Question

question 4
in which of the following scenarios would the distribution of the sample mean x - bar be normally distributed? check all that apply.
a. we take repeated random samples of size 10 from a population of unknown shape.
b. we take repeated random samples of size 15 from a population that is normally distributed.
c. we take repeated random samples of size 50 from a population of unknown shape.
d. we take repeated random samples of size 25 from a population that of unknown shape.

Explanation:

Step1: Recall central - limit theorem

The central - limit theorem states that if the sample size \(n\) is large (usually \(n\geq30\)), the sampling distribution of the sample mean \(\bar{x}\) is approximately normal regardless of the shape of the population. Also, if the population is normally distributed, the sampling distribution of the sample mean is normally distributed for any sample size \(n\).

Step2: Analyze option A

Sample size \(n = 10<30\) and population shape is unknown. So, the distribution of the sample mean is not necessarily normal.

Step3: Analyze option B

The population is normally distributed. So, for any sample size (here \(n = 15\)), the distribution of the sample mean \(\bar{x}\) is normally distributed.

Step4: Analyze option C

Sample size \(n=50\geq30\) and regardless of the population shape, by the central - limit theorem, the distribution of the sample mean \(\bar{x}\) is approximately normal.

Step5: Analyze option D

Sample size \(n = 25<30\) and population shape is unknown. So, the distribution of the sample mean is not necessarily normal.

Answer:

B. We take repeated random samples of size 15 from a population that is normally distributed.
C. We take repeated random samples of size 50 from a population of unknown shape.