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Question

# A mutual fund is a professionally managed investment that can be sold to the general public. Mutual funds can be specialized into different categories such as healthcare-related or technology-related mutual funds. The following table lists the 3-month rates of return, in percent, for samples of health-care related and technology-related mutual funds, as reported by Morningstar, an independent investment research company. \$\begin{array}{ccccc|ccccc} \hline & {\text { Healthcare }} & {\text { }} &{\text { }} &{\text { }} &{\text { Technology }} \\ \hline 9.7 & 9.5 & 7.6 & 6.8 & 6.7 & 6.5 & 6.4 & 6.4 & 6.2 & 4.9 \\ 6.7 & 6.6 & 4.6 & 3.9 & 3.9 & 4.6 & 4.6 & 4.2 & 3.6 & 3.2 \\ 3.7 & 3.3 & 3.0 & 0.8 & -9.9 & 2.9 & 2.8 & 2.7 & 2.5 & \\ \hline \end{array}$ Use the technology of your choice to answer the following questions. Explain your answers. a. If you had to choose between the use of pooled t-procedures and nonpooled t -procedures here, which would you choose? b. Is it reasonable to use the type of procedure that you selected in part (a)?

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(a) The mean is the sum of all values divided by the number of values:

\begin{align*} \overline{x}_1&=\dfrac{9.7+9.5+...+0.8-9.9}{15}=4.46 \\ \overline{x}_2&=\dfrac{6.5+6.4+...+2.7+2.5}{14}=4.3929 \end{align*}

The variance is the sum of squared deviations from the mean divided by $n-1$. The standard deviation is the square root of the variance:

\begin{align*} s_1&=\sqrt{\dfrac{\begin{matrix}(9.7-4.46)^2+(9.5-4.46)^2+...+(-9.9-4.46)^2\end{matrix}}{6-1}}\approx 4.6837 \\ s_2&=\sqrt{\dfrac{\begin{matrix}(6.5-4.3929)^2+(6.4-4.3929)^2+...+(2.5-4.3929)^2\end{matrix}}{7-1}}\approx 1.5041 \end{align*}

We note that one of the standard deviations is more than twice the other standard deviation and thus the population standard deviations appear to be not equal.

When the population standard deviations appear to be not equal, then it is appropriate to use the $\textbf{nonpooled $t$-procedures}$.

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