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Statistics

Statistics. Inferences About Population Variances. Contents. Inference about a Population Variance. Inferences about the Variances of Two Populations. STATISTICS in PRACTICE. The U.S. General Accounting Office (GAO) evaluators studied a Department of Interior program

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Statistics

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  1. Statistics Inferences About Population Variances

  2. Contents • Inference about a Population Variance • Inferences about the Variances of Two Populations

  3. STATISTICSin PRACTICE • The U.S. General Accounting Office (GAO) evaluators studied a Department of Interior program established to help clean up the nation’s rivers and lakes. • The audits reviewed sample data on the oxygen content, the pH level, and the amount of suspended solids in the effluent.

  4. STATISTICSin PRACTICE • The hypothesis test was conducted about the variance in pH level for the population of effluent. The population variance in pH level expected at a properly functioning plant. • In this chapter you will learn how to conduct statistical inferences about the variances of one and two populations.

  5. Inferences About a Population Variance • Chi-Square Distribution(2) • Interval Estimation of 2 • Hypothesis Testing

  6. Chi-Square Distribution • The chi-square distribution is the sum of • squared standardized normal random • variables such as • The chi-square distribution is based on • samplingfrom a normal population.

  7. Chi-Square Distribution • Probability density function • where • Mean: k • Variance: 2k

  8. Chi-Square Distribution • The sampling distribution of (n - 1)s2/ 2 • has a chi-square distribution whenever a simple • random sample of sizenis selected from a • normal population. • We can use the chi-square distribution to • developinterval estimates and conduct hypothesis • tests about a population variance.

  9. Examples of Sampling Distribution of (n - 1)s2/ 2 With 2 degrees of freedom With 5 degrees of freedom With 10 degrees of freedom 0

  10. 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, there is a .95 probability of obtaining a (chi-square) value such that Chi-Square Distribution

  11. Chi-Square Distribution • A Chi-Square Distribution with 19 Degrees of Freedom

  12. Chi-Square Distribution • Selected Values form the Chi-Square Distribution Table

  13. Chi-Square Distribution

  14. Interval Estimation of 2 .025 .025 95% of the possible 2 values 2 0

  15. Interval Estimation of 2 • There is a (1 –a) probability of obtaining a c2value such that • Substituting (n– 1)s2/s 2 for the c2 we get • Performing algebraic manipulation we get

  16. Interval Estimation of 2 • Interval Estimate of a Population Variance where thevalues are based on a chi-square distribution withn - 1 degrees of freedom and where 1 -is the confidence coefficient.

  17. 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 confidenceinterval for the population standard deviation.

  18. 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 The rmoRitewere 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.

  19. 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

  20. 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

  21. Interval Estimation of 2 For n - 1 = 10 - 1 = 9 d.f. and a =.05 .025 Area in Upper Tail = .975 2 0 2.700

  22. 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

  23. Interval Estimation of 2 n - 1 = 10 - 1 = 9 degrees of freedom and a = .05 Area in Upper Tail = .025 .025 2 0 19.023 2.700

  24. 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

  25. whereis the hypothesized value for the population variance Hypothesis TestingAbout a Population Variance • Left-Tailed Test • Hypotheses • Test Statistic

  26. RejectH0if whereis based on a chi-square distribution withn- 1 d.f. Hypothesis TestingAbout a Population Variance • Left-Tailed Test (continued) • Rejection Rule Critical value approach: p-Value approach: Reject H0if p-value<a

  27. whereis the hypothesized value for the population variance Hypothesis TestingAbout a Population Variance • Right-Tailed Test • Hypotheses • Test Statistic

  28. RejectH0if whereis based on a chi-square distribution withn - 1 d.f. Hypothesis TestingAbout a Population Variance • Right-Tailed Test (continued) • Rejection Rule Critical value approach: RejectH0if p-value<a p-Value approach:

  29. whereis the hypothesized value for the population variance Hypothesis TestingAbout a Population Variance • Two-Tailed Test • Hypotheses • Test Statistic

  30. RejectH0if whereandare based on a chi-square distribution withn - 1 d.f. Hypothesis TestingAbout a Population Variance • Two-Tailed Test (continued) • Rejection Rule Critical value approach: p-Valueapproach: RejectH0ifp-value<a

  31. 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 thermo- stat 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”.

  32. Hypothesis TestingAbout a Population Variance • Example: Buyer’s Digest (B) Using the 10 readings, we will conduct a hypothesis test (witha= .10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”. Thermostat1 2 3 4 5 6 7 8 9 10 Temperature67.4 67.8 68.2 69.3 69.5 67.0 68.1 68.6 67.9 67.2

  33. Hypothesis TestingAbout a Population Variance • Hypotheses • Rejection Rule RejectH0ifc 2>14.684

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

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

  36. Hypothesis TestingAbout a Population Variance • Test Statistic The sample variances2= 0.7 • Conclusion Becausec2= 12.6 is less than 14.684, we cannotrejectH0. The sample variances2= .7 is insufficientevidence to conclude that the temperature variancefor ThermoRitethermostats is unacceptable.

  37. Using Excel to Conduct a Hypothesis Testabout a Population Variance • Using the p-Value • The rejection region for the ThermoRite • thermostat example is in the upper tail; thus, the • appropriate p-value is less than .90 (c2 = 4.168) • and greater than .10 (c2 = 14.684). • Because the p –value > a = .10, we cannot • reject the null hypothesis. • The sample variance of s2 = .7 is insufficient • evidence to conclude that the temperature • variance is unacceptable (>.5).

  38. Hypothesis TestingAbout a Population Variance • Summary

  39. Hypothesis Testing About theVariances of Two Populations • One-Tailed Test • Hypotheses Denote the population providing the larger sample variance as population 1. • Test Statistic

  40. Hypothesis Testing About theVariances of Two Populations • One-Tailed Test (continued) • Rejection Rule Critical value approach: Reject H0if F>F where the value ofFis based on an Fdistribution withn1- 1 (numerator) andn2 - 1 (denominator) d.f. p-Valueapproach: RejectH0 ifp-value<a

  41. Hypothesis Testing About theVariances of Two Populations • Two-Tailed Test • Hypotheses Denote the population providing the larger sample variance as population 1. • Test Statistic

  42. Hypothesis Testing About theVariances of Two Populations • Two-Tailed Test (continued) • Rejection Rule Critical value approach: RejectH0if F>F/2 where the value ofF/2 is based on an Fdistribution withn1- 1 (numerator) andn2 - 1 (denominator) d.f. p-Valueapproach: RejectH0ifp-value<a

  43. F Distribution

  44. F Distribution • Probability density function • where

  45. F Distribution • mean • for d2 > 2 • Variance • ford2 > 2

  46. F Distribution with (20, 20) Degrees of Freedom

  47. F Distribution • Selected Values From the F Distribution Table

  48. F Distribution Table

  49. Hypothesis Testing About theVariances of Two Populations • Example: Buyer’s Digest (C) 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 on the next slide.

  50. Hypothesis Testing About theVariances of Two Populations • Example: Buyer’s Digest (C) We will conduct a hypothesis test with= .10 to see if the variances are equal for ThermoRite’s thermostats and TempKing’s thermostats.

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