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 ( overline{x} ).
(b) assuming the normal model can be used, determine ( p(overline{x}<68.2) ).
(c) assuming the normal model can be used, determine ( p(overline{x}geq65.7) ).

(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.

Explanation:

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 shape of the population distribution. Here, \(n = 42\geq30\).

Answer:

B. Since the sample size is large enough, the population distribution does not need to be normal.