Question

a simple random sample of size ( n = 42 ) is obtained from a population with ( mu = 64 ) and ( sigma = 17 ).
(a) what must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample mean? assuming that this condition is true, describe the sampling distribution of ( \bar{x} ).
(b) assuming the normal model can be used, determine ( p(\bar{x}<68.2) ).
(c) assuming the normal model can be used, determine ( p(\bar{x} geq 65.7) ).
(a) what must be true regarding the distribution of the population?
a. the population must be normally distributed and the sample size must be large.
b. since the sample size is large enough, the population distribution does not need to be normal.
c. the sampling distribution must be assumed to be normal.
d. the population must be normally distributed.
assuming the normal model can be used, describe the sampling distribution ( \bar{x} ). choose the correct answer below.
a. approximately normal, with ( mu_{\bar{x}} = 64 ) and ( sigma_{\bar{x}}=\frac{17}{sqrt{42}} )
b. approximately normal, with ( mu_{\bar{x}} = 64 ) and ( sigma_{\bar{x}} = 17 )
c. approximately normal, with ( mu_{\bar{x}} = 64 ) and ( sigma_{\bar{x}}=\frac{42}{sqrt{17}} )

Explanation:

• For part (a), according to the Central Limit Theorem, if the sample size \(n\geq30\), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal regardless of the population distribution. Here \(n = 42\geq30\).
• For part (b), the mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}=\mu\) and the standard deviation \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Given \(\mu = 64\), \(\sigma=17\) and \(n = 42\).

Answer:

• (a) B. Since the sample size is large enough, the population distribution does not need to be normal.
• (b) A. Approximately normal, with \(\mu_{\bar{x}} = 64\) and \(\sigma_{\bar{x}}=\frac{17}{\sqrt{42}}\)