Question

In a large high school, 18% of sophomores and 67% of seniors have part-time jobs. Suppose random samples of 32 sophomores and 31 seniors from this high school are asked if they have a part-time job. Let p₁ be the sample proportion of sophomores and p2 be the sample proportion of seniors who have part-time jobs. Which of the following is the correct shape and justification of the sampling distribution of p₁ - p2 ?

a) Bimodal because one population proportion is centered at 0.18, while the other is centered at 0.67.
b) Approximately normal because the expected numbers of successes and failures for each sample are all at least 10.
c) Not approximately normal because the expected numbers of successes and failures for each sample are all at least 10.
d) Not approximately normal because the expected numbers of successes and failures for the sophomores group are not both at least 10.

Answer 1

The shape and justification of the sampling distribution of p₁ - p2 is approximately normal because of the central limit theorem.

Step-by-step explanation:

The correct shape and justification of the sampling distribution of p₁ - p2 is b) Approximately normal because the expected numbers of successes and failures for each sample are all at least 10.

When we have random samples with enough successes and failures within each sample, the sampling distribution of the difference in sample proportions will be approximately normal. This is because the central limit theorem states that as the sample size increases, the distribution of the sample proportions approaches a normal distribution.