journalists have "low" ethical standards is calculated for repeated random samples of 500 American adults, approximately 95\% of these intervals will contain the true population proportion.
D
3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion question 2
A survey conducted by a national research center asked a random sample of 920 teenagers in the United States how often they use a video streaming service. From the sample, 59% answered that they use a video streaming service every day.
Based on the confidence interval in part (a), do the sample data provide convincing statistical evidence that the proportion of all teenagers in the United States who would respond that they use a video streaming service every day is not 0.5 ? Justify your answer.
The 95% confidence interval of (0.558, 0.622) indicates that any value between 0.558 and 0.622 is a plausible value for the proportion of all teenagers in the United States who use video streaming every day. Because the value 0.5 is not contained in the interval, the sample data provide convincing statistical evidence that the proportion of all teenagers in the United States who would use a streaming service every day is not 0.5.
Scoring:
Essentially correct (E) if the response provides both:
1. A correct conclusion that there is convincing evidence, consistent with the interval calculated in part (a).
2. Correct justification based on whether 0.5 is contained in the interval calculated in part (a).
Partially correct (P) if the response satisfies only one of the two components.
Incorrect (I) if the response does not meet the criteria for E or P.
3.4 Justifying a Claim Based on a Confidence Interval for a Population Proportion question 3
[Maximum number: 4]
To increase business, the owner of a restaurant is running a promotion in which a customer's bill can be randomly selected to receive a discount. When a customer's bill is printed, a program in the cash register randomly determines whether the customer will receive a discount on the bill. The program was written to generate a discount with a probability of 0.2, that is, giving 20 percent of the bills a discount in the long run. However, the owner is concerned that the program has a mistake that results in the program not generating the intended long-run proportion of 0.2.
The owner selected a random sample of bills and found that only 15 percent of them received discounts. A confidence interval for p, the proportion of bills that will receive a discount in the long run, is 0.15±0.06. All conditions for inference were met.
Question (a)
(a)
Consider the confidence interval 0.15±0.06.
Question (i)
(i)
Does the confidence interval provide convincing statistical evidence that the program is not working as intended? Justify your answer.
No. The confidence interval is (0.09, 0.21), which includes the value of 0.20. Therefore, it is plausible that the computer program is generating discounts with a probability of 0.20, and the confidence interval does not provide convincing statistical evidence that the program is not working as intended.
Question (ii)
(ii)
Does the confidence interval provide convincing statistical evidence that the program generates the discount with a probability of 0.2 ? Justify your answer.
A second random sample of bills was taken that was four times the size of the original sample. In the second sample 15 percent of the bills received the discount.
No. The confidence interval includes values from 0.09 to 0.21, so any value in that interval is a plausible value for the probability that the computer is using to generate discounts.
Question (b)
(b)
Determine the value of the margin of error based on the second sample of bills that would be used to compute an interval for p with the same confidence level as that of the original interval.
Part (b):
The formula for computing the margin of error for a proportion includes the square root of the sample size in the denominator. For a random sample that is four times the size of the original sample, the margin of error can be determined by dividing the margin of error of the original sample by two. Therefore, the new margin of error is 0.03.
Question (c)
(c)
Based on the margin of error in part (b) that was obtained from the second sample, what do you conclude about whether the program is working as intended? Justify your answer.
Part (c):
Using the margin of error of 0.03 obtained from the second sample, the confidence interval for p is 0.15 ± 0.03 or (0.12, 0.18). The interval does not include 0.20, and therefore, there is convincing evidence that the computer program is not working as intended and is not generating discounts with a probability of 0.20.
All question bank results loaded