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This problem explores the question of estimating (> 0) using a confidence interval based on a random sample X₁, X2, ..., Xn from the distribution with pdf fo(x) = { 5 exp(-) if=> 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating & using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a) % confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and x₁ = 1. : 1 the (e) Is the 95% confidence interval for o you obtained in (d) when n = = 1 and 21 shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.

This problem explores the question of estimating (> 0) using a confidence interval based on a random sample X₁, X2, ..., Xn from the distribution with pdf fo(x) = { 5 exp(-) if=> 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating & using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a) % confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and x₁ = 1. : 1 the (e) Is the 95% confidence interval for o you obtained in (d) when n = = 1 and 21 shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter2: Exponential, Logarithmic, And Trigonometric Functions
Section2.CR: Chapter 2 Review
Problem 111CR: Respiratory Rate Researchers have found that the 95 th percentile the value at which 95% of the data...
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This problem explores the question of estimating o(> 0) using a confidence interval based on a random sample X1, X2, ..., Xn from the distribution with pdf fo(x) = {& exp(- 20 if x > 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a)% confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and ₁ = 1. (e) Is the 95% confidence interval for a you obtained in (d) when n = 1 and ₁ = 1 the shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.
Transcribed Image Text:This problem explores the question of estimating o(> 0) using a confidence interval based on a random sample X1, X2, ..., Xn from the distribution with pdf fo(x) = {& exp(- 20 if x > 0 otherwise. (a) Find the pdf of X2 by finding the cdf and differentiating it. (b) Using the result of (a) and techniques we developed in class, propose a pivot for estimating using a confidence interval and find the distribution of your pivot. Review Slides 38-41 of the power points for Chapter 7.1 to remind yourself how a pivot is used to construct a confidence interval. (c) Using the pivot obtained in (b) and the fact that o> 0, write down the formula for finding a short 100(1-a)% confidence interval for o. (d) Using the formula obtained in (c), compute the width of the resulting 95% confidence interval when n = 1 and ₁ = 1. (e) Is the 95% confidence interval for a you obtained in (d) when n = 1 and ₁ = 1 the shortest possible? If your answer is yes, explain why. If the answer is no, give a counterexample.
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