Question

Following are the data on the percentage of investments in energy securities and tax efficiency$95\%$$,\alpha =0.05$ . find and interpret a confidence interval, at the specified confidence level, for the slope of the population regression line that relates the response variable to the predictor variable.

Short answer

$95\%$certain that the decline in mean tax efficiencies increase in energy securities for mutual fund portfolio investments is between $-6.245$ and$-4.282$.

Step-by-step solution

Given Information

Given table is

Tax efficiency$95\%and\alpha =0.05$we have to find and interpret a confidence interval, at the specified confidence level.

Explanation

STEP-1 : For a $95\%$confidence interval $\alpha =0.05$

$$\begin{gathered} df=n-2 \\ =10-2 \\ =8 \end{gathered}$$

Using technology$t_{a/2}=t_{0.05/2}=t_{0.025}=2.306$

Table of computation

$$\begin{gathered} S_{xY}=\sum x_i{y}_j-\left(\sum x_j\right)\left(\sum y_i\right)/n \\ =4376.95-(55.9)(832.5)/10 \\ =-276.725 \end{gathered}$$

$$\begin{gathered} S_{xx}=\sum x_i^2-{\left(\sum x_j\right)}^2/n \\ =365.05-(55.9{)}^2/10 \\ =52.569 \end{gathered}$$

SST is given by

$$\begin{gathered} S_{yy}=\sum y_i^2-{\left(\sum y_i\right)}^2/n \\ =70838.49-(832.5{)}^2/10 \\ =1532.865 \end{gathered}$$

SSR is given by

$$\begin{gathered} SSR=\frac{S_{xy}^2}{S_{xx}} \\ =\frac{(-276.725{)}^2}{52.569} \\ =1456.689791 \end{gathered}$$

$$\begin{gathered} SSE=SST-SSR \\ =1532.865-1456.689791 \\ =76.17520896 \end{gathered}$$

$b_1=-5.264033936$

$s_e\approx 3.0857$

The formula for calculating the end points of confidence interval for ${\beta }_1$is

$b_1\pm t_{\alpha /2}\frac{s_e}{\sqrt{S_{xx}}}$

$$\begin{gathered} -5.264033936\pm 2.306\times \frac{3.0857}{\sqrt{52.569}} \\ Or-5.264033936\pm 0.981423107,Or-6.245to-4.282 \end{gathered}$$

$95\%$certain that the decline in mean tax efficiencies increase in energy securities for mutual fund portfolio investments is between$-6.245and-4.282$.