This document discusses hypothesis testing for population variances. It covers testing for a single population variance, testing if two population variances are equal, and how to conduct these tests. The chi-square distribution is used to develop critical values and rejection regions for variance hypothesis tests. An example compares the variances of temperature readings from two thermostat models, testing if one has an "acceptable" variance below 0.5 and if their variances are equal.

1. Chapter 11- Inferences
About Population Variances

2. Chapter 11 Inferences About Population Variances
11.1 - Inference about a Population Variance
11.2 - Inferences about Two Populations Variances
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3. Chi-Square Distribution(1 of 2)
• The chi-square distribution is based on sampling from a normal
population.
• We can use the chi-square distribution to develop interval estimates and
conduct hypothesis tests about a population variance.
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4. Hypothesis Testing About a Population Variance
(1 of 8)
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5. Hypothesis Testing About a Population Variance(2 of 8)
For each type of test,
• the chi-square critical values are based on a chi-square distribution with
𝑛 − 1 degrees of freedom.
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6. Hypothesis Testing About a Population Variance (3 of 8)
Example: Buyer’s Digest (B)
Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s
Digest gives an “acceptable” rating to a thermostat with a temperature
variance of 0.5 or less.
Using the 10 readings, we will conduct a hypothesis test (with a = 0.10)
to determine whether the ThermoRite thermostat’s temperature
variance is “acceptable”.
Thermostat 1 2 3 4 5 6 7 8 9 10
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
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7. Hypothesis Testing About a Population Variance (4 of 8)
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8. Hypothesis Testing About a Population Variance(5 of 8)
For n – 1 = 10 – 1 = 9 df and a = 0.10
Selected Values from the Chi-Square Distribution Table
Degrees of
Freedom
.99 Area
in Upper
Tail
.975 Area
in Upper
Tail
.95 Area
in Upper
Tail
.90 Area
in Upper
Tail
.10 Area
in Upper
Tail
.05 Area
in Upper
Tail
.025 Area
in Upper
Tail
.01 Area
in Upper
Tail
5 0.554 0.831 1.145 1.610 9.236 11.070 12.832 15.086
6 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812
7 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475
8 1.647 2.180 2.733 3.490 13.362 15.507 17.535 20.090
9 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666
10 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209
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9. Hypothesis Testing About a Population Variance(6 of 8)
Rejection Region
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10. Hypothesis Testing About a Population Variance (7 of 8)
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11. Hypothesis Testing About a Population Variance (8 of 8)
Using the p-Value
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12. Hypothesis Testing About two Population Variances (1 of 2)
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13. Hypothesis Testing About a Population Variance(2 of 2)
For each type of test,
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14. Hypothesis Testing About the Variances of Two Populations(1 of 5)
Example: Buyer’s Digest (C)
Buyer’s Digest has conducted the same test, as described earlier, on
another 10 thermostats, this time manufactured by TempKing. We will
conduct a hypothesis test with α = 0.10 to see if the variances are equal for
ThermoRite’s thermostats and TempKing’s thermostats.
ThermoRite Sample
Thermostat 1 2 3 4 5 6 7 8 9 10
Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2
TempKing Sample
Thermostat 1 2 3 4 5 6 7 8 9 10
Temperature 67.7 66.4 69.2 70.1 69.5 69.7 68.1 66.6 67.3 67.5
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15. Hypothesis Testing About the Variances of Two Populations(2 of 5)
Hypotheses
Rejection Rule
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16. Hypothesis Testing About the Variances of Two Populations(4 of 5)
Test Statistic:
TempKing’s sample variance is 1.768. ThermoRite’s sample variance is
0.7.
Conclusion:
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