Question

Tax Efficiency. In Exercise 4.58, you determined a regression equation that relates the variables percentage of investments in energy securities and tax efficiency for mutual fund portfolios.

a. Should that regression equation be used to predict the tax efficiency of a mutual fund portfolio with $6.4\%$of its investments in energy securities? with $15\%$of its investments in energy securities? Explain your answers.

b. For which percentages of investments in energy securities is use of the regression equation to predict tax efficiency reasonable?

Short answer

a. In the example data,$6.4\%$of the investments in a mutual fund portfolio were made in energy securities outside the range of energy securities. Using the above regression equation, it would not be possible to predict the tax efficiency of the portfolio.

For a mutual fund portfolio with $15\%$of its investments in energy securities, we cannot use the above regression equation to predict tax efficiency.

b. The tax efficiency of a mutual fund portfolio is predicted using regression equations between $3.1$and $10.6$.

Step-by-step solution

Part (a) Step 1: Given Information

As a result of Exercise 4.58, the following table shows the ten mutual fund portfolios.

$X$and $Y$represent the tax efficiency and percentage of investment in energy securities in the ten mutual fund portfolios given in Exercise 4.58 .

Part (a)  Step 2: Explanation

For tax efficiency $\left(y\right)$to be predicted from investments in energy securities$\left(x\right)$, the regression equation is as follows:

$\hat{y}=112.6758-5.2640x$

Concept:

Predicting the outcome with the regression line is reasonable if the value of the predictor variable is within the given $x$-value range.

According to the regression line, $6.4\%$of investments in energy securities fall within the interval of given $x$-values. Therefore, the regression line can be used to predict text efficiency.

It is not a reasonable assumption to predict text efficiency using the calculated regression line because $15\%$ of investments in energy securities fall outside the interval of given $x$-values.

Part (b) Step 3: Given Information

The goal is to determine the interval of the percentage of investments in energy securities that can be used to predict tax efficiency using the regression equation.

Part (b) Step 4: Explanation 

Assuming the given percentage is invested in energy securities

The minimum is $3.1\%$

The maximum is $10.6\%$.

Based on the calculated regression equation, it is possible to predict the tax efficiency, if the investments in energy security are between$3.2\%$ and $15\%$.