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Econ 3790: Business and Economic Statistics Instructor: Yogesh Uppal

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Chapter 11 Inferences About Population Variances Inference about a Population Variance Chi-Square Distribution Interval Estimation of 2 Hypothesis Testing

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Chi-Square Distribution 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. For example, Chi-squared value with 5 degrees of freedom (df) at =0.05 is

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95% of the possible 2 values 95% of the possible 2 values 22 2 Interval Estimation of 2 = 11.07

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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 1 - is the confidence coefficient.

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Interval Estimation of 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.

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Example: Buyer’s Digest (A): 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 shown on the next slide. Interval Estimation of 2

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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. Example: Buyer’s Digest (A) Temperature Thermostat

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

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Sample variance s 2 provides a point estimate of 2. Interval Estimation of 2.33 < 2 < 2.33 A 95% confidence interval for the population variance is given by:

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

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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.

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

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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

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

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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.

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Example: Buyer’s Digest (B): Recall that Buyer’s Digest is rating ThermoRite thermostats. Buyer’s Digest gives an “acceptable” rating to a thermostat with a temperature variance of 0.5 or less. Hypothesis Testing About a Population Variance We will conduct a hypothesis test (with =.10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”.

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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 Using the 10 readings, we will conduct a hypothesis test (with =.10) to determine whether the ThermoRite thermostat’s temperature variance is“acceptable”. Example: Buyer’s Digest (B) Temperature Thermostat

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

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

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22 2 Area in Upper Tail =.10 Area in Upper Tail =.10 Hypothesis Testing About a Population Variance Rejection Region Reject H 0

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Test Statistic Hypothesis Testing About a Population Variance 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. Conclusion The sample variance s 2 = 0.7

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