## Sample Tests for a Population Mean

This section talks about using the central limit theorem to test a population mean when the sample size is large. It also addresses how to interpret the test results in the application background. Then, it discusses testing a population mean when the sample size is small, outlines a five-step testing procedure, and illustrates the procedure with an example. Study the example carefully and complete the relevant exercises and applications. Finally, it talks about large sample tests for a population proportion. The critical value and p-value approach are introduced based on a standardized test statistic.

### Large Sample Tests for a Population Mean

1. a. $Z \leq-1.645$

b. $Z \leq-1.96$ or $Z \geq 1.96$

c. $Z \geq 1.28$

d. $Z \leq-1.645$ or $Z \geq 1.645$

3. a. $Z \leq-0.84$

b. $Z \leq-1.645$

c. $Z \leq-1.96$ or $Z \geq 1.96$

d. $Z \geq 3.1$

5. a. $Z=2.235$

b. $Z=2.592$

c. $Z=-2.122$

d. $Z=3.002$

7. a. $Z=-2.18,-z_{0.10}=-1.28$, reject $H_{0}$.

b. $Z=1.61, z_{0.05}=1.645$, do not reject $H_{0}$.

c. $Z=-4.47,-z_{0.01}=-2.33$, reject $H_{0}$.

9. $Z=-2.86,-z_{0.01}=-2.33$, reject $H_{0}$.

11. $Z=-1.69,-z_{0.025}=-1.96$, do not reject $H_{0}$.

13. $Z=8.33, z_{0.05}=1.645$, reject $H_{0}$.

15. $Z=.2.02, z_{0.10}=1.28$, reject $H_{0}$.

17. $Z=-2.08,-z_{0.01}=-2.33$, do not reject $H_{0}$.

19. a. $Z=2.54, z_{0.05}=1.645$, reject $H_{0}$;

b. $Z=2.54, z_{0.10}=1.28$, reject $H_{0}$.

21. $H_{0}: \mu=1510$ vs. $H_{a}: \mu > 1510$. Test Statistic: $Z=2.7882$. Rejection Region: $[2.33, \infty).$ Decision: Reject $H_{0}$

23. a. $\mu_{0}=1528.74$

b. $H_{0}: \mu=1510$ vs. $H_{a}: \mu > 1510$. Test Statistic: $Z=-1.41.$ Rejection Region: $[1.28, \infty)$. Decision: Fail to reject $H_{0}$.

c. No, it is a Type II error.