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AP Statistics 3.7 Carrying Out a Test for a Population Proportion- Exam Style Questions - MCQs - New Syllabus

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Question

It is claimed that more than 65% of parents begin reading to their child aloud when the child is an infant. A random sample of 256 parents in a school district were selected, and 180 began reading to their child aloud when their child was an infant. A z-test was conducted to test the claim and the p-value was 0.04. Which of the following is an appropriate interpretation of the p-value?

(A) 0.04 is the probability that 180 out of 256 randomly selected parents began reading to their child aloud when the child was an infant if the population proportion actually is 0.65.
(B) 0.04 is the probability that 180 or more out of 256 randomly selected parents began reading to their child aloud when the child was an infant if the population proportion actually is 0.65.
(C) 0.04 is the probability that 70% or more parents began reading to their child aloud when the child was an infant if the sample proportion actually was 0.65.
(D) 0.04 is the probability that 70% of parents began reading to their child aloud when the child was an infant if the population proportion actually is more than 0.65.
(E) 0.04 is the probability that 65% or more randomly selected parents begin reading to their child aloud when the child was an infant if the population proportion actually is 0.70.

▶️ Answer/Explanation

A p-value is the probability of obtaining a sample result at least as extreme as the one observed, assuming the null hypothesis is true.

For this test, the null hypothesis is:

\(H_0:p=0.65\)

The observed sample proportion is:

\(\hat{p}=\frac{180}{256}\approx0.703\)

Therefore, the p-value of 0.04 represents the probability of obtaining a sample result this large or larger (180 or more parents out of 256, or equivalently a sample proportion of about 0.703 or greater) if the true population proportion is actually 0.65.

This matches the interpretation given in choice (B).

✅ Answer: (B)

Question

Last January, the State of Iowa organized an optional Day of Service. At a large high school in Jasper County, Iowa, only 12% of the students participated in the Day of Service activities at the school. As part of the check-in process at the high school, AP Statistics students gave each participant a blank piece of 8.5″×11″ paper and asked the student to fold the paper in half. Eighteen of the 80 participants folded the paper in half lengthwise (18/80 = 0.225).

Students wish to test the hypotheses \(H_0:p=0.25\) vs. \(H_A:p<0.25\), where \(p\) is the proportion of all students at this high school who would have folded the piece of paper in half lengthwise. Which of the following statements is correct?

(A) A one-sample z-test is appropriate since all conditions are met.
(B) The independence condition is not satisfied, so a one-sample z-test is not appropriate.
(C) There are too few successes and failures, so a one-sample z-test is not appropriate.
(D) The sample size is not large enough, so a one-sample z-test is not appropriate.
(E) The significance level is not specified, so a one-sample z-test is not appropriate.

▶️ Answer/Explanation

For a one-sample z-test for a population proportion, one important condition is the independence condition. The sample should be randomly selected, or the sample size should be less than 10% of the population when sampling without replacement.

In this study, the participants were volunteers in an optional Day of Service. Only 12% of students participated, meaning the sample was not randomly selected from the student population. Because the participants are a self-selected group, the responses may not be representative of all students at the school.

The success-failure condition is satisfied since:

\(np_0 = 80(0.25)=20\)

\(n(1-p_0)=80(0.75)=60\)

Both values are at least 10. Therefore, the main issue is the lack of independence/random selection, making a one-sample z-test inappropriate.

✅ Answer: (B)

Question 

530 infants who were at high risk for peanut allergies were randomly divided into two groups: the peanut avoidance group and the peanut consumption group. Caregivers of infants in the avoidance group were told to avoid feeding their children peanut protein until they reached the age of 8 months. Caregivers of infants in the consumption group were told to feed their children at least 6 g of peanut protein per week.

At the end of the study, when the children were five years old, the participants were tested to see if they had developed a peanut allergy. A hypothesis test was performed to determine if there was a difference in the proportion of peanut allergies between the two groups; the researchers reported a p-value < 0.001. Which of the following is the best interpretation of the small p-value?

(A) The probability of developing a peanut allergy is very low regardless of whether children avoid peanut protein or consume peanut protein regularly.
(B) The probability that peanut avoidance/consumption during childhood is associated with a difference in the proportion who develop peanut allergies is very small.
(C) It is likely that peanut avoidance/consumption during childhood is associated with a difference in the proportion who develop peanut allergies, but the size of the effect is very small.
(D) It would be unlikely to get a difference in proportions as or more extreme than observed in this study if peanut avoidance/consumption really had no effect on the proportion who develop peanut allergies.
(E) It would be unlikely to get a difference in proportions as or more extreme than observed in this study if peanut avoidance/consumption really had a substantial effect on the proportion who develop peanut allergies.

▶️ Answer/Explanation

A p-value measures the probability of obtaining results at least as extreme as those observed, assuming the null hypothesis is true.

In this study, the null hypothesis states that there is no difference in the proportion of peanut allergies between the avoidance and consumption groups. A p-value less than 0.001 indicates that the observed difference would be extremely unlikely if there truly were no effect.

Therefore, the correct interpretation is that obtaining a difference in sample proportions this large (or larger) would be very unlikely if peanut avoidance and consumption had no effect on allergy development.

Notice that a p-value does not measure the probability that a hypothesis is true, nor does it describe the size of the effect.

✅ Answer: (D)

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