1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 11 Inferences About Population Variances n Inference about a Population Variance n Inferences about Two Population Variances

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Inferences About a Population Variance n Chi-Square Distribution n Interval Estimation n Hypothesis Testing

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Chi-Square Distribution We can use the chi-square distribution to develop We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests interval estimates and conduct hypothesis tests about a population variance. about a population variance. The sampling distribution of ( n - 1) s 2 /  2 has a chi- The sampling distribution of ( n - 1) s 2 /  2 has a chi- square distribution whenever a simple random sample square distribution whenever a simple random sample of size n is selected from a normal population. of size n is selected from a normal population. The chi-square distribution is based on sampling The chi-square distribution is based on sampling from a normal population. from a normal population. n The chi-square distribution is the sum of squared standardized normal random variables such as standardized normal random variables such as ( z 1 ) 2 +( z 2 ) 2 +( z 3 ) 2 and so on. ( z 1 ) 2 +( z 2 ) 2 +( z 3 ) 2 and so on.

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Examples of Sampling Distribution of ( n - 1) s 2 /  2 0 0 With 2 degrees of freedom of freedom With 2 degrees of freedom of freedom With 5 degrees of freedom of freedom With 5 degrees of freedom of freedom With 10 degrees of freedom of freedom With 10 degrees of freedom of freedom

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Chi-Square Distribution For example, there is a.95 probability of obtaining a  2 (chi-square) value such that For example, there is a.95 probability of obtaining a  2 (chi-square) value such that We will use the notation to denote the value for the chi-square distribution that provides an area of  to the right of the stated value. We will use the notation to denote the value for the chi-square distribution that provides an area of  to the right of the stated value.

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved 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 8 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  2 Substituting ( n – 1) s 2 /  2 for the  2 we get Substituting ( n – 1) s 2 /  2 for the  2 we get n Performing algebraic manipulation we get There is a (1 –  ) probability of obtaining a  2 value There is a (1 –  ) probability of obtaining a  2 value such that such that

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved n Interval Estimate of a Population Variance Interval Estimation of  2 where the    values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 -  is the confidence coefficient.

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  n Interval Estimate of a Population Standard Deviation Taking the square root of the upper and lower Taking the square root of the upper and lower limits of the variance interval provides the confidence interval for the population standard deviation.

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved 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 The temperature readings of the ten thermostats are shown on the next slide. Interval Estimation of  2 n Example: Buyer’s Digest (A)

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  2 We will use the 10 readings below to We will use the 10 readings below to develop a 95% confidence interval estimate of the population variance. n Example: Buyer’s Digest (A) Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Thermostat 1 2 3 4 5 6 7 8 9 10

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  2 Selected Values from the Chi-Square Distribution Table Our value For n - 1 = 10 - 1 = 9 d.f. and  =.05 For n - 1 = 10 - 1 = 9 d.f. and  =.05

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  2 22 22 0 0.025 Area in Upper Tail =.975 2.700 For n - 1 = 10 - 1 = 9 d.f. and  =.05 For n - 1 = 10 - 1 = 9 d.f. and  =.05

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved Interval Estimation of  2 Selected Values from the Chi-Square Distribution Table For n - 1 = 10 - 1 = 9 d.f. and  =.05 For n - 1 = 10 - 1 = 9 d.f. and  =.05 Our value

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved 22 22 0 0.025 2.700 Interval Estimation of  2 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 n - 1 = 10 - 1 = 9 degrees of freedom and  =.05 19.023 Area in Upper Tail =.025 Area in Upper Tail =.025

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved Sample variance s 2 provides a point estimate of  2. Sample variance s 2 provides a point estimate of  2. Interval Estimation of  2.33 <  2 < 2.33 n A 95% confidence interval for the population variance is given by:

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved n Left-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved n Left-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if p -value <  p -Value approach: Critical value approach: Rejection Rule Rejection Rule Reject H 0 if where is based on a chi-square distribution with n - 1 d.f.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved n Right-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved n Right-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if Reject H 0 if p -value <  where is based on a chi-square distribution with n - 1 d.f. p -Value approach: Critical value approach: Rejection Rule Rejection Rule

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Tailed Test Hypothesis Testing About a Population Variance where is the hypothesized value for the population variance Test Statistic Test Statistic Hypotheses Hypotheses

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Tailed Test (continued) Hypothesis Testing About a Population Variance Reject H 0 if p -value <  p -Value approach: Critical value approach: Rejection Rule Rejection Rule Reject H 0 if where are based on a chi-square distribution with n - 1 d.f.

24. 24 Slide © 2008 Thomson South-Western. All Rights Reserved Recall that Buyer’s Digest is rating Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable” rating to a thermo- stat with a temperature variance of 0.5 or less. Hypothesis Testing About a Population Variance n Example: Buyer’s Digest (B) We will conduct a hypothesis test (with We will conduct a hypothesis test (with  =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”.

25. 25 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis Testing About a Population Variance Using the 10 readings, we will Using the 10 readings, we will conduct a hypothesis test (with  =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. n Example: Buyer’s Digest (B) Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 Thermostat 1 2 3 4 5 6 7 8 9 10

26. 26 Slide © 2008 Thomson South-Western. All Rights Reserved n Hypotheses Hypothesis Testing About a Population Variance Reject H 0 if  2 > 14.684 n Rejection Rule

27. 27 Slide © 2008 Thomson South-Western. All Rights Reserved Selected Values from the Chi-Square Distribution Table For n - 1 = 10 - 1 = 9 d.f. and  =.10 For n - 1 = 10 - 1 = 9 d.f. and  =.10 Hypothesis Testing About a Population Variance Our value

28. 28 Slide © 2008 Thomson South-Western. All Rights Reserved 22 22 0 0 14.684 Area in Upper Tail =.10 Area in Upper Tail =.10 Hypothesis Testing About a Population Variance n Rejection Region Reject H 0

29. 29 Slide © 2008 Thomson South-Western. All Rights Reserved n Test Statistic Hypothesis Testing About a Population Variance Because  2 = 12.6 is less than 14.684, we cannot Because  2 = 12.6 is less than 14.684, we cannot reject H 0. The sample variance s 2 =.7 is insufficient evidence to conclude that the temperature variance for ThermoRite thermostats is unacceptable. n Conclusion The sample variance s 2 = 0.7

30. 30 Slide © 2008 Thomson South-Western. All Rights Reserved n Using the p -Value The sample variance of s 2 =.7 is The sample variance of s 2 =.7 is insufficient evidence to conclude that the insufficient evidence to conclude that the temperature variance is unacceptable (>.5). temperature variance is unacceptable (>.5). Because the p –value >  =.10, we Because the p –value >  =.10, we cannot reject the null hypothesis. cannot reject the null hypothesis. The rejection region for the ThermoRite The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the thermostat example is in the upper tail; thus, the appropriate p -value is less than.90 (  2 = 4.168) appropriate p -value is less than.90 (  2 = 4.168) and greater than.10 (  2 = 14.684). and greater than.10 (  2 = 14.684). Hypothesis Testing About a Population Variance A precise p -value can be found using Minitab or Excel. A precise p -value can be found using Minitab or Excel.

31. 31 Slide © 2008 Thomson South-Western. All Rights Reserved n One-Tailed Test Test Statistic Test Statistic Hypotheses Hypotheses Hypothesis Testing About the Variances of Two Populations Denote the population providing the larger sample variance as population 1.

32. 32 Slide © 2008 Thomson South-Western. All Rights Reserved n One-Tailed Test (continued) Reject H 0 if p -value <  where the value of F  is based on an F distribution with n 1 - 1 (numerator) and n 2 - 1 (denominator) d.f. p -Value approach: Critical value approach: Rejection Rule Rejection Rule Hypothesis Testing About the Variances of Two Populations Reject H 0 if F > F 

33. 33 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Tailed Test Test Statistic Test Statistic Hypotheses Hypotheses Hypothesis Testing About the Variances of Two Populations Denote the population providing the larger sample variance as population 1.

34. 34 Slide © 2008 Thomson South-Western. All Rights Reserved n Two-Tailed Test (continued) Reject H 0 if p -value <  p -Value approach: Critical value approach: Rejection Rule Rejection Rule Hypothesis Testing About the Variances of Two Populations 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.

35. 35 Slide © 2008 Thomson South-Western. All Rights Reserved 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 on the next slide. Hypothesis Testing About the Variances of Two Populations n Example: Buyer’s Digest (C) We will conduct a hypothesis test with  =.10 to see We will conduct a hypothesis test with  =.10 to see if the variances are equal for ThermoRite’s thermostats and TempKing’s thermostats.

36. 36 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis Testing About the Variances of Two Populations n Example: Buyer’s Digest (C) ThermoRite Sample TempKing Sample Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2 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 Thermostat 1 2 3 4 5 6 7 8 9 10

37. 37 Slide © 2008 Thomson South-Western. All Rights Reserved n Hypotheses Hypothesis Testing About the Variances of Two Populations Reject H 0 if F > 3.18 The F distribution table (on next slide) shows that with with  =.10, 9 d.f. (numerator), and 9 d.f. (denominator), F.05 = 3.18. (Their variances are not equal) (TempKing and ThermoRite thermostats have the same temperature variance) n Rejection Rule

38. 38 Slide © 2008 Thomson South-Western. All Rights Reserved Selected Values from the F Distribution Table Hypothesis Testing About the Variances of Two Populations

39. 39 Slide © 2008 Thomson South-Western. All Rights Reserved n Test Statistic Hypothesis Testing About the Variances of Two Populations We cannot reject H 0. F = 2.53 < F.05 = 3.18. There is insufficient evidence to conclude that the population variances differ for the two thermostat brands. Conclusion Conclusion = 1.768/.700 = 2.53 TempKing’s sample variance is 1.768 ThermoRite’s sample variance is.700

40. 40 Slide © 2008 Thomson South-Western. All Rights Reserved n Determining and Using the p -Value Hypothesis Testing About the Variances of Two Populations Because  =.10, we have p -value >  and therefore Because  =.10, we have p -value >  and therefore we cannot reject the null hypothesis. we cannot reject the null hypothesis. But this is a two-tailed test; after doubling the upper- But this is a two-tailed test; after doubling the upper- tail area, the p -value is between.20 and.10. (A precise tail area, the p -value is between.20 and.10. (A precise p -value can be found using Minitab or Excel.) p -value can be found using Minitab or Excel.) Because F = 2.53 is between 2.44 and 3.18, the area Because F = 2.53 is between 2.44 and 3.18, the area in the upper tail of the distribution is between.10 in the upper tail of the distribution is between.10 and.05. and.05. Area in Upper Tail.10.05.025.01 F Value (df 1 = 9, df 2 = 9) 2.44 3.18 4.03 5.35

41. 41 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 11