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Chapter 11 - Inferences About Population Variances Business...

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1 Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 1 Business Statistics II ECO 362: Sections A & B Chapter 11 Inferences About Population Variances Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 2 Inferences About Population Variances square6 Inference about a Population Variance square6 Inferences about the Variances of Two Populations Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 3 Inferences About a Population Variance square6 Chi-Square Distribution square6 Interval Estimation of σ 2 square6 Hypothesis Testing Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 4 Characteristics of The Chi-Square Distribution square6 A random variable that has a chi-square distribution is the sum of squared standardized normal random variables such as (z1) 2 +(z2) 2 +(z3) 2 etc. square6 Based on sampling from a normal population. square6 The sampling distribution of ( n - 1) s 2 / σ 2 has a chi- square distribution whenever a simple random sample of size n is selected from a normal population. square6 Use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance. Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 5 Examples of Sampling Distribution of (n - 1)s 2 / σ 2 0 With 2 degrees of freedom 2 2 ( 1) n s σ - With 5 degrees of freedom With 10 degrees of freedom Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 6 2 2 2 .975 .025 χ χ χ Chi-Square Distribution square6 For example, there is a .95 probability of obtaining a χ 2 (chi-square) value such that square6 We will use the notation χ α 2 to denote the value for the chi-square distribution that provides an area of α to the right of the stated χ α 2 value.
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2 Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 7 95% of the possible χ 2 values χ 2 0 .025 2 .025 χ .025 2 .975 χ Interval Estimation of σ 2 2 2 2 .975 .025 2 ( 1) n s χ χ σ - Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 8 Interval Estimation of σ 2 ( ) ( ) / ( / ) n s n s - - - 1 1 2 2 2 2 2 1 2 2 χ σ χ α α ) n 2 2 χ σ χ 2 2 2 (1 / 2) / 2 α α χ χ χ - 2 2 2 (1 /2) /2 2 ( 1) n s α α χ χ σ - - square6 Substituting ( n – 1) s 2 / σ 2 for the χ 2 we get square6 Performing algebraic manipulation we get square6 There is a (1 – α ) probability of obtaining a χ 2 value such that Spring 2011 Dr. Kameliia Petrova School of Business and Economics SUNY Platsburgh Slide 9 square6 Interval Estimate of a Population Variance Interval Estimation of σ 2 ( ) ( ) / ( / ) n s n s - - - 1 1 2 2 2 2 2 1 2 2 χ σ χ α α ) / ( / ) s - 2 2 2 1 2 2 σ α α where the χ 2 values are based on a chi-square distribution with n - 1 degrees of freedom and where 1 - α is the confidence coefficient.
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