Question

a simple random sample of size ( n = 15 ) is obtained from a population with ( mu = 97 ) and ( sigma = 23 ).
pick the best statement from the options below that must be true regarding the sampling
distribution of the sampling mean of ( x ) in order to use the normal model.

the underlying distribution must be normal.
the underlying distribution only needs to be approximately normal.

Explanation:

The Central Limit Theorem (CLT) states that if the sample size \(n\) is large (\(n\geq30\)), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal regardless of the population distribution. Here, \(n = 15<30\). So, for the sampling distribution of the sample mean \(\bar{x}\) to be normal (so that we can use the normal model), the underlying population distribution must be normal. If the population is normal, then the sampling distribution of \(\bar{x}\) is normal for any sample size \(n\).

Answer:

The underlying distribution must be normal.