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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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
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.