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Math Help - Sampling distribution

  1. #1
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    Sampling distribution

    Suppose that X1, X2,....Xm and Y1,Y2,....,Yn are independent random samples, with the variables Xi normally distributed with mean \mu _1 and variance \sigma{_1}^{2} and the variables Yi normally distributed with mean \mu_2 and variance \sigma{_2}^{2}. The difference between the sample means, \bar{X} - \bar{Y} is then a linear combination of m + n normally distributed random variables and is itself normally distributed.

    Suppose that \sigma{_1}^{2} = 2, \sigma{_2}^{2} = 2.5, and m=n. Find the sample sizes so that ( \bar{X} - \bar{Y}) will be within 1 unit of ( \mu _1 - \mu _2) with probability .95.

    I don't know how to solve this one. I've tried to set up P(( \mu _1 - \mu _2) - 1  \preceq  (\bar{X} - \bar{Y} )  \preceq  (\mu _1 - \mu _2) + 1) = .95

    but I don't if that's right. And, if so, how do I go on?
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  2. #2
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    Quote Originally Posted by mirrormirror View Post
    Suppose that X1, X2,....Xm and Y1,Y2,....,Yn are independent random samples, with the variables Xi normally distributed with mean \mu _1 and variance \sigma{_1}^{2} and the variables Yi normally distributed with mean \mu_2 and variance \sigma{_2}^{2}. The difference between the sample means, \bar{X} - \bar{Y} is then a linear combination of m + n normally distributed random variables and is itself normally distributed.

    Suppose that \sigma{_1}^{2} = 2, \sigma{_2}^{2} = 2.5, and m=n. Find the sample sizes so that ( \bar{X} - \bar{Y}) will be within 1 unit of ( \mu _1 - \mu _2) with probability .95.

    I don't know how to solve this one. I've tried to set up P(( \mu _1 - \mu _2) - 1  \preceq  (\bar{X} - \bar{Y} )  \preceq  (\mu _1 - \mu _2) + 1) = .95

    but I don't if that's right. And, if so, how do I go on?
    The variance of \bar X-\bar Y is \sigma_1^2/m+\sigma_2^2/n and its mean is \mu_1-\mu_2

    RonL
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