A factory operates with two machines of type A and one

Chapter 8, Problem 93E

(choose chapter or problem)

A factory operates with two machines of type A and one machine of type B. The weekly repair costs \( X\) for type A machines are normally distributed with mean \(\mu_{1}\) and variance \(\sigma^{2}\) The weekly repair costs \(Y\) for machines of type  are also normally distributed but with mean  \(\mu_{2}\) and variance \(3 \sigma^{2}\). The expected repair cost per week for the factory is thus  \(2 \mu_{1}+\mu_{2}\). If you are given a random sample \(X_{1}, X_{2}, \ldots, X_{n}\) on costs of type A machines and an independent random sample Y_{1}, Y_{2}, \ldots, Y_{m} on costs for type  machines, show how you would construct a  confidence interval for \(2 \mu_{1}+\mu_{2}\)

a if \(\sigma^{2}\) is known.

b if \(\sigma^{2}\) is not known.

Equation Transcription:

 

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Text Transcription:

Y_,Y_2,...,Y_m

 X

mu_1

sigma^2

Y

mu_2

3sigma^2.

2mu_1+mu_2

X_1,X_2,...,X_n