The result in Exercise 7.58 holds even if the sample sizes

Chapter 7, Problem 60E

(choose chapter or problem)

Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) and \(Y_{1}, Y_{2}, \ldots, Y_{n}\) are independent random samples from populations with means \(\mu_{1} \text { and } \mu_{2}\) and variances  \(\sigma_{1}^{2} \text { and } \sigma_{2}^{2}\) , respectively. Show that the random variable

                                \(U_{n}=\frac{(\bar{X}-\bar{Y})-\left(\mu_{1}-\mu_{2}\right)}{\sqrt{\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right) / n}}\)

satisfies the conditions of Theorem 7.4 and thus that the distribution function of Un converges

to a standard normal distribution function as \(n \rightarrow \infty\). [Hint: Consider

\(W_{i}=X_{i}-Y_{i}\) , for \(i=1,2, \ldots, n .\).]

Equation Transcription:

 

 

   

Text Transcription:

x1, x2,...,xn

y1, y2,...,yn

\mu_1 and \mu_2

\sigma_1^2 and \sigma_2^2

U_n=(\bar X-\bar Y)-(\mu_1-\mu_2\sqrt(\sigma_1^2+\sigma_2^2 / n

n \rightarrow \infty

Wi=Xi-Yi    

i=1,2...,n