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.

Small Sample Tests for a Population Mean

1. a. $Z \leq-1.645$
b. $T \leq-2.571$ or $T \geq 2.571$
c. $T \geq 1.319$
d. $Z \leq-1645$ or $Z \geq 1.645$

3. a. $T \leq-0.855$
b. $Z \leq-1.645$
c. $T \leq-2.201$ or $T \geq 2.201$
d. $T \geq 3.435$

5. a. $T=-2.690, d f=19,-t_{0.005}=-2.861$, do not reject $H_{0}$.
b. $0.01 < p$-value $< 0.02, \alpha=0.01$, do not reject $H_{0}$.

7. a. $T=2.398, d f=7, t_{0.05}=1.895$, reject $H_{0}$.
b. $0.01 < p$-value $< 0.025, \alpha=0.05$, reject $H_{0}$

9. $T=-7.560, d f=12,-t_{0.10}=-1.356$, reject $H_{0}$.

11. $T=-7.076, d f=5,-t_{0.0005}=-6.869$, reject $H_{0}$

13. a. $T=-1.483, d f=14,-t_{0.05}=-1.761$, do not reject $H_{0}$;
b. $T=-1.483, d f=14,-t_{0.10}=-1.345$, reject $H_{0}$;

15. a. $T=2.069, d f=6, t_{0.10}=1.44$, reject $H_{0}$;
b. $T=2.069, d f=6, t_{0.05}=1.943$, reject $H_{0}$.

17. $T=4.472, d f=4, t_{0.10}=1.533$, reject $H_{0}$.

19. $T=0.798, d f=24, t_{0.10}=1.318$, do not reject $H_{0}$

21. a. $T=-1.773, d f=4,-t_{0.05}=-2.132$, do not reject $H_{0}$.
b. $0.05 < p$-value $< 0.10$
c. $\alpha=0.05$, do not reject $\mathrm{H}_{0}$