1/71 Statistics Inferences About Population Variances

Contents Inference about a Population Variance n Inferences about the Variances of Two Populations

STATISTICS in PRACTICE The U.S. General Accounting Office (GAO) evaluators studied a Department of Interior program established to help clean up the nation’s rivers and lakes. The audits reviewed sample data on the oxygen content, the pH level, and the amount of suspended solids in the effluent.

STATISTICS in PRACTICE The hypothesis test was conducted about the variance in pH level for the population of effluent. The population variance in pH level expected at a properly functioning plant. In this chapter you will learn how to conduct statistical inferences about the variances of one and two populations.

Inferences About a Population Variance Chi-Square Distribution(  2 ) Interval Estimation of  2 Hypothesis Testing

Chi-Square Distribution The chi-square distribution is based on sampling from a normal population. The chi-square distribution is the sum of squared standardized normal random variables such as

Chi-Square Distribution where Mean: k Variance: 2k n Probability density function

Chi-Square Distribution We can use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance. 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.

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

Chi-Square Distribution n For example, there is a.95 probability of obtaining a (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.

Chi-Square Distribution A Chi-Square Distribution with 19 Degrees of Freedom

Chi-Square Distribution Selected Values form the Chi-Square Distribution Table

Chi-Square Distribution

95% of the possible  2 values 95% of the possible  2 values 22 2 Interval Estimation of  2

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 such that

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.

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.

Buyer’s Digest rates thermostats manufactured for home temperature control. In a recent test, 10 thermostats manufactured by The rmoRite 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 shown on the next slide. Interval Estimation of  2 n Example: Buyer’s Digest (A)

Interval Estimation of  2 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 Thermostat

Selected Values from the Chi-Square Distribution Table Our value For n - 1 = = 9 d.f. and  =.05 Interval Estimation of  2

22 2 Area in Upper Tail = For n - 1 = = 9 d.f. and  =.05 Interval Estimation of  2

Selected Values from the Chi-Square Distribution Table For n - 1 = = 9 d.f. and  =.05 Our value Interval Estimation of  2

22 2 n - 1 = = 9 degrees of freedom and  = Area in Upper Tail =.025 Area in Upper Tail =.025 Interval Estimation 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:

Left-Tailed Test Hypothesis Testing About a Population Variance n Test Statistic n Hypotheses where is the hypothesized value for the population variance

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

n Right-Tailed Test n Test Statistic n Hypotheses where is the hypothesized value for the population variance Hypothesis Testing About a Population Variance

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

n Two-Tailed Test n Test Statistic n Hypotheses where is the hypothesized value for the population variance Hypothesis Testing About a Population Variance

Two-Tailed Test (continued) Reject H 0 if p -value <  p-Value approach: Critical value approach: n Rejection Rule Reject H 0 if where and are based on a chi-square distribution with n - 1 d.f. Hypothesis Testing About a Population Variance

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. n Example: Buyer’s Digest (B) We will conduct a hypothesis test (with  =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. Hypothesis Testing About a Population Variance

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 Thermostat Hypothesis Testing About a Population Variance

Hypotheses Reject H 0 if  2 > n Rejection Rule Hypothesis Testing About a Population Variance

Selected Values from the Chi-Square Distribution Table For n - 1 = = 9 d.f. and  =.10 Our value Hypothesis Testing About a Population Variance

22 2 Area in Upper Tail =.10 Area in Upper Tail =.10 n Rejection Region Reject H 0 Hypothesis Testing About a Population Variance

Test Statistic Because  2 = 12.6 is less than , 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 Hypothesis Testing About a Population Variance

The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p-value is less than.90 (c 2 = 4.168) and greater than.10 (c 2 = ). Using Excel to Conduct a Hypothesis Test about a Population Variance Using the p-Value The sample variance of s 2 =.7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5). Because the p –value > a =.10, we cannot reject the null hypothesis.

Hypothesis Testing About a Population Variance Summary

n One-Tailed Test Test Statistic Hypotheses Hypothesis Testing About the Variances of Two Populations Denote the population providing the larger sample variance as population 1.

Reject H 0 if F > F  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 Hypothesis Testing About the Variances of Two Populations

Two-Tailed Test Test Statistic Hypotheses Denote the population providing the larger sample variance as population 1. Hypothesis Testing About the Variances of Two Populations

Two-Tailed Test (continued) Reject H 0 if p-value <  p-Value approach: Critical value approach : Rejection Rule 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. Hypothesis Testing About the Variances of Two Populations

F Distribution

Probability density function n where

F Distribution n mean for d 2 > 2 n Variance for d 2 > 2

F Distribution with (20, 20) Degrees of Freedom

F Distribution Selected Values From the F Distribution Table

F Distribution Table

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. n Example: Buyer’s Digest (C) Hypothesis Testing About the Variances of Two Populations

n Example: Buyer’s Digest (C) We will conduct a hypothesis test with  =.10 to see if the variances are equal for ThermoRite’s thermostats and TempKing’s thermostats. Hypothesis Testing About the Variances of Two Populations

n Example: Buyer’s Digest (C) ThermoRite Sample TempKing Sample Temperature Thermostat Temperature Thermostat Hypothesis Testing About the Variances of Two Populations

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

Selected Values from the F Distribution Table Hypothesis Testing About the Variances of Two Populations

Test Statistic TempKing’s sample variance is ThermoRite’s sample variance is.700 = 1.768/.700 = 2.53 Hypothesis Testing About the Variances of Two Populations

We cannot reject H 0. F = 2.53 < F.05 = There is insufficient evidence to conclude that the population variances differ for the two thermostat brands. Conclusion Hypothesis Testing About the Variances of Two Populations

Determining and Using the p-Value Because  =.10, we have p-value >  and therefore we cannot reject the null hypothesis. But this is a two-tailed test; after doubling the upper-tail area, the p-value is between.20 and.10. Because F = 2.53 is between 2.44 and 3.18, the area in the upper tail of the distribution is between.10 and.05. Area in Upper Tail F Value (df 1 = 9, df 2 = 9) Hypothesis Testing About the Variances of Two Populations

Summary