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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 a to the right of the stated value. For example, Chi-squared value with 5 degrees of freedom (df) at a =0.05 is

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**Interval Estimation of 2**

.05 95% of the possible 2 values 2 = 11.07

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**Interval Estimation of 2**

Interval Estimate of a Population Variance 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 limits of the variance interval provides the confidence interval for the population standard deviation.

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**Interval Estimation of 2**

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 68oF. The temperature readings of the ten thermostats are shown on the next slide.

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**Interval Estimation of 2**

Example: Buyer’s Digest (A) We will use the 10 readings below to develop a 95% confidence interval estimate of the population variance. Thermostat Temperature

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**Selected Values from the Chi-Square Distribution Table**

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

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**Interval Estimation of 2**

Sample variance s2 provides a point estimate of 2. A 95% confidence interval for the population variance is given by: .33 < 2 < 2.33

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**Hypothesis Testing about a Population Variance**

Left-Tailed Test Hypotheses where is the hypothesized value for the population variance Test Statistic

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**Hypothesis Testing About a Population Variance**

Left-Tailed Test (continued) Rejection Rule Critical value approach: Reject H0 if p-Value approach: Reject H0 if p-value < a where is based on a chi-square distribution with n - 1 d.f.

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**Hypothesis Testing About a Population Variance**

Right-Tailed Test Hypotheses where is the hypothesized value for the population variance Test Statistic

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**Hypothesis Testing About a Population Variance**

Right-Tailed Test (continued) Rejection Rule Critical value approach: Reject H0 if p-Value approach: Reject H0 if p-value < a where is based on a chi-square distribution with n - 1 d.f.

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**Hypothesis Testing About a Population Variance**

Two-Tailed Test Hypotheses where is the hypothesized value for the population variance Test Statistic

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**Hypothesis Testing About a Population Variance**

Two-Tailed Test (continued) Rejection Rule Critical value approach: Reject H0 if p-Value approach: Reject H0 if p-value < a where are based on a chi-square distribution with n - 1 d.f.

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**Hypothesis Testing About a Population Variance**

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. We will conduct a hypothesis test (with a = .10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”.

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**Hypothesis Testing About a Population Variance**

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

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**Hypothesis Testing About a Population Variance**

Hypotheses Rejection Rule Reject H0 if c 2 >

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**Selected Values from the Chi-Square Distribution Table**

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

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**Hypothesis Testing About a Population Variance**

Rejection Region Area in Upper Tail = .10 2 14.684 Reject H0

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**Hypothesis Testing About a Population Variance**

The sample variance s 2 = 0.7 Test Statistic Conclusion Because c2 = 12.6 is less than , we cannot reject H0. The sample variance s2 = .7 is insufficient evidence to conclude that the temperature variance for ThermoRite thermostats is unacceptable.

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