Tax Efficiency. Tax efficiency is a measure, ranging from 0 to 100, of how much tax due to capital gains stock or mutual funds investors pay on their investments each year; the higher the tax efficiency, the lower is the tax. In the article "At the Mercy of the Manager" (Financial Planning, Vol. 30(5), pp. 54-56), C. Israel Sen examined the relationship between investments in mutual fund portfolios and their associated tax efficiencies. The following table shows percentage of investments in energy securities $\left(x\right)$and tax efficiency $\left(y\right)$for 10 mutual fund portfolios. For part $\left(g\right)$. predict the tax efficiency of a mutual fund portfolio with $5.0\%$of its investments in energy securities and one with $7.4\%$of its investments in energy securities.

1. find the regression equation for the data points.
2. graph the regression equation and the data points.
3. describe the apparent relationship between the two variables under consideration.
4. interpret the slope of the regression line.
5. identify the predictor and response variables.
6. identify outliers and potential influential observations.
7. predict the values of the response variable for the specified values of the predictor variable, and interpret your results.

Given in the question is a table, which shows the percentage of investments in energy securities and tax efficiency for 10 mutual fund portfolios.

We have to find the regression equation for the data points.

Let's use the regression beta coefficients for the calculation.

$\hat{y}=b_0+b_{1}x$

According to the information,$n=10$

We have to find the necessary sum as below:

$\sum x_i=3.1+3.2+\dots +10.6=55.9$

$\sum x_i^2=3.1^2+3.2^2+\dots +10.6^2=365.05$

$\sum y_i=98.1+94.7+\dots +53.5=832.5$

$\sum x_i{y}_i=(3.1)(98.1)+(3.2)(94.7)+\dots +(10.6)(53.5)=4736.95$

Let's find $s_{xy}$as follow:

$s_{xy}=\sum \left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n$

$$\begin{gathered} s_{xy}=4736.95-\frac{(55.9)(832.5)}{10} \\ =-276.725 \end{gathered}$$

Then, find $s_{xx}$as follow:

$s_{xx}=\sum {\left(x_i-\overline{x}\right)}^2=\sum x_i^2-{\left(\sum x_i\right)}^2/n$

$$\begin{gathered} s_{xx}=365.05-\frac{(55.9)(55.9)}{10} \\ =52.569 \end{gathered}$$

We can find the averages by using the given formula:

$\overline{x}=\frac{\sum x_i}{n}$

$\overline{y}=\frac{\sum y_i}{n}$

Here, the value of $\overline{x}$is:

$\overline{x}=\frac{55.9}{10}=5.59$

Then, find the value of $\overline{y}$is:

$\overline{y}=\frac{832.5}{10}=83.25$

Therefore, the parameters are:

role="math" localid="1653219984894" $$\begin{gathered} b_1=\frac{s_{xy}}{s_{xx}} \\ =\frac{-276.725}{52.569} \\ =-5.2640 \end{gathered}$$

$$\begin{gathered} b_0=\overline{y}-b_1\overline{x} \\ =83.25-(-5.2640)\times 5.59 \\ =112.6758 \end{gathered}$$

Given in the question that,

We have to graph the regression equation and the data points.

When investing $3.1\%$in energy security, the tax efficiency is:

$$\begin{gathered} {\hat{y}}_{3.1}=112.6758-5.2640(3.1) \\ =96.3574 \end{gathered}$$

The anticipated values for the provided data are also presented in the table below.

$\begin{array}{|l|}\hline xy\hat{y}\\ 3.198.196.3574\\ 3.294.795.8310\\ 3.792.093.1990\\ 4.389.890.0406\\ 4.087.591.6198\\ 5.585.083.7238\\ 6.782.077.4070\\ 7.477.873.7222\\ 7.472.173.7222\\ 10.653.556.8774\\ \hline\end{array}$

The given points and the fitted regression line are represented in the graph below.

The apparent relationship between the two variables under investigation must be described.

The regression line slopes downward, implying that as the share of investments in energy securities $\left(x\right)$ increases, the tax efficiency $\left(y\right)$ tends to decline.

Given in the question that,

We have to interpret the slope of the regression line

The average increase in $y$per unit increase in $x$is represented by the slope.

The slope was determined in section (a) to be $-5.2640$.

The predictor and response variables must be identified.

The response variable seems to be the one that will be measured.

The predictor variable is a variable that is used to estimate the output of the response variable.

The percentage of investments in energy security is used to illustrate tax efficiency. As a result, the response variable is tax efficiency, and the predictor variable is the percentage of investments in energy securities.

Outliers and potentially influential observations must be identified.

An outlier is just a piece of evidence that is far off the regression line.

An impactful observation is one where the removal of a point causes a significant change in the regression equation. That instance, removing a point creates a significant shift in the regression line's direction.

All of the points in component (b) are closed to the regression line, indicating that there are no outliers in the dataset. There are no major changes in the orientation of the regression line when a point is removed, hence there are no influencing observations.

We must forecast the response variable's values based on the values of the predictor variable and evaluate the results.

It was determined that,

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

When $5\%$of investments in energy security are made, the tax efficiency is:

$$\begin{gathered} {\hat{y}}_{5.0}=112.6758-5.2640(5.0) \\ =86.3558\% \end{gathered}$$

When $7.4\%$of investments in energy security, the tax efficiency is:

$$\begin{gathered} {\hat{y}}_{7.4}=112.6758-5.2640(7.4) \\ =73.7222\% \end{gathered}$$

A mutual fund portfolio with $5\%$of its investments in energy securities has a forecast tax efficiency of $86.3558$, whereas a mutual fund portfolio with $7.4\%$of its investments in energy securities has a predicted tax efficiency of $73.7222$. As a result, growing investments in energy commodities will lower the tax efficiency of a mutual fund portfolio.