Question

Following are the data on percentage of investment in energy securities and tax efficiency from Exercise \(14.22\).

a. Obtain a point estimate for the mean tax efficiency of all mutual fund portfolios with \(6%\) of their investments in energy securities.

b. Determine a \(95%\) confidence interval for the mean tax efficiency of all mutual fund portfolios with \(6%\) of their investments in energy securities.

c. Find the predicted tax efficiency of a mutual fund portfolio with \(6%\) of its investment in energy securities.

d. Determine a \(95%\) prediction interval for the tax efficiency of a mutual fund portfolio with \(6%\) of its investments in energy securities.

e. Draw graph similar to those in Fig. \(14.11\) on page \(576\), showing both the \(95%\) confidence interval from part (b) and the \(95%\) prediction interval from part (d).

f. Why is the prediction interval wider than the confidence interval?

Short answer

Part a. The point estimate is \(\hat{y_{p}}=81.093\)

Part b. We can be \(95%\) confident that the mean tax efficiency of all mutual fund portfolios is somewhere between \(78.806\) to \(83.378\).

Part c. The point estimate is \(\hat{y_{p}}=81.093\)

Part d. We can be \(95%\) confident that the mean tax efficiency of all mutual fund portfolios is somewhere between \(73.618\) to \(88.566\).

Part e.

Part f.Theerror in the estimate is because of the population regression line which is being estimated by sample regression line and the error in the prediction is because of the error in estimating the mean tax efficiency plus the variation in tax efficiency.

Step-by-step solution

Part a. Step 1. Given information

Given,

Part a. Step 2. Calculation

Computation of table:

\(S_{xy}=\sum x_{i}y_{i}-(\sum x_{i})(\sum y_{i})/n\)

\(=4376.95-(55.9)(832.5)/10\)

\(=4376.95-46536.75/10\)

\(=4376.95-4653.675\)

\(=-276.20\)

\(S_{xx}=\sum x^{2}_{i}-(\sum x_{i})^{2}/n\)

\(=365.05-(55.9)^{2}/10\)

\(=365.05-3124.81/10\)

\(=365.05-312.481\)

\(=52.569\)

The total sum of squares SST is given by,

\(S_{yy}=\sum y^{2}_{i}-(\sum y_{i})^{2}/n\)

\(=70838.49-(832.5)^{2}/10\)

\(=70838.49-693056.25/10\)

\(=70838.49-69305.625\)

\(=52.569\)

The regression sum of squares SSR is given by,

\(SSR=\frac{S_{xy}^{2}}{S_{xx}}\)

\(=\frac{(-276.725)^{2}}{52.569}=\frac{76576.72562}{52.569}=1456.689791\)

\(SSE=SST-SSR\)

\(=1532.865-1456.689791\)

\(=76.17520896\)

The formula for calculating the standard error of the estimate is,

\(s_{e}=\sqrt{\frac{SSE}{n-2}}\)

\(=\sqrt{\frac{76.17520896}{10-2}}\)

\(=3.085757787\)

\(\approx 3.0857\)

The formula for calculating the slope of the regression line is.

\(b_{1}=\frac{S_{xy}}{S_{xx}}\)

\(=\frac{-276.725}{52.569}\)

\(=-5.26\)

The formula for calculating the value of y-intercept is

\(b_{0}=\frac{1}{n}(\sum y_{i}-b\sum x_{i})\)

\(=\frac{1}{10}(832.5+5.26(55.9))\)

\(=\frac{1}{10}(1126.534)\)

\(=112.653\)

So, the regression equation is \(\hat{y_{p}}=112.653 -5.26x_{p}\)

The formula for calculating the value of the point estimate is obtained by substituting the value of \(x_{p}=6\) in the regression equation.

\(\hat{y_{p}}=1112.653 -5.26x_{p}\)

\(=112.653 -5.26(6)\)

\(=81.093\)

The point estimate is \(\hat{y_{p}}=81.093\)

Part b. Step 1. Calculation

STEP 1: For a \(95%\) confidence interval, \(\alpha=0.05\). Because \(n=10\),

\(df=n-2\)

\(=10-2\)

\(=8\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=2.306\)

STEP 2:

The formula for calculating the end points of the confidence interval for the conditional mean of the response variable are

\(\hat{y_{p}}\pm t_{\alpha/2}\times s_{e}\sqrt{\frac{1}{n}+\frac{(x_{p}-\sum x_{i}/n)^{2}}{S_{xx}}}\)

We have, \(x_{p}=6\),

\(\hat{y_{p}} =81.093\),

\(s_{e}=3.0857\),

\(S_{xx}=52.569\).

So, \(81.093\pm 2.306\times (3.0857) \sqrt{\frac{1}{10}+\frac{(6-55.9/10)^{2}}{52.569}}\)

\(81.093\pm 7.1156242 \sqrt{0.1+0.003197702}\)

Or \(81.093\pm 2.28585152\)

Or \(78.806\) to \(83.378\)

Therefore, the \(95%\) confidence interval for the conditional mean is \(78.806\) to \(83.378\).

Part c. Step 1. Calculation

The regression equation is \(\hat{y_{p}}=112.653-5.26x_{p}\)

The predicted value is obtained by substituting the value of \(x_{p}=6\) in the regression equation.

\(\hat{y_{p}}=112.653-5.26x_{p}\)

\(=112.653-5.26(2)\)

\(=81.093\)

The predicted value is \(\hat{y_{p}}= 81.093\)

Part d. Step 1. Calculation

STEP 1: For a \(95%\) confidence interval, \(\alpha=0.05\). Because \(n=10\),

\(df=n-2\)

\(=10-2\)

\(=8\)

From technology, \(t_{\alpha/2}=t_{0.05/2}=t_{0.025}=2.306\)

STEP 2:

The formula for calculating the end points of the confidence interval for the conditional mean of the response variable are

\(\hat{y_{p}}\pm t_{\alpha/2}\times s_{e}\sqrt{\frac{1}{n}+\frac{(x_{p}-\sum x_{i}/n)^{2}}{S_{xx}}}\)

We have, \(x_{p}=6\),

\(\hat{y_{p}} =81.093\),

\(s_{e}=3.0857\),

\(S_{xx}=52.569\).

So, \(81.093\pm 2.306\times (3.0857) \sqrt{1+\frac{1}{10}+\frac{(6-55.9/10)^{2}}{52.569}}\)

\(81.093\pm 7.1156242 \sqrt{1+0.1+0.003197702}\)

Or \(81.093\pm 2.28585152\)

Or \(73.618\) to \(88.566\)

Therefore, the \(95%\) confidence interval for the conditional mean is \(73.618\) to \(88.566\).

Part e. Step 1. Calculation

The graph showing \(90%\) prediction interval

The graph showing \(90%\) confidence interval

Therefore, the graph showing \(90%\) confidence interval and \(90%\) prediction interval is drawn.

Part f. Step 1. Calculation

The prediction interval is wider than the confidence interval for the following reason:

The error in the estimate of the mean tax efficiency of all mutual fund portfolios with 6% of their investments in energy securities is due only to the fact that the population regression line is being estimated by sample regression line, whereas the error in the prediction of all mutual fund portfolios with \(6%\) of their investments in energy securities is due to the error in estimating the mean tax efficiency plus the variation in tax efficiencies of all mutual fund portfolios with \(6%\) of their investments in energy securities.

Therefore, theerror in the estimate is because of the population regression line which is being estimated by sample regression line and the error in the prediction is because of the error in estimating the mean tax efficiency plus the variation in tax efficiency.