Question

which of the following statements about the sampling distribution of the sample mean, x - bar, is not true?
a) the distribution is normal regardless of the shape of the population distribution, as long as the sample
size, n, is large enough.
b) the distribution is normal regardless of the sample size, as long as the population distribution is normal.
c) the distributions mean is the same as the population mean.
d) the distributions standard deviation is smaller than the population standard deviation.
e) none of the above are not true; all of the above statements are true.

Explanation:

• Option A: The sampling distribution of the sample mean \(\bar{x}\) is normal only if the population is normal or the sample size \(n\) is large enough (Central Limit Theorem). It's not normal regardless of the population distribution shape. So this statement is false.
• Option B: By the Central Limit Theorem, if the population is normal, the sampling distribution of \(\bar{x}\) is normal for any sample size. If the population is non - normal, when \(n\) is large enough (usually \(n\geq30\)), the sampling distribution of \(\bar{x}\) is approximately normal. So this statement is true.
• Option C: The mean of the sampling distribution of \(\bar{x}\), \(\mu_{\bar{x}}=\mu\) (where \(\mu\) is the population mean). So the sampling distribution's mean is the same as the population mean. This statement is true.
• Option D: The standard deviation of the sampling distribution (standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\) (\(\sigma\) is the population standard deviation, \(n\) is the sample size). Since \(\sqrt{n}> 1\) for \(n > 1\), the sampling distribution's standard deviation is smaller than the population standard deviation. This statement is true.

Answer:

A. The distribution is normal regardless of the shape of the population distribution, as long as the sample size, \(n\), is large enough.