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IS 310 Business Statistics CSU Long Beach

IS 310 Business Statistics CSU Long Beach. Inferences About Population Variances. Previously, we have made inferences about population means and proportions . We can make similar inferences about population variances . Examples:

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IS 310 Business Statistics CSU Long Beach

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  1. IS 310 Business Statistics CSU Long Beach

  2. Inferences About Population Variances Previously, we have made inferences about population means and proportions. We can make similar inferences about population variances. Examples: A manufacturing plant produces a part whose length should be 1.2”. Some variation of length is allowed. We want to determine that the variation of length is no more than 0.05”. The amount of rainfall varies from state to state. A researcher wants to know if the variation in the amount of rainfall in a specific state is less than 3”. In these cases, we are dealing with population variances (not means or proportions)

  3. Inferences About Population Variances • Inference about a Population Variance

  4. Inferences About a Population Variance • Chi-Square Distribution • Interval Estimation • Hypothesis Testing

  5. Inferences About a Population Variance In studying population variances, the following quantity is important 2 2 (n – 1) s / σ This quantity follows a distribution that is called chi-square distribution (with n – 1 degree of freedom) As the degree of freedom increases with sample size, the shape of the chi-square distribution resembles that of a normal distribution. See Figure 11.1 (10-page 437; 11-page 450)

  6. 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

  7. Chi-Square Distribution The chi-square distribution can be used o To estimate a population variance o Perform hypothesis tests about population variances

  8. 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. Chi-Square Distribution • For example, there is a .95 probability of obtaining a c2 (chi-square) value such that

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

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

  11. 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 where 1 -  is the confidence coefficient.

  12. 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.

  13. Interval Estimation of 2 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. • Example: Buyer’s Digest (A)

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

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

  21. 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:

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

  23. 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:

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

  25. 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

  26. Hypothesis TestingAbout a Population Variance 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. • Example: Buyer’s Digest (B) We will conduct a hypothesis test (with a = .10) to determine whether the ThermoRite thermostat’s temperature variance is “acceptable”.

  27. 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

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

  29. 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

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

  31. Hypothesis TestingAbout a Population Variance • Test Statistic The sample variance s 2 = 0.7 • 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.

  32. Hypothesis TestingAbout 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 (c 2 = 4.168) • and greater than .10 (c 2 = 14.684). A precise p-value can be found using Minitab or Excel. • Because the p –value > a = .10, we • cannot reject the null hypothesis. • The sample variance of s 2 = .7 is • insufficient evidence to conclude that the • temperature variance is unacceptable (>.5).

  33. Sample Problem Problem # 4 (10-Page 443; 11-Page 457) Given: 2 n = 18 s = 0.36 The 90% confidence interval estimate for population variance is: 2 2 2 2 2 (n – 1) s / Χ ≤ σ ≤ (n – 1) s /Χ .05 .95 [(18 – 1) (0.36)/27.587 < < [(18 – 1) (0.36)/8.672)

  34. Sample Problem Problem # 9 (10-Page 445; 11-Page 459) 2 Given: n = 30 s = 0.0005  = 0.05 2 2 H : σ ≤ 0.0004 H : σ > 0.0004 0 a 2 2 2 Test statistic, Χ = (n – 1) s / σ = (29) (.0005) / .0004 = 36.25 0 2 2 Reject Null Hypothesis if Χ > Χ 2 2 0.05, 29 Since Χ (=36.25) < Χ (=42.557), we do not reject the Null Hypothesis 0. 05, 29 Test indicates that the population variance specification is not being violated.

  35. End of Chapter 11

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