Module 25 confidence intervals and hypothesis tests for variances for one sample
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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. The Situation.

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

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Module 25 confidence intervals and hypothesis tests for variances for one sample

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


The situation

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.


Module 25 confidence intervals and hypothesis tests for variances for one sample

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


Module 25 confidence intervals and hypothesis tests for variances for one sample

For a 95% confidence interval, or  = 0.05, we use

For hypothesis tests we calculate

and compare the results to the χ2 tables.


Module 25 confidence intervals and hypothesis tests for variances for one sample

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


Other samples

Other Samples

From the Population of weights, for n = 5, we had

s2 = 5.4

s3 = 18.6

s4 = 8.1

s5 = 7.7


Module 25 confidence intervals and hypothesis tests for variances for one sample

95% CI for 2, n = 5, df = 4

Length = 230.52 lbs2

Length = 2,734.98 lbs2


Module 25 confidence intervals and hypothesis tests for variances for one sample

Length = 518.68 lbs2

Length = 468.72 lbs2


Module 25 confidence intervals and hypothesis tests for variances for one sample

For n = 20, we had

s1 = 10.2

s2 = 8.4

s3 = 11.4

s4 = 11.5

s5 = 8.4


Module 25 confidence intervals and hypothesis tests for variances for one sample

95% CIs for 2, n = 20, df = 19

Length = 161.76 lbs2


Module 25 confidence intervals and hypothesis tests for variances for one sample

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


Module 25 confidence intervals and hypothesis tests for variances for one sample

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.


The question

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.


Module 25 confidence intervals and hypothesis tests for variances for one sample

  • 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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