Suppose that we are to observe two independent random

Chapter 5, Problem 161SE

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QUESTION:

Suppose that we are to observe two independent random samples:\(Y_{1}, Y_{2} \ldots \ldots . Y_{n}\) denoting a random sample from a normal đistribution with mean \(\mu_{1}\) and variance \(\sigma_{1}^{2}\); and \(X_{1}, X_{2}, \ldots, X_{m}\) denoting a random sample from another normal distribution with mean \(\mu_{2}\) and variance \(\sigma_{2}^{2}\) An approximation for \(\mu_{1}-\mu_{2}\) is given by \(Y^{-}-X^{-}\), the difference between the sample means. Find \(E\left(Y^{-}-X^{-}\right)\) and \(V\left(Y^{-}-X^{-}\right)\).

Equation Transcription:

Text Transcription:

Y_1,Y_2........Y_n

mu1

sigma_1^2

X_1,X_2,...,X_m

sigma_2^2

mu_1-mu_2

Y^- - X^-

E(Y^- - X^-)

V(Y^- - X^-)

Questions & Answers

QUESTION:

Suppose that we are to observe two independent random samples:\(Y_{1}, Y_{2} \ldots \ldots . Y_{n}\) denoting a random sample from a normal đistribution with mean \(\mu_{1}\) and variance \(\sigma_{1}^{2}\); and \(X_{1}, X_{2}, \ldots, X_{m}\) denoting a random sample from another normal distribution with mean \(\mu_{2}\) and variance \(\sigma_{2}^{2}\) An approximation for \(\mu_{1}-\mu_{2}\) is given by \(Y^{-}-X^{-}\), the difference between the sample means. Find \(E\left(Y^{-}-X^{-}\right)\) and \(V\left(Y^{-}-X^{-}\right)\).

Equation Transcription:

Text Transcription:

Y_1,Y_2........Y_n

mu1

sigma_1^2

X_1,X_2,...,X_m

sigma_2^2

mu_1-mu_2

Y^- - X^-

E(Y^- - X^-)

V(Y^- - X^-)

ANSWER:

Answer:

Step 1 of 1:

Suppose that we are to observe two independent random samples:  denote a random sample from a normal distribution with mean  and variance  and  denoting a random sample from another normal distribution with mean  and variance

An approximation for  is given by , the difference between the sample means.

We need to find the value of

We can write  as,

……(1)

The mean of a sample of measured responses  is given by

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