Let and S2Y be the respective sample means and unbiased

Chapter 7, Problem 8E

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QUESTION:

Let \(\bar{X}, \bar{Y}, S_{X}^{2}\), and \(S_{Y}^{2}\) be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\), where \(\mu_X\), \(\mu_{Y}, \sigma_{X}^{2}\), and \(\sigma_{Y}^{2}\) are unknown. If \(\sigma_{X}^{2} / \sigma_{Y}^{2}=d\), a known constant,

(a) Argue that \(\frac{(\bar{X}-\bar{Y})-\left(\mu_{X}-\mu_{Y}\right)}{\sqrt{d \sigma_{Y}^{2} / n+\sigma_{Y}^{2} / m}}\) is N(0, 1).

(b) Argue that \(\frac{(n-1) S_{X}^{2}}{d \sigma_{Y}^{2}}+\frac{(m-1) S_{Y}^{2}}{\sigma_{Y}^{2}}\) is \(\chi^{2}(n+m-2)\).

(c) Argue that the two random variables in (a) and (b) are independent.

(d) With these results, construct a random variable (not depending upon \(\sigma_{Y}^{2}\)) that has a t distribution and that can be used to construct a confidence interval for \(\mu_{X}-\mu_{Y}\).

Questions & Answers

QUESTION:

Let \(\bar{X}, \bar{Y}, S_{X}^{2}\), and \(S_{Y}^{2}\) be the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the normal distributions \(N\left(\mu_{X}, \sigma_{X}^{2}\right)\) and \(N\left(\mu_{Y}, \sigma_{Y}^{2}\right)\), where \(\mu_X\), \(\mu_{Y}, \sigma_{X}^{2}\), and \(\sigma_{Y}^{2}\) are unknown. If \(\sigma_{X}^{2} / \sigma_{Y}^{2}=d\), a known constant,

(a) Argue that \(\frac{(\bar{X}-\bar{Y})-\left(\mu_{X}-\mu_{Y}\right)}{\sqrt{d \sigma_{Y}^{2} / n+\sigma_{Y}^{2} / m}}\) is N(0, 1).

(b) Argue that \(\frac{(n-1) S_{X}^{2}}{d \sigma_{Y}^{2}}+\frac{(m-1) S_{Y}^{2}}{\sigma_{Y}^{2}}\) is \(\chi^{2}(n+m-2)\).

(c) Argue that the two random variables in (a) and (b) are independent.

(d) With these results, construct a random variable (not depending upon \(\sigma_{Y}^{2}\)) that has a t distribution and that can be used to construct a confidence interval for \(\mu_{X}-\mu_{Y}\).

ANSWER:

Step 1 of 5

Given:

,  and  are the respective sample means and unbiased estimates of the variances obtained from independent samples of sizes n and m from the

normal distributions  and , where  and  are unknown.

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