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Module 25: Confidence Intervals and Hypothesis Tests for Variances for One Sample

This module offers an in-depth exploration of confidence intervals and hypothesis tests for variances in a one-sample scenario. It elaborates on how to compute point estimates, confidence intervals, and hypothesis tests based on sample variance using the chi-squared distribution. Key concepts include testing against known population parameters, interpreting sample data (e.g., sample means and variances), and establishing critical values for hypothesis testing. Practical examples are provided to demonstrate the application of these statistical methods.

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Module 25: Confidence Intervals and Hypothesis Tests for Variances for One Sample

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  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. 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 lbs2,  = 10 lbs.

  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 s2 and the chi-squared distribution for

  4. For a 95% confidence interval, or  = 0.05, we use For hypothesis tests we calculate and compare the results to the χ2 tables.

  5. Population of Weights Example n = 5, = 153.0, s = 12.9, s2 = 166.41 s2 = 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

  6. Other Samples From the Population of weights, for n = 5, we had s2 = 5.4 s3 = 18.6 s4 = 8.1 s5 = 7.7

  7. 95% CI for 2, n = 5, df = 4 Length = 230.52 lbs2 Length = 2,734.98 lbs2

  8. Length = 518.68 lbs2 Length = 468.72 lbs2

  9. For n = 20, we had s1 = 10.2 s2 = 8.4 s3 = 11.4 s4 = 11.5 s5 = 8.4

  10. 95% CIs for 2, n = 20, df = 19 Length = 161.76 lbs2

  11. Example: For the first sample from the samples with n = 5, we had s2 = 166.41. Test whether or not 2 = 200. 1. The hypothesis: H0: 2 = 200, vs H1: 2 ≠ 200 2. The assumptions: Independent observations normal distribution 3. The α-level: α = 0.05

  12. 4. The test statistic: 5. The critical region:Reject H0: σ2 = 200 if the value calculated for χ2 is not between χ20.025 (4) =0 .484, and χ20.975 (4) =11.143 6. The Result: 7. The conclusion: Accept H0: 2 = 200.

  13. The Question 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.

  14. 1. The hypothesis: H0: 2 = 100, vs H1: 2 ≠ 100 • The assumptions: Independence, normal distribution 3. The α-level: α = 0.05 • The test statistic: • 5. The critical region:Reject H0: σ2 = 100 if the value • calculated for χ2 is not between • χ20.025 (549) = 615.82, and • χ20.975 (549) = 485.97 • The Result: • 7. The conclusion: Reject H0: 2 = 100.

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