1 / 22

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.

Download Presentation

Econ 3790: Business and Economic Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related