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Econ 3790: Business and Economic Statistics

Econ 3790: Business and Economic Statistics. Instructor: Yogesh Uppal Email: yuppal@ysu.edu. Chapter 11 Inferences About Population Variances. Inference about a Population Variance. Chi-Square Distribution Interval Estimation of  2 Hypothesis Testing.

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Econ 3790: Business and Economic Statistics

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  1. Econ 3790: Business and Economic Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu

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

  3. 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 11.07. Chi-Square Distribution

  4. Interval Estimation of 2 .05 95% of the possible 2 values 2 0 = 11.07

  5. 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.

  6. 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.

  7. 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.

  8. 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. Thermostat1 2 3 4 5 6 7 8 9 10 Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

  9. Our value Interval Estimation of 2 For n - 1 = 10 - 1 = 9 d.f. and a = .05 Selected Values from the Chi-Square Distribution Table

  10. 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

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

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

  13. where is the hypothesized value for the population variance Hypothesis TestingAbout a Population Variance • Right-Tailed Test • Hypotheses • Test Statistic

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

  15. where is the hypothesized value for the population variance Hypothesis TestingAbout a Population Variance • Two-Tailed Test • Hypotheses • Test Statistic

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

  17. Hypothesis TestingAbout 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”.

  18. Hypothesis TestingAbout 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”. Thermostat1 2 3 4 5 6 7 8 9 10 Temperature 67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

  19. Hypothesis TestingAbout a Population Variance • Hypotheses • Rejection Rule Reject H0 if c 2> 14.684

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

  21. Hypothesis TestingAbout a Population Variance • Rejection Region Area in Upper Tail = .10 2 14.684 0 Reject H0

  22. Hypothesis TestingAbout a Population Variance The sample variance s 2 = 0.7 • Test Statistic • Conclusion Because c2 = 12.6 is less than 14.684, 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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