Concept explainers
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.
a.
Write the fitted regression equation.
Answer to Problem 62CE
The regression equation is,
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
Where,
The total sum of squares is denoted as,
The regression sum of squares is denoted as,
The error sum of squares is denoted as,
From the regression the fitted line is denoted as,
From the output,
Hence, the regression equation is,
b.
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 62CE
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,
The degrees of freedom is,
For two tailed test, the critical value for t-test will be,
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.
c.
Make a conclusion about the slope.
Explanation of Solution
Let
Hypotheses:
Null hypothesis:
That is, the slope is zero.
Alternative hypothesis:
That is, the slope is not equal to zero.
Decision rule:
If
If
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,
Hence, by the decision rule reject the null hypothesis.
That is, the slope is significantly different from zero.
d.
Interpret the 95% confidence interval for the slope.
Explanation of Solution
The 95% confidence interval for the slope,
Where,
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.
e.
Verify
Explanation of Solution
Calculation:
From the output the F statistic is 66.97.
For the slope the t-statistic is 8.183.
Hence, it can be concluded that
f.
Describe the fit of the regression.
Explanation of Solution
Calculation:
From the output, the R-squared value is 0.519.
The coefficient of determination (
The
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Chapter 12 Solutions
Applied Statistics in Business and Economics
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