1. 25 - 1 Module 25: Confidence Intervals and Hypothesis Tests for Variances for One Sample This module discusses confidence intervals and hypothesis tests for variances for the one sample situation. Reviewed 19 July 05/ MODULE 25

2. 25 - 2 The Situation Earlier we selected from the population of weights numerous samples of sizes n = 5, 10, and 20 where we assumed we knew that the population parameters were:  = 150 lbs,  2 = 100 lbs 2,  = 10 lbs.

3. 25 - 3 For the population mean , point estimates, confidence intervals and hypothesis tests were based on the sample mean and the normal or t distributions. For the population variance  2, point estimates, confidence intervals and hypothesis tests are based on the sample variance s 2 and the chi-squared distribution for

4. 25 - 4 For a 95% confidence interval, or  = 0.05, we use For hypothesis tests we calculate and compare the results to the χ 2 tables.

5. 25 - 5 n = 5, = 153.0, s = 12.9, s 2 = 166.41 s 2 = 166.41 is sample estimate of  2 = 100 s = 12.9 is sample estimate of  = 10 For a 95% confidence interval, we use df = n - 1 = 4 Population of Weights Example

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7. 25 - 7 From the Population of weights, for n = 5, we had s 2 = 5.4 s 3 = 18.6 s 4 = 8.1 s 5 = 7.7 Other Samples

8. 25 - 8 Length = 230.52 lbs 2 Length = 2,734.98 lbs 2 95% CI for  2, n = 5, df = 4

9. 25 - 9 Length = 518.68 lbs 2 Length = 468.72 lbs 2

10. 25 - 10 s 1 = 10.2 s 2 = 8.4 s 3 = 11.4 s 4 = 11.5 s 5 = 8.4 For n = 20, we had

11. 25 - 11 Length = 161.76 lbs 2 95% CIs for  2, n = 20, df = 19

12. 25 - 12 Example: For the first sample from the samples with n = 5, we had s 2 = 166.41. Test whether or not  2 = 200. 1. The hypothesis: H 0 :  2 = 200, vs H 1 :  2 ≠ 200 2. The assumptions: Independent observations normal distribution 3. The α-level: α = 0.05

13. 25 - 13 4. The test statistic: 5. The critical region : Reject H 0 : σ 2 = 200 if the value calculated for χ 2 is not between χ 2 0.025 (4) =0.484, and χ 2 0.975 (4) =11.143 6. The Result: 7. The conclusion: Accept H 0 :  2 = 200.

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17. 25 - 17 Table 3 indicates that the mean Global Stress Index for Lesbians is 16 with SD = 6.8. Suppose that previous work in this area had indicated that the SD for the population was about  = 10. Hence, we would be interested in testing whether or not  2 = 100. The Question

18. 25 - 18 1. The hypothesis: H 0 :  2 = 100, vs H 1 :  2 ≠ 100 2. The assumptions: Independence, normal distribution 3. The α-level: α = 0.05 4. The test statistic: 5. The critical region : Reject H 0 : σ 2 = 100 if the value calculated for χ 2 is not between χ 2 0.025 (549) = 615.82, and χ 2 0.975 (549) = 485.97 6. The Result: 7. The conclusion: Reject H 0 :  2 = 100.