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

ANSWERS

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.