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Math Help - Variable of two populations

  1. #1
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    Variable of two populations

    Can someone help me understand how to get started on this problem?

    A variable of two populations has a mean of 40 and a standard deviation of 12 for one of the populations and a mean of 40 and a standard deviation of 6 for the other population. Moreover, the variable is normally distributed on each of the two populations.

    a. For independent samples of sizes 9 and 4 respectively, determine the mean and standard deviation of mean1 - mean2.
    b. Can you conclude that the variable mean1-mean2 is normally distributed?
    c. Determine the percentage of all pairs of independent samples of sizes 9 and 4 respectively from the two populations that have the property that the difference between the sample means is between -10 and 10.
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  2. #2
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    Quote Originally Posted by sjenkins View Post
    Can someone help me understand how to get started on this problem?

    A variable of two populations has a mean of 40 and a standard deviation of 12 for one of the populations and a mean of 40 and a standard deviation of 6 for the other population. Moreover, the variable is normally distributed on each of the two populations.

    a. For independent samples of sizes 9 and 4 respectively, determine the mean and standard deviation of mean1 - mean2.
    b. Can you conclude that the variable mean1-mean2 is normally distributed?
    c. Determine the percentage of all pairs of independent samples of sizes 9 and 4 respectively from the two populations that have the property that the difference between the sample means is between -10 and 10.
    Where are you stuck?

    Let M be the random variable difference between sample means:

    M = M_1 - M_2

    where M_1 is mean of sample from population 1 and M_2 is mean of sample from population 2.

    If the mean and sd of population 1 is \mu_1 and \sigma_1 and the mean and sd of population 2 is \mu_2 and \sigma_2 then:

    \bar{M} = \mu_1 - \mu_2 and \text{Var} (M) = \frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}<br />

    where n_1 is the size of sample 1 and n_2 is the size of sample 2.
    Last edited by mr fantastic; July 28th 2008 at 10:42 PM.
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  3. #3
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    Can someone please explain part c of this problem "Determine the percentage of all pairs of independent samples of sizes 9 and 4 respectively, from the two populations that have the property that the difference between the sample means is between -10 and 10."
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    Quote Originally Posted by sjenkins View Post
    Can someone please explain part c of this problem "Determine the percentage of all pairs of independent samples of sizes 9 and 4 respectively, from the two populations that have the property that the difference between the sample means is between -10 and 10."
    The difference M between the two means follows a normal distribution with mean and sd given by the formulae I gave. This is because you can use the population variances instead of the sample variances.

    So calculate Pr(-10 < M < 10) and then multiply the result by 100 to convert into a percentage.
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    Ok, I get it now. Thank you!
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