Chapter 11 - 1 Spring 2011 Dr Kameli a Petrova School of...

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Unformatted text preview: 1 Spring 2011 Dr. Kameli a 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. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 2 Inferences About Population Variances s Inference about a Population Variance s Inferences about the Variances of Two Populations Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 3 Inferences About a Population Variance s Chi-Square Distribution s Interval Estimation of σ 2 s Hypothesis Testing Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 4 Characteristics of The Chi-Square Distribution s 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. s Based on sampling from a normal population. s 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. s Use the chi-square distribution to develop interval estimates and conduct hypothesis tests about a population variance. Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 5 Examples of Sampling Distribution of (n - 1)s 2 / σ 2 With 2 degrees With 2 degrees of freedom of freedom With 2 degrees With 2 degrees of freedom of freedom 2 2 ( 1) n s σ- With 5 degrees With 5 degrees of freedom of freedom With 5 degrees With 5 degrees of freedom of freedom With 10 degrees With 10 degrees of freedom of freedom With 10 degrees With 10 degrees of freedom of freedom Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 6 2 2 2 .975 .025 χ χ χ ≤ ≤ Chi-Square Distribution s For example, there is a .95 probability of obtaining a χ 2 (chi-square) value such that s 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. 2 Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 7 95% of the possible χ 2 values 95% of the possible χ 2 values χ 2 χ 2 .025 .025 2 .025 χ .025 .025 2 .975 χ Interval Estimation of σ 2 2 2 2 .975 .025 2 ( 1) n s χ χ σ- ≤ ≤ Spring 2011 Dr. Kameli a 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 s n s- ≤ ≤-- 1 1 2 2 2 2 2 1 2 2 χ σ χ α α 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-- ≤ ≤ s Substituting ( n – 1) s 2 / σ 2 for the χ 2 we get s Performing algebraic manipulation we get s There is a (1 – α ) probability of obtaining a χ 2 value such that Spring 2011 Dr. Kameli a Petrova School of Business and Economics SUNY Platsburgh Slide 9 s Interval Estimate of a Population Variance...
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This note was uploaded on 06/08/2011 for the course BUS STAT 362 taught by Professor - during the Spring '11 term at SUNY Plattsburgh.

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Chapter 11 - 1 Spring 2011 Dr Kameli a Petrova School of...

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