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AP Statistics 7.2 Constructing a Confidence Interval for a Population Mean - MCQs - Exam Style Questions

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Question

A program exists to encourage more middle school students to major in math and science when they go to college. The organizers of the program want to estimate the proportion of students who, after completing the program, go on to major in math or science in college. The organizers will select a sample of students from a list of all students who completed the program. Which of the following sampling methods describes a stratified random sample?
(A) Select all female students on the list.
(B) Randomly select 50 students on the list.
(C) Randomize the names on the list and then select every tenth student on the randomized list.
(D) Randomly select 25 names from the female students on the list and randomly select 25 names from the male students on the list.
(E) Randomly select 50 students on the list who are attending college.
▶️ Answer/Explanation
Detailed solution

1. Definition of Stratified Random Sampling:
This method involves two steps:
– First, divide the population into distinct, non-overlapping groups (strata).
– Second, take a simple random sample from each group.

2. Analysis of the Options:
– (A) is a census of one group, not a sample of the whole population.
– (B) is a simple random sample.
– (C) is a systematic random sample.
– (D) correctly follows the two steps. The strata are ‘female students’ and ‘male students’. A random sample is then drawn from each stratum.
– (E) is a biased sample, as it excludes students not attending college.

Therefore, (D) is the only option that describes a stratified random sample.
✅ Answer: (D)

Question

A medical center conducted a study to investigate cholesterol levels in people who have had heart attacks. A random sample of 16 people was obtained from the names of all patients of the medical center who had a heart attack in the previous year. Of the people in the sample, the mean cholesterol level was 264.70 milligrams per deciliter (mg/dL) with standard deviation 42.12 mg/dL. Assuming all conditions for inference were met, which of the following is a 90 percent confidence interval for the mean cholesterol level, in mg/dL, of all patients of the medical center who had a heart attack in the previous year?
(A) (242.26, 287.14)
(B) (244.06, 285.34)
(C) (246.24, 283.16)
(D) (247.38, 282.02)
(E) (260.09, 269.31)
▶️ Answer/Explanation
Detailed solution

1. Identify the Correct Interval Type:
Since the population standard deviation is unknown, we use a t-interval for a mean.
The formula is: $\bar{x} \pm t^* \frac{s}{\sqrt{n}}$.

2. List the Given Values:
– Sample mean $\bar{x} = 264.70$
– Sample standard deviation $s = 42.12$
– Sample size $n = 16$
– Confidence level = 90%

3. Find the Critical Value ($t^*$):
– Degrees of freedom ($df$) = $n-1 = 16-1 = 15$.
– For a 90% confidence interval with $df=15$, the critical value is $t^* = 1.753$.

4. Calculate the Confidence Interval:
– Margin of Error (ME) = $1.753 \times \frac{42.12}{\sqrt{16}} = 1.753 \times \frac{42.12}{4} \approx 18.46$
– Lower Bound = $264.70 – 18.46 = 246.24$
– Upper Bound = $264.70 + 18.46 = 283.16$

The interval is (246.24, 283.16).
✅ Answer: (C)

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