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

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

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This module discusses confidence intervals and hypothesis tests for variances for the one sample situation.

Reviewed 19 July 05/ MODULE 25

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.

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

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

For hypothesis tests we calculate

and compare the results to the χ2 tables.

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

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

s2 = 5.4

s3 = 18.6

s4 = 8.1

s5 = 7.7

95% CI for 2, n = 5, df = 4

Length = 230.52 lbs2

Length = 2,734.98 lbs2

Length = 518.68 lbs2

Length = 468.72 lbs2

For n = 20, we had

s1 = 10.2

s2 = 8.4

s3 = 11.4

s4 = 11.5

s5 = 8.4

95% CIs for 2, n = 20, df = 19

Length = 161.76 lbs2

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

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.

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.

- 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.