Question

a simple random sample of size ( n = 64 ) is obtained from a population that is skewed right with ( mu = 88 ) and ( sigma = 32 ). (a) describe the sampling distribution of ( overline{x} ). (b) what is ( p(overline{x}>95.4) )? (c) what is ( p(overline{x}leq78.6) )? (d) what is ( p(86<overline{x}<97.8) )? (a) choose the correct description of the shape of the sampling distribution of ( overline{x} ). ( \bigcirc ) a. the distribution is approximately normal. ( \bigcirc ) b. the distribution is uniform. ( \bigcirc ) c. the distribution is skewed left. ( \bigcirc ) d. the distribution is skewed right. ( \bigcirc ) e. the shape of the distribution is unknown.

Explanation:

Step1: Apply Central Limit Theorem

According to the Central Limit Theorem, for a sample size \(n = 64\) (\(n\geq30\)), the sampling distribution of \(\bar{x}\) is approximately normal. The mean of the sampling distribution \(\mu_{\bar{x}}=\mu = 88\), and the standard deviation \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}=\frac{32}{\sqrt{64}} = 4\).

Answer:

A. The distribution is approximately normal