374 Chapter 8 Confidence Intervals EXCEL Constructing a confidence interval for the mean when �� is unknown The CONFIDENCE.T command returns the margin of error for a confidence interval when the population standard deviation �� is unknown. Step 1. In an empty cell, select the Insert Function icon and highlight Statistical in the category field. Step 2. Click on the CONFIDENCE.T function and press OK. Step 3. Enter the value of �� (0.05) in the Alpha field, the sample standard deviation (9.84) in the Standard dev field, and the sample size (123) in the Size field. Step 4. Click OK (Figure G) to obtain the margin of error m. The confidence interval is given by ̄x − m < �� < ̄x + m. Figure G SECTION 8.2 Exercises Exercises 1– 6 are the Check Your Understanding exercises located within the section. Understanding the Concepts In Exercises 7 and 8, fill in each blank with the appropriate word or phrase. 7. When constructing a confidence interval for a population mean �� from a sample of size 12, the number of degrees of freedom for the critical value t��∕2 is . 11 8. When the number of degrees of freedom is large, the Student’s t distribution is close to the distribution. normal In Exercises 9 and 10, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement. 9. The Student’s t curve is less spread out than the standard normal curve. False 10. The Student’s t distribution should not be used to find a confidence interval for �� if outliers are present in a small sample. True Practicing the Skills 11. Find the critical value t��∕2 needed to construct a confidence interval of the given level with the given sample size. a. Level 95%, sample size 23 2.074 b. Level 90%, sample size 3 2.920 c. Level 98%, sample size 18 2.567 d. Level 99%, sample size 29 2.763 12. Find the critical value t��∕2 needed to construct a confidence interval of the given level with the given sample size. a. Level 90%, sample size 6 2.015 b. Level 98%, sample size 12 2.718 c. Level 95%, sample size 32 2.040 d. Level 99%, sample size 10 3.250 13. A sample of size n = 18 is drawn from a normal population. a. Find the critical value t��∕2 needed to construct a 95% confidence interval. 2.110 b. If the sample size were n = 25, would the critical value be smaller or larger? Smaller 14. A sample of size n = 22 is drawn from a normal population. a. Find the critical value t��∕2 needed to construct a 90% confidence interval. 1.721 b. If the sample size were n = 15, would the critical value be smaller or larger? Larger 15. A sample of size n = 12 is drawn from a normal population. a. Find the critical value t��∕2 needed to construct a 98% confidence interval. 2.718 b. If the sample size were n = 50, would it be necessary for the population to be approximately normal? No 16. A sample of size n = 61 is drawn. a. Find the critical value t��∕2 needed to construct a 95% confidence interval. 2.000 b. If the sample size were n = 15, what additional assumption would need to be made for the confidence interval to be valid? Population is approximately normal 17. A sample of size n = 15 has sample mean ̄x = 2.1 and sample standard deviation s = 1.7. a. Construct a 95% confidence interval for the population mean ��. (1.2, 3.0) b. If the sample size were n = 25, would the confidence interval be narrower or wider? Narrower 18. A sample of size n = 44 has sample mean ̄x = 56.9 and sample standard deviation s = 9.1. a. Construct a 98% confidence interval for the population mean ��. (53.6, 60.2) b. If the sample size were n = 30, would the confidence interval be narrower or wider? Wider 19. A sample of size n = 89 has sample mean ̄x = 87.2 and sample standard deviation s = 5.3. a. Construct a 95% confidence interval for the population mean ��. (86.1, 88.3)
navidi_monk_elementary_statistics_2e_ch7-9
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