Question

a simple random sample of size ( n = 20 ) is obtained from a population with ( mu = 154 ) and ( sigma = 47 ).
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:

When using the normal model for the sampling distribution of the sample mean \(\bar{x}\), if the sample size \(n\) is small (\(n = 20<30\) in this case), the underlying population distribution must be normal. The Central Limit Theorem (which allows us to use the normal model for the sampling distribution of \(\bar{x}\) when \(n\geq30\) regardless of the population distribution) does not apply for small sample sizes. So, for \(n = 20\), the underlying distribution must be normal.

Answer:

The underlying distribution must be normal.