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Statistics / Mathematical Statistics with Applications 7 / Chapter 5 / Problem 161SE

Mathematical Statistics with Applications | 7th Edition | ISBN: 9780495110811 | Authors: Dennis Wackerly; William Mendenhall; Richard L. Scheaffer

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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:

${Y}_{1},{Y}_{2}........{Y}_{n}$

${\mu{}}_{1}$

${\sigma{}}_{1}^{2}$

${X}_{1},{X}_{2},...,{X}_{m}$

${\sigma{}}_{2}^{2}$

${\mu{}}_{1}-{\mu{}}_{2}$

${Y}^{-}-{X}^{-}$

$E({Y}^{-}-{X}^{-})$

$V({Y}^{-}-{X}^{-})$

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:

${Y}_{1},{Y}_{2}........{Y}_{n}$

${\mu{}}_{1}$

${\sigma{}}_{1}^{2}$

${X}_{1},{X}_{2},...,{X}_{m}$

${\sigma{}}_{2}^{2}$

${\mu{}}_{1}-{\mu{}}_{2}$

${Y}^{-}-{X}^{-}$

$E({Y}^{-}-{X}^{-})$

$V({Y}^{-}-{X}^{-})$

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: ${Y}_{1},\ {Y}_{2},.......,{Y}_{n}$ denote a random sample from a normal distribution with mean ${\mu{}}_{1}$ and variance ${\sigma{}}_{1}^{2};$ and ${X}_{1},\ {X}_{2},.......,{X}_{n}$ 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 $\overline{Y}-\overline{X}$ , the difference between the sample means.

We need to find the value of $E(\overline{Y}-\overline{X})\ and\ V(\overline{Y}-\overline{X}).$

We can write $E(\overline{Y}-\overline{X})$ as,

$E(\overline{Y}-\overline{X})\ =E(\overline{Y})\ -E(\overline{X})$ ……(1)

The mean of a sample of $n\$ measured responses ${Y}_{1},\ {Y}_{2},.......,{Y}_{n}$ is given by

$\overline{Y}\ =\frac{1}{n}\sum\limits_{i\ =1}^{n}{Y}_{i}$

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