Question

it is recommended that adults get 8 hours of sleep each night. a researcher hypothesized college students got less than the recommended number of hours of sleep each night, on average. the researcher randomly sampled 20 college students and calculated a sample mean of 7.5 hours per night. if the researcher wanted to perform a one - sample t - test, which of the following is a correct statement?
choose the correct answer below.
○ a. the distribution of sample means will be normal even if the distribution of the data in the population is not normal
○ b. the number of hours of sleep per night for all college students must be normally distributed because the sample size is small.
○ c. by taking a random sample, the researcher has guaranteed that the distribution of sample means is normal
○ d. by taking a random sample, the researcher has guaranteed that the distribution of number of hours of sleep per night for all college students is normal

Explanation:

Step1: Check Central Limit Theorem

The Central Limit Theorem states that for a sample of size \(n\), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal if \(n\) is large (\(n\geq30\)) or if the population is normal. But for a \(t -\)test, when \(n\) is small (\(n = 20\) here), if the population is not normal, the sampling distribution of the sample mean is still approximately normal due to the properties of the \(t -\)distribution (which is robust to moderate departures from normality for inference about the mean).

Step2: Analyze each option

• Option A:

The \(t -\)distribution (used in one - sample \(t -\)test) is robust. So, the distribution of sample means will be normal (in the sense that the \(t -\)test is valid) even if the population is not normal (for moderate sample sizes and when we are doing inference about the mean).

• Option B:

The sample size \(n=20\) is small. We cannot say that the population (number of hours of sleep for all college students) must be normally distributed. The \(t -\)test is robust to non - normality for inference about the mean, not that the population must be normal.

• Option C:

Taking a random sample does not guarantee that the distribution of sample means is normal. The Central Limit Theorem (for large \(n\)) or the robustness of the \(t -\)distribution (for small \(n\) and inference about the mean) is what helps with the normality of the sampling distribution of the sample mean.

• Option D:

Taking a random sample does not guarantee that the population (number of hours of sleep for all college students) is normal. A random sample is for representativeness, not for normality of the population.

Answer:

A. The distribution of sample means will be normal even if the distribution of the data in the population is not normal