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 p hat Subscript under and p hat Subscript upper be the sample proportions of sophomores and seniors, respectively, who have part-time jobs. Which of the following is the correct shape and justification of the sampling distribution of P hat Subscript under Baseline minus p hat Subscript upper ? bimodal because one population proportion is centered at 0.18, while the other is centered at 0.67 approximately Normal because the expected numbers of successes and failures for each sample are all at least 10 not approximately Normal because the expected numbers of successes and failures for each sample are all at least 10 not approximately Normal because the expected numbers of successes and failures for the sophomores group are not both at least 10

Answer 1

Using the Central Limit Theorem, it is found that the correct statement is given by:

bimodal because one population proportion is centered at 0.18, while the other is centered at 0.67.

What does the Central Limit Theorem state?

It states that for a proportion p in a sample of size n, the sampling distribution of sample proportion is approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1 - p)}{n}}$, as long as $np \geq 10$ and $n(1 - p) \geq 10$

In this problem, for sophomores, we have that:

np = 32 x 0.18 = 5.76 < 10.

Hence the distribution cannot be normal, which means that the correct statement is:

bimodal because one population proportion is centered at 0.18, while the other is centered at 0.67.

More can be learned about the Central Limit Theorem at https://brainly.com/question/24663213

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