Question

A variable of a population has mean μ and standard deviation$\sigma$. that For a large sample size n, answer the following questions.

a. Identify the distribution of$\overline{x}.$

b. Does your answer to part (a) depend on n being large? Explain your answer.

c. Identify the mean and the standard deviation of$\overline{x}.$

d. Does your answer to part (c) depend on the sample size being large? Why or why not?

Short answer

Part a) By application of CLT, sample mean follows Normal distribution.

Part b) Yes, if the sample is not a large sample then we cannot apply CLT to approximate the distribution of the sample mean as Normal distribution.

Part c) Mean of $\overline{x}=\mu$And standard deviation of $\overline{x}=\frac{\sigma }{\sqrt{n}}.$

Part d) No, the expression for mean and S. d of $\overline{X}$remains same the large samples also.

Step-by-step solution

Given information

Population variable has the mean $\mu$and standard deviation$\sigma$

Sample size $n$is large.

Part a) Step 1:

By application of CLT, sample mean follows Normal distribution.

Part b) Step 1:

Yes, since we can approximate the distribution of sample mean by Normal distribution in case of unknown or Non normal population distribution iff the sample size $n$is large. That is if the sample is not a large sample then we cannot apply CLT to approximate the distribution of the sample mean as Normal distribution.

Part c) Step 1:

Mean of $\overline{x}=\mu$and standard deviation of$\overline{x}=\frac{\sigma }{\sqrt{n}}.$

Part d) Step 1:

No, the expression for mean and S. d of $\overline{X}$remains same the large samples also.

i.e. For large sample also${\mu }_{\overline{\mathrm{x}}}=\mu and{\sigma }_{\overline{\mathrm{x}}}=\frac{\sigma }{\sqrt{n}}.$