Question

checkpoint question: suppose you draw an srs of size n from any population with mean μ and standard deviation σ. the central limit theorem (clt) says that when n is sufficiently large, ______________. the sampling distribution of (overline{x}) is approximately normal. the sampling distribution of (overline{x}) is symmetric but not normal. the sampling distribution of (hat{p}) is approximately normal. the sampling distribution of (hat{p}) is symmetric but not normal. the observations in the sample can be always viewed as independent.

Explanation:

The Central Limit Theorem (CLT) states that for a sufficiently large sample size \(n\), the sampling distribution of the sample mean \(\bar{x}\) (not the sample proportion \(\hat{p}\)) is approximately normal, regardless of the shape of the population distribution. The options about the sampling distribution of \(\hat{p}\) are incorrect as the CLT in this context (for mean) is about \(\bar{x}\). The option about symmetry but not normal is wrong because CLT gives an approximate normal distribution. The option about observations always being independent is not what CLT addresses in terms of distribution shape (CLT is about distribution of \(\bar{x}\) being normal for large \(n\), independence is a condition for sampling but not the main point here).

Answer:

the sampling distribution of \(\bar{x}\) is approximately Normal.