# 1 1 Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western/Thomson Learning 

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1 1 Slide Slides Prepared by JOHN S. LOUCKS St. Edward’s University © 2002 South-Western/Thomson Learning 

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

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

4 4 Slide Chi-Square Distribution n The chi-square distribution is the sum of squared standardized normal random variables such as ( z 1 ) 2 +( z 2 ) 2 +( z 3 ) 2 and so on. n The chi-square distribution is based on sampling from a normal population. The sampling distribution of ( n - 1) s 2 /  2 has a chi- square distribution whenever a simple random sample of size n is selected from a normal population. The sampling distribution of ( n - 1) s 2 /  2 has a chi- square distribution whenever a simple random sample of size n is selected from a normal population. n We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance.

5 5 Slide Interval Estimation of  2 n 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 6 Slide Interval Estimation of  n 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 7 Slide n Chi-Square Distribution With Tail Areas of.025 95% of the possible  2 values 95% of the possible  2 values 22 22 0 0.025 Interval Estimation of  2

8 8 Slide 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 68 o F. 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. 1 2 3 4 5 6 7 8 9 10 Temp. 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

9 9 Slide Example: Buyer’s Digest Interval Estimation of  2 Interval Estimation of  2 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 22 22 0 0.025

10 Slide Interval Estimation of  2 Interval Estimation of  2 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 22 22 0 0.025 2.70 Example: Buyer’s Digest Area in Upper Tail =.975

11 Slide Example: Buyer’s Digest Interval Estimation of  2 Interval Estimation of  2 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 22 22 0 0 Area in Upper Tail =.025 Area in Upper Tail =.025.025 2.70 19.02

12 Slide Interval Estimation of  2 Interval Estimation of  2 Sample variance s 2 provides a point estimate of  2. A 95% confidence interval for the population variance is given by:.33 <  2 < 2.33.33 <  2 < 2.33 Example: Buyer’s Digest

13 Slide n Left-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if (where is based on a chi-square distribution with n - 1 d.f.) or Reject H 0 if p -value <  Hypothesis Testing About a Population Variance

14 Slide n Right-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if (where is based on a chi-square distribution with n - 1 d.f.) or Reject H 0 if p -value <  Hypothesis Testing About a Population Variance

15 Slide n Two-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if (where are based on a chi-square distribu- tion with n - 1 d.f.) or Reject H 0 if p -value <  Hypothesis Testing About a Population Variance

16 Slide n One-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if F > F  where the value of F  is based on an F distribution with n 1 - 1 (numerator) and n 2 - 1 (denominator) d.f. Hypothesis Testing About the Variances of Two Populations

17 Slide n Two-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Rejection Rule Rejection Rule Reject H 0 if F > F  /2 where the value of F  /2 is based on an F distribution with n 1 - 1 (numerator) and n 2 - 1 (denominator) d.f. Hypothesis Testing About the Variances of Two Populations

18 Slide 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.12345678910 Temp.66.467.868.270.3 69.5 68.0 68.1 68.6 67.9 66.2 Example: Buyer’s Digest

19 Slide n Hypothesis Testing About the Variances of Two Populations Hypotheses Hypotheses (ThermoRite and TempKing thermo- stats have same temperature variance) (ThermoRite and TempKing thermo- stats have same temperature variance) (Their variances are not equal) (Their variances are not equal) Rejection Rule Rejection Rule The F distribution table shows that with  =.10, 9 d.f. (numerator), and 9 d.f. (denominator), F.05 = 3.18. Reject H 0 if F > 3.18 Example: Buyer’s Digest

20 Slide n Hypothesis Testing About the Variances of Two Populations Test Statistic Test Statistic ThermoRite’s sample variance is.70. TempKing’s sample variance is 1.52. F = 1.52/.70 = 2.17 Conclusion Conclusion We cannot reject H 0. There is insufficient evidence to conclude that the population variances differ for the two thermostat brands. Example: Buyer’s Digest

21 Slide End of Chapter 11

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