The Mean, Standard Deviation, and Sampling Distribution of the Sample Mean

This section gives several concrete examples of calculating the exact distributions of the sample mean. The corresponding means and standard deviations are computed for demonstration based on these distributions. Next, it discusses sampling distributions of sample means when the sample size is large. It also considers the case when the population is normal. Finally, it uses the central limit theorem for large sample approximations.

The Sampling Distribution of the Sample Mean

Answers

1. a. \mu_{X}=128, \sigma_{\bar{X}}=3.67

b. 0.9936

3. a. \mu_{\bar{X}}=73.5, \sigma_{\bar{X}}=0.456

b. 0.0005

5. a. 0.0918
b. \mu_{\bar{X}}=25.6, \sigma_{\bar{X}}=1.1

c. 0.0000

7. a. \mu_{X}=557, \sigma_{X}=4.9497

b. 0.0043

9. a. 0.5818
b. \mu_{\bar{X}}=1214, \sigma_{\bar{X}}=24.4

c. 0.9998

11. a. \mu_{\bar{X}}=72, \sigma_{\bar{X}}=0.8944

b. 0.0250

13. 0.9940

15. 0.9994

17. a. 0.8036

b. 1.0000

19. 0.9994

21. a. 0.2955
b. 0.8977

23. 0.9251

25. 0.9982