Refer to Exercise 7.60. Suppose that n1 = n2 = n, and find

Chapter 7, Problem 61E

(choose chapter or problem)

The result in Exercise 7.58 holds even if the sample sizes differ. That is, if

\(X_{1}, X_{2}, \ldots, X_{n 1}\) and

 \(Y_{1}, Y_{2}, \ldots, Y_{n 2}\) constitute independent random samples from populations with means \(\mu_{1} \text { and } \mu_{2}\) and variances \(\sigma_{1}^{2} \text { and } \sigma_{2}^{2}\), respectively, then \(\bar{X}-\bar{Y}\) will be approximately normally distributed, for large \(n_{1} \text { and } n_{2}\), with mean \(\mu_{1}-\mu_{2}\)  and variance

\(\left(\sigma_{1}^{2} / n_{1}\right)+\left(\sigma_{2}^{2} / n_{2}\right)\)

The flow of water through soil depends on, among other things, the porosity (volume proportion of voids) of the soil. To compare two types of sandy soil, \(\mathrm{n} 1=50\) measurements are to be taken on the porosity of soil A and \(n_{2}=100\) measurements are to be taken on soil B.

Assume that \(\sigma_{1}^{2}=.01 \text { and } \sigma_{2}^{2}=.02\). Find the probability that the difference between the sample means will be within .05 unit of the difference between the population means \(\mu_{1}-\mu_{2}\)

Equation Transcription:

 

 

 

Text Transcription:

X1, X2,...,Xn1

Y1, Y2,...,Yn2

\mu_1 and \mu_2

\sigma_1^2 and  \sigma_2^2

\bar X-\bar Y

n_1 and  n_2

\mu_1-\mu_2

(\sigma_1^2 / n_1)+(\sigma_2^2 / n_2

n 1=50

n_2=100

\sigma_1^2=.01  and \sigma_2^2=.02

\mu_1-\mu_2