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.

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.