Q 7.64. - Elementary Statistics

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?

Step-by-Step Solution

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 1 : 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 ${\mu }_{\overline{\mathrm{x}}}=\mu and {\sigma }_{\overline{\mathrm{x}}}=\frac{\sigma }{\sqrt{n}}.$