Chapter 12, Problem 68CE

In the following regression, X = total assets ($ billions), Y = total revenue ($ billions), and n = 64 large banks. (a) Write the fitted regression equation. (b) State the degrees of freedom for a two-tailed test for zero slope, and use Appendix D to find the critical value at α = .05. (c) What is your conclusion about the slope? (d) Interpret the 95 percent confidence limits for the slope. (e) Verify that F= t² for the slope. (f) In your own words, describe the fit of this regression.

To determine

Write the fitted regression equation.

Answer to Problem 68CE

The regression equation is, $\text{Total Revenue}=6.5763+0.0452\text{Total assets}$

Explanation of Solution

Calculation:

An output of a regression is given. The X variable is the total assets and Y be the total revenue.

Regression:

Suppose $x_1\mathrm{...}{x}_n$ be n sample values of independent variable and the corresponding dependent variable values are $y_1\mathrm{...}{y}_n$. The slope and the intercept of ordinary least square can be defined as $b_0=\overline{y}-b_1\overline{x}$ and $b_1=\frac{S{S}_{xy}}{S{S}_{xx}}$.

Where, $S{S}_{xx},S{S}_{yy},S_{xy}$ are the sum of squares due to x, y andxy, respectively. $\overline{x}\text{and}\overline{y}$ are the sample mean of the independent and dependent variables, respectively.

The total sum of squares is denoted as, $SST={\displaystyle \sum _{i=1}^n\left(y_i-\overline{y}\right)}$.

The regression sum of squares is denoted as, $SSR={\displaystyle \sum _{i=1}^n\left({\widehat{y}}_i-\overline{y}\right)}$.

The error sum of squares is denoted as, $SSE={\displaystyle \sum _{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$.

From the regression the fitted line is denoted as, $\widehat{y}=b_0+b_{1}x$.

From the output, $b_0=6.5763$$b_1=0.0452$

Hence, the regression equation is, $\text{Total Revenue}=6.5763+0.0452\text{Total assets}$

To determine

Find the degrees of freedom for the two-tailed test for zero slope.

Find the critical value at 0.05 level of significanceusing Appendix D.

Answer to Problem 68CE

The degrees of freedom for the two-tailed test for zero slop is62.

The critical value at 0.05 level of significanceis 2.000.

Explanation of Solution

Calculation:

Critical value:

Here from the output, the sample size, $n=64$.

The degrees of freedom is,

$\begin{array}{c}df=n-2\\ =64-2\\ =62\end{array}$

For two tailed test, the critical value for t-test will be, $t_{\frac{\alpha }{2},\left(n-2\right)}$.

From the Appendix D: STUDENT’S t CRITICAL VALUES:

• • Since 62 is not in the table, so locate the value 60in the column of degrees of freedom.
• • Locate the 0.025 in level of significance.
• • The intersecting value that corresponds to the degrees of freedom 60 with level of significance 0.025 is 2.000.

Thus, the critical-valueusing Appendix D is 2.000.

To determine

Make a conclusion about the slope.

Explanation of Solution

Let ${\beta }_1$ be the slope parameter.

Hypotheses:

Null hypothesis:

$H_0:{\beta }_1=0$

That is, the slope is zero.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$

That is, the slope is not equal to zero.

Decision rule:

If $t\text{-statistic>critical value}$, reject the null hypothesis.

If $t\text{-statistic}\le \text{critical value}$, fail to reject the null hypothesis.

From the output, the t-statistics is 8.183 and from part (b), the critical value at 0.05 level of significanceis 2.000.

Conclusion:

Here the t-statistics is greater than the critical value at 0.05 level of significance.

That is, $8.183\left(=t\text{-statistic}\right)>2.000\left(=\text{critical value}\right)$ .

Hence, by the decision rule reject the null hypothesis.

That is, the slope is significantly different from zero.

To determine

Interpret the 95% confidence interval for the slope.

Explanation of Solution

The 95% confidence interval for the slope, ${\beta }_1$ is defined as,

$b_1-t_{\frac{\alpha }{2}}{s}_{b_1}\le {\beta }_1\le b_1+t_{\frac{\alpha }{2}}{s}_{b_1}$

Where, $b_1$ is the slope, $t_{\frac{\alpha }{2}}$ be the table value of t-distribution for two-tailed with $\alpha$ level of significance and $s_{b_1}$ is the standard error of the slope denoted as, $s_{b_1}=\frac{s_e}{\sqrt{{\displaystyle \sum _{i=1}^n\left(x_i-\overline{x}\right)}}},s_e\text{ is the standard error}\text{.}$$s_e=\sqrt{\frac{SSE}{n-2}}$ with sample size n and the sum of squares due to error.

From the output, the 95% confidence interval of the slope is (0.0342, 0.0563).

Interpretation:

From the confidence interval it can be concluded that there is 95% confident that the slope will lie between 0.0342 and 0.0563.

To determine

Verify $F=t^2$ for the slope.

Explanation of Solution

Calculation:

From the output the F statistic is 66.97.

For the slope the t-statistic is 8.183.

$\begin{array}{c}{t}^2={\left(8.183\right)}^2\\ =66.96\\ =F\end{array}$

Hence, it can be concluded that $F=t^2$.

To determine

Describe the fit of the regression.

Explanation of Solution

Calculation:

From the output, the R-squared value is 0.519.

$R^2$(R-squared):

The coefficient of determination ($R^2$) is defined as the proportion of variation in the observed values of the response variable that is explained by the regression. The squared correlation gives fraction of variability of response variable (y) accounted for by the linear regression model.

The $R^2$ value is 52%, which means that the percentage of variation in the observed values of total revenue that is explained by the regression is 52%, which indicates that 52% of the variability in total revenue is explained by the total assets with a linear relationship.