## 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

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. $0.9982$