Question

A variable of two populations has a mean of $7.9$and standard deviation of $5.4$for one of the populations and a mean of $7.1$and a standard deviation of $4.6$for the other population.

a. For independent samples of sizes $3$and $6$respectively find the mean and standard deviation of $\overline{x_1}-\overline{x_2}$

b. Must the variable under consideration be normally distributed on each of the two populations for you to answer part (a)? Explain your answer.

b. Can you conclude that the variable localid="1652696712667" $\overline{x_1}-\overline{x_2}$is normally distributed? Explain your answer.

Short answer

Part a. For the variable $\overline{x_1}-\overline{x_2}$, the mean is $0.8$and the standard deviation is $3.64$.

Part b. The variable under consideration may or may not be normally distributed on each of the two populations for us to answer part (a).

Part c. No, it cannot be concluded that the variable$\overline{x_1}-\overline{x_2}$ is normally distributed.

Step-by-step solution

Part (a) Step 1. Given Information

We are given data of two populations:

For the first population, the sample size is $n_1=3$, mean is ${\mu }_1=7.9$, and the standard deviation is ${\sigma }_1=5.4$.

For the second population, the sample size is$n_2=6$, mean is${\mu }_2=7.1$, and the standard deviation is ${\sigma }_2=4.6$.

Part (a) Step 2. Find the mean and standard deviation

The mean for the variable $\overline{x_1}-\overline{x_2}$is given as

$$\begin{gathered} {\mu }_{\overline{x_1}-\overline{x_2}}={\mu }_1-{\mu }_2 \\ {\mu }_{\overline{x_1}-\overline{x_2}}=7.9-7.1 \\ {\mu }_{\overline{x_1}-\overline{x_2}}=0.8 \end{gathered}$$

And the standard deviation is given as

$$\begin{gathered} {\sigma }_{\overline{x_1}-\overline{x_2}}=\sqrt{\frac{{{\sigma }_1}^2}{n_1}+\frac{{{\sigma }_2}^2}{n_2}} \\ {\sigma }_{\overline{x_1}-\overline{x_2}}=\sqrt{\frac{5.4^2}{3}+\frac{4.6^2}{6}} \\ {\sigma }_{\overline{x_1}-\overline{x_2}}=\sqrt{9.72+3.53} \\ {\sigma }_{\overline{x_1}-\overline{x_2}}\approx 3.64 \end{gathered}$$

Part (b) Step 1. Tell whether variables are normally distributed

The formulas in part (a) can be used regardless of the distributions of the variables of the two populations.

So the variables under consideration may or may not be normally distributed on each of the two populations.

Part (c) Step 1. Is the variable x1¯-x2¯ normally distributed

If the variable is normally distributed on each of the two populations then only it can be concluded that the variable $\overline{x_1}-\overline{x_2}$is normally distributed.

Also, the sample size is small so the distribution type matters.

Thus it cannot be concluded for sure that the variable $\overline{x_1}-\overline{x_2}$is normally distributed.