Question

ch 18 in a test of hypothesis for a mean, which of the following is correct?
the population distribution can be any distribution because the sample mean is always normally distributed.
as long as the population distribution consists only of positive values, sample size does not matter.
the test requires that the sample mean is normally distributed; this requires that either the population distribution is normal or the sample size exceeds 30.
the population must be normal because that is the only case when the sample mean is normally distributed.

Explanation:

• The Central Limit Theorem (CLT) states that for a large sample size ($n > 30$), the sampling distribution of the sample mean is approximately normal, regardless of the population distribution. If the population is normally distributed, the sampling distribution of the sample mean is normal for any sample size.
• The first option is incorrect because the sample mean is not always normally distributed. For example, if the population is highly skewed and the sample size is small ($n<30$), the sample - mean distribution may not be normal.
• The second option is wrong. The positivity of population values does not make the sample - size irrelevant. The CLT still depends on sample size for non - normal populations.
• The fourth option is incorrect. The sample mean can be normally distributed either when the population is normal (for any $n$) or when the sample size is large ($n > 30$) even if the population is non - normal.

Answer:

The test requires that the sample mean is normally distributed; this requires that either the population distribution is Normal or the sample size exceeds 30.