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

ANSWERS

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}.