Question: Let \(X_{1}, X_{2}, \ldots, X_{m}\) be an independent random sample of size \(\mathrm{m}\) from \(N\left(\mu_{1}, \sigma_{1}\right)\), and let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) be another independent random sample of size n from \(N\left(\mu_{2}, \sigma_{2}\right)\) and The two samples are independent. The difference between the sample means, \(\bar{X}-\bar{Y}\), is then a linear combination of \(\mathrm{m}+\mathrm{n}\) normal random variables.

1. Find \(E(\bar{X}-\bar{Y})\)
2. Find \(\operatorname{Var}(\bar{X}-\bar{Y})\)
3. Use the \(m \cdot g, f^{\prime} s\) to show that the distribution \(o \bar{X}-\bar{Y} f\) is normal with mean and standard deviation equal to the answers of part a and b respectively.
4. Suppose that \(\sigma_{1}=\sqrt{2}\) and \(\sigma_{2}=\sqrt{2.5}\) and \(m=n .\) Find the sample sizes such that \(\bar{X}-\bar{Y}\) will be within one unit of \(\mu_{1}-\mu_{1}\) with a probability of 0.95

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