1. 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 =
s2 = 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

7. Other Samples From the Population of weights, for n = 5, we had

8. 95% CI for 2, n = 5, df = 4 Length = 230.52 lbs2

9. Length = lbs2
Length = lbs2

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

11. 95% CIs for 2, n = 20, df = 19
Length = lbs2

12. 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

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

17. The Question
Table 3 indicates that the mean Global Stress Index for Lesbians is 16 with SD = 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.

18. 1. The hypothesis: H0: 2 = 100, vs H1: 2 ≠ 100
The assumptions: Independence, normal distribution 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
χ (549) = , and
χ (549) =
The Result:
7. The conclusion: Reject H0: 2 = 100.