## 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 Proportion

1. a. $Z=2.277$
b. $Z=2.277$
c. $Z=-1.435$

3. a. $Z \geq 1.645$; reject $H_{0}$.
b. $Z \leq-1.96$ or $Z \geq 1.96$; reject $H_{0}$.
c. $Z \leq-1.645 ;$ do not reject $H_{0}$.

5. a. $p$-value $=0.0116, \alpha=0.05$; reject $H_{0}$.
b. $p$-value $=0.0232, \alpha=0.05 ;$ reject $H_{0}$.
c. $p$-value $=0.0749, \alpha=0.05$; do not reject $H_{0}$.

7. a. $Z=1.74, z_{0.05}=1.645$, reject $H_{0}$.
b. $Z=-1.98,-z_{0.005}=-2.576$, do not reject $H_{0}$

9. a. $Z=2.24, p$-value $=0.025, \alpha=0.005$, do not reject $H_{0}$.
b. $Z=2.92, p$-value $=0.0018, \alpha=0.05$, reject $H_{0}$.

11. $z=1.11, z_{0.025}=1.96$, do not reject $H_{0}$.

13. $Z=1.93, z_{0.10}=1.28$, reject $H_{0}$.

15. $Z=-0.523, \pm z_{0.05}=\pm 1.645$, do not reject $H_{0}$.

17. a. $Z=-1.798,-z_{0.05}=-1.645$, reject $H_{0}$;
b. $p-$-value $=0.0359$.

19. a. $Z=-8.92,-z_{0.01}=-2.33$, reject $H_{0}$;
b. $p$-value $\approx 0$.

21. $Z=3.04, z_{0.01}=2.33$, reject $H_{0}$.

23. $H_{0}: p=1 / 6$ vs. $H_{a}: p \neq 1 / 6$. Test Statistic: $Z=-0.76$. Rejection Region: $(-\infty,-1.28] \cup[1.28, \infty)$. Decision: Fail to reject $H_{0}$.

25. $H_{0}: p=0.25$ vs. $H_{a}: p < 0.25$. Test Statistic: $Z=-1.17$. Rejection Region: $(-\infty,-1.28].$ Decision: Fail to reject $H_{0}$.