Question

A variable of a population has a mean of $\mu =35$and a standard deviation of $\sigma =42.$

a. If the variable is normally distributed, identify the sampling distribution of the sample mean for samples of size $9.$

b. Can you answer part (a) if the distribution of the variable under consideration is unknown? Explain your answer.

c. Can you answer part (a) if the distribution of the variable under consideration is unknown but the sample size is $36$instead of $9$?

Why or why not?

Short answer

Part a) The sampling distribution of the sample mean for samples of size $9$ is$14.$

Part b) No, because the sample size is fewer than 30, it cannot be considered a representative sample.

Part c) Yes, we can find the distribution of sample mean in case of sample size $36.$

Step-by-step solution

Part a) Step 1: 

Population mean $\mu =35$

Population S.D. $\sigma =42$

If the population variable is normal then the sample mean is also follows with mean ${\mu }_{\overline{x}}=\mu$and S.D ${\sigma }_{\overline{X}}=\frac{\sigma }{\sqrt{n}},n=$sample size.

Therefore,

$$\begin{gathered} {\mu }_{\overline{x}}=\mu =35 \\ {\sigma }_{\overline{x}}=\frac{42}{\sqrt{9}} \\ =\frac{42}{3} \\ {\sigma }_{\overline{x}}=14 \end{gathered}$$

So, the Sample mean is normally distributed with mean$35$and S.D $=14.$

In notation$\overline{X}~N\left(35,{14}^2\right).$

part b) Step 1: Explanation

No, since the sample size is less than 30 we can not consider it as a large sample. So if the population distribution is unknown then we can not answer part (a) i.e. can not find the distribution of sample mean because we can not apply CLT here.

Part c) Step 1: Explanation

Yes, we can find the distribution of sample mean in case of sample size 36 . If the population distribution is unknown. Here the sample size is 36 , which is greater than 30 . We can consider it as a large sample. Hence, by using CLT we can approximate the distribution of sample mean as Normal.