231hw8solutionsF11

231hw8solutionsF11 - Will Landau October 3 2011 STAT 231...

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Unformatted text preview: Will Landau October 3, 2011 STAT 231 Problem Set 8 Solutions Exercise 8.1. I begin with the following table from homework 7, where X1,...,X5, Xnew are standard normal random variables. If you don’t know how to generate this table, see problem 5 of the posted solutions for homework 7. For each row, we want to use our sample of n = 5 standard normal draws draws (X1, ... X5) to create a tolerance interval that captures 99% of the sampling distribution with 95% confidence. From page 291 in Devore, this interval is of the form, X ± T · σ , where X = X 1 + X 2 + X 3 + X 4 + X 5 5 , σ = q ∑ 5 i =1 ( X i- X ) 2 5- 1 , and T is the desired tolerance critical value. To find the tolerance critical value, we look at table A.6 and use sample size n = 5, confidence level = 95%,and % of population captured ≥ 99%. We find that T = 6 . 634, which agrees with what Prof. Vardeman wrote in the homework set. With the above figured out, we go back to our JMP table and make two more columns: Ltol, the column containing the lower tolerance limits of all the rows, and Utol, the column containing the upper tolerance limits of all the rows: 1 For Ltol, I use the formula: ”Mean() - 6.634*(Std Dev(X1,X2,X3,X4,X5))” For Utol, I use the formula: ”Mean() + 6.634*(Std Dev(X1,X2,X3,X4,X5))” More specifically, I went through these steps for Ltol: And similarly for Utol, yielding the following table: 2 But we’re not done. Next, we need to know what fraction of the standard normal density is covered by each of the tolerance intervals: i.e.,normal density is covered by each of the tolerance intervals: i....
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231hw8solutionsF11 - Will Landau October 3 2011 STAT 231...

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