1. 統計學 Spring 2004
授課教師：統計系余清祥
日期：2004年3月16日
第五週：比較變異數

2. Chapter 11 Inferences About Population Variances
Inference about a Population Variance
Inferences about the Variances of Two Populations

3. Inferences About a Population Variance
Chi-Square Distribution
Interval Estimation of 2
Hypothesis Testing

4. Chi-Square Distribution
The chi-square distribution is the sum of squared standardized normal random variables such as
(z1)2+(z2)2+(z3)2 and so on.
The chi-square distribution is based on sampling from a normal population.
The sampling distribution of (n - 1)s2/ 2 has a chi-square distribution whenever a simple random sample of size n is selected from a normal population.
We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance.

5. Interval Estimation of 2
Interval Estimate of a Population Variance
where the values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 -  is the confidence coefficient.

6. Interval Estimation of 
Interval Estimate of a Population Standard Deviation
Taking the square root of the upper and lower limits of the variance interval provides the confidence interval for the population standard deviation.

7. Interval Estimation of 2
Chi-Square Distribution With Tail Areas of .025
.025
.025
95% of the
possible 2 values 2

8. Example: Buyer’s Digest
Buyer’s Digest rates thermostats manufactured
for home temperature control. In a recent test, 10
thermostats manufactured by ThermoRite were selected
and placed in a test room that was maintained at a
temperature of 68oF. The temperature readings of the
ten thermostats are listed below.
We will use the 10 readings to develop a 95%
confidence interval estimate of the population variance.
Therm
Temp

9. Example: Buyer’s Digest
Interval Estimation of 2
n - 1 = = 9 degrees of freedom and a = .05
.025
.025 2

10. Example: Buyer’s Digest
Interval Estimation of 2
n - 1 = = 9 degrees of freedom and a = .05
.025
Area in
Upper Tail
= .975 2
2.70

11. Example: Buyer’s Digest
Interval Estimation of 2
n - 1 = = 9 degrees of freedom and a = .05
.025
Area in Upper
Tail = .025 2
2.70
19.02

12. Example: Buyer’s Digest
Interval Estimation of  2
Sample variance s2 provides a point estimate of  2.
A 95% confidence interval for the population variance is given by:
.33 < 2 < 2.33

13. Hypothesis Testing About a Population Variance
Left-Tailed Test
Hypotheses
Test Statistic
Rejection Rule
Reject H0 if (where is based on
a chi-square distribution with n - 1 d.f.) or
Reject H0 if p-value < a

14. Hypothesis Testing About a Population Variance
Right-Tailed Test
Hypotheses
Test Statistic
Rejection Rule
Reject H0 if (where is based on
a chi-square distribution with n - 1 d.f.) or
Reject H0 if p-value < a

15. Hypothesis Testing About a Population Variance
Two-Tailed Test
Hypotheses
Test Statistic
Rejection Rule
Reject H0 if (where are based on a chi-square distribu- tion with n - 1 d.f.) or Reject H0 if p-value < a

16. Hypothesis Testing About the Variances of Two Populations
One-Tailed Test
Hypotheses
Test Statistic
Rejection Rule
Reject H0 if F > F where the value of F is based on an F distribution with n1 - 1 (numerator) and n2 - 1 (denominator) d.f.

17. Hypothesis Testing About the Variances of Two Populations
Two-Tailed Test
Hypotheses
Test Statistic
Rejection Rule
Reject H0 if F > F/2 where the value of F/2 is based on an F distribution with n (numerator) and n2 - 1 (denominator) d.f.

18. Example: Buyer’s Digest
Buyer’s Digest has conducted the same test, as was
described earlier, on another 10 thermostats, this time
manufactured by TempKing. The temperature readings
of the ten thermostats are listed below.
We will conduct a hypothesis test with  = .10 to see
if the variances are equal for ThermoRite’s thermostats
and TempKing’s thermostats.
Therm
Temp

19. Example: Buyer’s Digest
Hypothesis Testing About the Variances of Two Populations
Hypotheses
(ThermoRite and TempKing thermo stats have same temperature variance)
(Their variances are not equal)
Rejection Rule
The F distribution table shows that with  = .10,
9 d.f. (numerator), and 9 d.f. (denominator),
F.05 = 3.18.
Reject H0 if F > 3.18

20. Example: Buyer’s Digest
Hypothesis Testing About the Variances of Two Populations
Test Statistic
ThermoRite’s sample variance is .70.
TempKing’s sample variance is 1.52.
F = 1.52/.70 = 2.17
Conclusion
We cannot reject H0. There is insufficient
evidence to conclude that the population
variances differ for the two thermostat brands.

21. End of Chapter 11