1 1 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

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Inferences About a Population Variance n Chi-Square Distribution n Interval Estimation n Hypothesis Testing

3 3 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.

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Examples of Sampling Distribution of ( n - 1) s 2 /  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

5 5 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.

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved 95% of the possible  2 values 95% of the possible  2 values 22 2 Interval Estimation of  2

7 7 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

8 8 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.

9 9 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.

10 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

11 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.

12 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

13 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

14 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

15 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.

16 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.

17 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 (numerator) and n (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 

18 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.

19 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 (numerator) and n (denominator) d.f.