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1 1 Slide © 2003 South-Western /Thomson Learning™ Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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2 2 Slide © 2003 South-Western /Thomson Learning™ Chapter 11 Inferences about Population Variances n Inference about a Population Variance n Inferences about the Variances of Two Populations

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3 3 Slide © 2003 South-Western /Thomson Learning™ Inferences about a Population Variance n Chi-Square Distribution Interval Estimation of 2 Interval Estimation of 2 n Hypothesis Testing

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4 4 Slide © 2003 South-Western /Thomson Learning™ 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.

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5 5 Slide © 2003 South-Western /Thomson Learning™ 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.

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6 6 Slide © 2003 South-Western /Thomson Learning™ 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.

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7 7 Slide © 2003 South-Western /Thomson Learning™ n Chi-Square Distribution With Tail Areas of.025 95% of the possible 2 values 95% of the possible 2 values 22 22 0 0.025 Interval Estimation of 2

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8 8 Slide © 2003 South-Western /Thomson Learning™ 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 Example: Buyer’s Digest (A)

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9 9 Slide © 2003 South-Western /Thomson Learning™ Example: Buyer’s Digest (A) 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 22 22 0 0.025

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10 Slide © 2003 South-Western /Thomson Learning™ 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 22 22 0 0.025 2.70 Example: Buyer’s Digest (A) Area in Upper Tail =.975

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11 Slide © 2003 South-Western /Thomson Learning™ Example: Buyer’s Digest (A) 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 22 22 0 0 Area in Upper Tail =.025 Area in Upper Tail =.025.025 2.70 19.02

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12 Slide © 2003 South-Western /Thomson Learning™ 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 (A)

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13 Slide © 2003 South-Western /Thomson Learning™ Using Excel to Construct an Interval Estimate of a Population Variance n Formula Worksheet

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14 Slide © 2003 South-Western /Thomson Learning™ n Value Worksheet Using Excel to Construct an Interval Estimate of a Population Variance

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15 Slide © 2003 South-Western /Thomson Learning™ n Right-Tailed Test Hypotheses Hypotheses where is the hypothesized value for the population variance Test Statistic Test Statistic Hypothesis Testing about a Population Variance

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16 Slide © 2003 South-Western /Thomson Learning™ Hypothesis Testing about a Population Variance n Right-Tailed Test (continued) Rejection Rule Rejection Rule Using test statistic: Using test statistic: Using p -value: Using p -value: where is based on a chi-square distribution with n - 1 d.f. Reject H 0 if Reject H 0 if p -value <

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17 Slide © 2003 South-Western /Thomson Learning™ n Left-Tailed Test Hypotheses Hypotheses where is the hypothesized value for the population variance Test Statistic Test Statistic Hypothesis Testing about a Population Variance

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18 Slide © 2003 South-Western /Thomson Learning™ n Left-Tailed Test (continued) Rejection Rule Rejection Rule Using test statistic: Using test statistic: Using p -value: Using p -value: where is based on a chi-square distribution with n - 1 d.f. Hypothesis Testing about a Population Variance Reject H 0 if Reject H 0 if p -value < Reject H 0 if p -value <

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19 Slide © 2003 South-Western /Thomson Learning™ n Two-Tailed Test Hypotheses Hypotheses Test Statistic Test Statistic Hypothesis Testing about a Population Variance

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20 Slide © 2003 South-Western /Thomson Learning™ Hypothesis Testing about a Population Variance n Two-Tailed Test (continued) Rejection Rule Rejection Rule Using test statistic: Reject H 0 if Using p -value: Using p -value: Reject H 0 if p -value < Reject H 0 if p -value < where are based on a where are based on a chi-square distribution with n - 1 d.f.

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21 Slide © 2003 South-Western /Thomson Learning™ Example: Buyer’s Digest (B) Buyer’s Digest is rating ThermoRite thermostats made for home temperature control. Buyer’s Digest gives an “acceptable” rating to a thermostat with a temperature variance of 0.5 or less. In a recent test, 10 ThermoRite thermostats were selected and placed in a test room that was maintained at a temperature of 68 o F. The temperature readings of the thermostats are listed below. Using the 10 readings, we will conduct a hypothesis test (with =.05) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. 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

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22 Slide © 2003 South-Western /Thomson Learning™ Example: Buyer’s Digest (B) n Hypotheses n Rejection Rule Reject H 0 if 2 > 14.6837 Reject H 0 if 2 > 14.6837

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23 Slide © 2003 South-Western /Thomson Learning™ Example: Buyer’s Digest (B) n Rejection Region 22 22 0 0.10 14.6837 Reject H 0

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24 Slide © 2003 South-Western /Thomson Learning™ n Test Statistic The sample variance s 2 = 0.7 n Conclusion Because 2 = 12.6 is less than 14.6837, 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. Example: Buyer’s Digest (B)

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25 Slide © 2003 South-Western /Thomson Learning™ Using Excel to Conduct a Hypothesis Test about a Population Variance n Formula Worksheet

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26 Slide © 2003 South-Western /Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about a Population Variance

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27 Slide © 2003 South-Western /Thomson Learning™ Using Excel to Conduct a Hypothesis Test about a Population Variance n Using the p -Value The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p -value is.1816. The rejection region for the ThermoRite thermostat example is in the upper tail; thus, the appropriate p -value is.1816. Because.1816 > =.10, we cannot reject the null hypothesis. Because.1816 > =.10, we cannot reject the null hypothesis. The sample variance of s 2 =.7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5) The sample variance of s 2 =.7 is insufficient evidence to conclude that the temperature variance is unacceptable (>.5)

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28 Slide © 2003 South-Western /Thomson Learning™ n One-Tailed Test Hypotheses Hypotheses Denote the population providing the Denote the population providing the larger sample variance as population 1. larger sample variance as population 1. Test Statistic Test Statistic Hypothesis Testing about the Variances of Two Populations

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29 Slide © 2003 South-Western /Thomson Learning™ n One-Tailed Test (continued) Rejection Rule Rejection Rule Using test statistic: where the value of F is based on an F distribution with n 1 - 1 (numerator) and n 2 - 1 (denominator) d.f. Using p -value: Hypothesis Testing about the Variances of Two Populations Reject H 0 if F > F Reject H 0 if p -value <

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30 Slide © 2003 South-Western /Thomson Learning™ n Two-Tailed Test Hypotheses Hypotheses Denote the population providing the Denote the population providing the larger sample variance as population 1. larger sample variance as population 1. Test Statistic Test Statistic Hypothesis Testing about the Variances of Two Populations

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31 Slide © 2003 South-Western /Thomson Learning™ n Two-Tailed Test (continued) Rejection Rule Rejection Rule Using test statistic: where the value of F /2 is based on an F distribution with n 1 - 1 (numerator) and n 2 - 1 (denominator) d.f. Using p -value: Hypothesis Testing about the Variances of Two Populations Reject H 0 if F > F /2 Reject H 0 if p -value <

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32 Slide © 2003 South-Western /Thomson Learning™ 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 (C)

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33 Slide © 2003 South-Western /Thomson Learning™ n Hypothesis Testing about the Variances of Two Populations Hypotheses Hypotheses (TempKing and ThermoRite thermo- stats have same temperature variance) (TempKing and ThermoRite 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 (C)

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34 Slide © 2003 South-Western /Thomson Learning™ n Hypothesis Testing about the Variances of Two Populations Test Statistic Test Statistic TempKing’s sample variance is 1.52. ThermoRite’s sample variance is.70. = 1.52/.70 = 2.17 = 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 (C)

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35 Slide © 2003 South-Western /Thomson Learning™ n Excel’s F-Test Two-Sample for Variances Tool Step 1: Select the Tools pull-down menu Step 2: Choose the Data Analysis option Step 3: When the Data Analysis dialog box appears: Choose F-Test Two Sample for Variances Click OK … continue Using Excel to Conduct a Hypothesis Test about the Variances of Two Populations

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36 Slide © 2003 South-Western /Thomson Learning™ n Excel’s F-Test Two-Sample for Variances Tool Step 4: When the F-Test Two-Sample for Variances dialog box appears: Enter A1:A11 in the Variable 1 Range box Enter B1:B11 in the Variable 2 Range box Select Labels Enter.05 in the Alpha box Select Output Range Enter C1 in the Output Range box Using Excel to Conduct a Hypothesis Test about the Variances of Two Populations

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37 Slide © 2003 South-Western /Thomson Learning™ n Value Worksheet Using Excel to Conduct a Hypothesis Test about the Variances of Two Populations

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38 Slide © 2003 South-Western /Thomson Learning™ Using Excel to Conduct a Hypothesis Test about the Variances of Two Populations n Determining and Using the p -Value The output labeled P(F<=f) one-tail, 0.1314, can be used to determine the p -value for the hypothesis test. The output labeled P(F<=f) one-tail, 0.1314, can be used to determine the p -value for the hypothesis test. If the thermostat example had been a one-tailed hypothesis test, this would have been the p -value. If the thermostat example had been a one-tailed hypothesis test, this would have been the p -value. Because the thermostat example is a two-tailed test, we must multiply the 0.1314 value by 2 to obtain the correct p -value, 0.2628. Because the thermostat example is a two-tailed test, we must multiply the 0.1314 value by 2 to obtain the correct p -value, 0.2628. Because.2628 > =.10, we cannot reject the null hypothesis. Because.2628 > =.10, we cannot reject the null hypothesis.

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39 Slide © 2003 South-Western /Thomson Learning™ End of Chapter 11

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