Variable of two populations

**sjenkins** · July 27th 2008, 07:24 PM

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.

**mr fantastic** · July 27th 2008, 08:02 PM

Originally Posted by sjenkins

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.

**sjenkins** · July 28th 2008, 07:14 PM

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."

**mr fantastic** · July 28th 2008, 07:37 PM

Originally Posted by sjenkins

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.

**sjenkins** · July 28th 2008, 07:43 PM

Ok, I get it now. Thank you!

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