Question

a simple random sample of size ( n = 15 ) is obtained from a population with ( mu = 62 ) and ( sigma = 19 ).
(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} < 65.7) ).
(c) assuming the normal model can be used, determine ( p(overline{x} geq 63.3) ).
(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 population must be normally distributed.
d. the sampling distribution must be assumed to be normal.

Explanation:

Step1: Central Limit Theorem

The Central Limit Theorem (CLT) states that if the sample size \(n\) is large (\(n\geq30\) is a common rule - of - thumb, but for some populations with less skewness, smaller \(n\) can also work). When using the normal model for the sampling distribution of the sample mean \(\bar{x}\), if the population is not known to be normal, we need to consider the sample size.
Here, \(n = 15\) which is not a large sample (\(n<30\)). So, if we want to use the normal model for the sampling distribution of \(\bar{x}\), the population must be normally distributed.

Answer:

C. The population must be normally distributed.