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Let X11, X12, Xin, and X21, X22, X2n₂ be two independent random samples of size ₁ and ₂ from two normal populations N(μ₁, of) and N(2, 2) respectively. 1 Xij, S² = Σ(Xij-X₁)² for i = 1, 2. n₂ Let X₁ = = n₁ (a) Using an appropriate statistic and a parameter from the above specifications, define a function/quantity that follows a chi-square distribution with (n₁-1) degrees of freedom. (b) Argue that the above chi-square statistic is a pivotal quantity and can be used to define a (1-a) x 100% confidence interval for the relevant population variance. Derive the confidence interval for the population variance, clearly

Let X11, X12, Xin, and X21, X22, X2n₂ be two independent random samples of size ₁ and ₂ from two normal populations N(μ₁, of) and N(2, 2) respectively. 1 Xij, S² = Σ(Xij-X₁)² for i = 1, 2. n₂ Let X₁ = = n₁ (a) Using an appropriate statistic and a parameter from the above specifications, define a function/quantity that follows a chi-square distribution with (n₁-1) degrees of freedom. (b) Argue that the above chi-square statistic is a pivotal quantity and can be used to define a (1-a) x 100% confidence interval for the relevant population variance. Derive the confidence interval for the population variance, clearly

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Let X11, X12,..., Xin, and X21, X22, X2n₂ be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 0) and N(2, 2) respectively. n₂ Let X₂ Xij, S² = Σ(Xij- Xi)² for i=1,2. j=1 n₂ j=1 ni - (a) Using an appropriate statistic and a parameter from the above specifications, define a function/quantity that follows a chi-square distribution with (n₁ - 1) degrees of freedom. (b) Argue that the above chi-square statistic is a pivotal quantity and can be used to define a (1-a) x 100% confidence interval for the relevant population variance. Derive the confidence interval for the population variance, clearly stating its limits.
Transcribed Image Text:Let X11, X12,..., Xin, and X21, X22, X2n₂ be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 0) and N(2, 2) respectively. n₂ Let X₂ Xij, S² = Σ(Xij- Xi)² for i=1,2. j=1 n₂ j=1 ni - (a) Using an appropriate statistic and a parameter from the above specifications, define a function/quantity that follows a chi-square distribution with (n₁ - 1) degrees of freedom. (b) Argue that the above chi-square statistic is a pivotal quantity and can be used to define a (1-a) x 100% confidence interval for the relevant population variance. Derive the confidence interval for the population variance, clearly stating its limits.
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