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If X1,X2, ..., Xn are independent and unbiased

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 23SE Chapter 3

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 23SE

If X1,X2, ..., Xn are independent and unbiased measurements of true values μ1, μ2, ⋯, μn and

U(X1,X2,⋯, Xn) is a nonlinear function of X1, X2,..., Xn then in general U(X1, X2,..., Xn) is a biased estimate of the true value U(μ1, μ2, ⋯, μn). A bias-corrected estimate is

 When air enters a compressor at pressure P1 and leaves at pressure P2, the intermediate pressure is given by  Assume that P1, = 8.1 ± 0.1 MPa and P2 = 15.4 ± 0.2 MPa.

 a. Estimate P3, and find the uncertainty in the estimate, without bias correction.

 b. Compute the bias-corrected estimate of P3.

 c. Compare the difference between the bias- corrected and non-bias-corrected estimates to the uncertainty in the non-bias-corrected estimate. Is bias correction important in this case? Explain.

Step-by-Step Solution:

Solution :

Step 1 of 3:

Let are independent and are unbiased measurement values.

Then a bias-correct estimate is

Where is non linear function .

Given,

A compressor at pressureand

Leaves at pressure .

Then the intermediate pressure is given by

.

We assume that MPa and MPa.

Our goal is :

a). We need to estimate and we have to find uncertainty in the estimate , without bias correction.

b). We need to compute the bias-corrected estimate of .

e). We need to find is bias correction important in this case? Explain.

a).

Now we have to estimate and we have to find uncertainty in the estimate , without bias correction.

Then the intermediate pressure is is

   

We know that and

MPa

Therefore the intermediate pressure is is 11.1687 MPa.

Given MPa

We consider and

And MPa.

We consider and .

Here we are differentiating with respect to .

Here

Then,

 

 

 

 

 

Therefore is 0.6894

Here we are differentiating with respect to .

Here

Then,

 

 

 

 

 

Therefore is 0.3626

0.10

Therefore MPa.


Step 2 of 3

Chapter 3, Problem 23SE is Solved
Step 3 of 3

Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. The full step-by-step solution to problem: 23SE from chapter: 3 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. This full solution covers the following key subjects: bias, estimate, corrected, pressure, uncertainty. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. Since the solution to 23SE from 3 chapter was answered, more than 280 students have viewed the full step-by-step answer. The answer to “If X1,X2, ..., Xn are independent and unbiased measurements of true values ?1, ?2, ?, ?n andU(X1,X2,?, Xn) is a nonlinear function of X1, X2,..., Xn then in general U(X1, X2,..., Xn) is a biased estimate of the true value U(?1, ?2, ?, ?n). A bias-corrected estimate is When air enters a compressor at pressure P1 and leaves at pressure P2, the intermediate pressure is given by Assume that P1, = 8.1 ± 0.1 MPa and P2 = 15.4 ± 0.2 MPa. a. Estimate P3, and find the uncertainty in the estimate, without bias correction.________________ b. Compute the bias-corrected estimate of P3.________________ c. Compare the difference between the bias- corrected and non-bias-corrected estimates to the uncertainty in the non-bias-corrected estimate. Is bias correction important in this case? Explain.” is broken down into a number of easy to follow steps, and 128 words.

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If X1,X2, ..., Xn are independent and unbiased