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 twotailed
test for zero slope, and use Appendix D to find the critical value at á = .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.
R2 0.519
Std. Error 6.977
n 64
ANOVA table
Source SS df MS F p-value
Regression 3,260.0981 1 3,260.0981 66.97 1.90E-11
Residual 3,018.3339 62 48.6828
Total 6,278.4320 63
Regression output confidence interval
variables coefficients std. error t (df = 62) p-value 95% lower 95% upper
Intercept 6.5763 1.9254 3.416 .0011 2.7275 10.4252
X1 0.0452 0.0055 8.183 1.90E-11 0.0342 0.0563
(a) The fitted regression equation is Y = 6.5763 + 0.0452X.
(b) The degrees of freedom for a two-tailed test for zero slope are df = 62. The critical value at α = 0.05 can be found in Appendix D.
(c) Since the p-value for the slope is 1.90E-11 (very small), we can conclude that there is a significant relationship between total assets and total revenue.
(d) The 95% confidence interval for the slope is (0.0342, 0.0563). This means that we are 95% confident that the true slope of the regression line falls within this range.
(e) To verify that F = t^2 for the slope, we need to compare the F-value (66.97) from the ANOVA table to the square of the t-value (8.183)^2. If they are approximately equal, then the condition is satisfied.
(f) The fit of this regression can be described as moderately strong (R^2 = 0.519) since 51.9% of the variation in total revenue can be explained by the variation in total assets.
(a) The fitted regression equation is:
Y = 6.5763 + 0.0452*X
(b) The degrees of freedom for a two-tailed test for zero slope is df = n-2 = 64-2 = 62. From Appendix D, the critical value for α = 0.05 is 2.00.
(c) The p-value for the slope is 1.90E-11, which is less than the significance level of 0.05. Therefore, we can reject the null hypothesis of a zero slope and conclude that there is a significant relationship between total assets and total revenue for large banks.
(d) The 95% confidence interval for the slope is 0.0342 to 0.0563. This means that we are 95% confident that the true slope value falls within this range.
(e) To verify that F = t^2 for the slope, we need to compare the F-value from the ANOVA table to the square of the t-value for the slope. The F-value is 66.97 and the t-value squared is (8.183)^2 = 67.01. Since these values are almost equal, we can conclude that F = t^2 for the slope.
(f) The R^2 value, which represents the proportion of variation in the dependent variable (total revenue) explained by the independent variable (total assets), is 0.519. This suggests that around 51.9% of the variation in total revenue can be explained by the total assets of large banks. The standard error of 6.977 indicates the average distance between the observed and predicted values. Overall, the regression equation provides a reasonably good fit to the data.
(a) The fitted regression equation is Y = 6.5763 + 0.0452X, where Y is the total revenue and X is the total assets.
(b) To find the degrees of freedom for a two-tailed test for zero slope, we need to subtract 1 from the number of observations. In this case, n = 64, so the degrees of freedom would be 63. To find the critical value at α = 0.05, you can refer to Appendix D of your statistical textbook or use a statistical software or online calculator.
(c) To draw a conclusion about the slope, we need to analyze the p-value. The p-value for the slope is 1.90E-11, which is less than α = 0.05. Therefore, we can reject the null hypothesis of zero slope and conclude that there is a significant linear relationship between total assets and total revenue.
(d) The 95 percent confidence limits for the slope are 0.0342 and 0.0563. This means that we can be 95 percent confident that the true value of the slope falls between these two values.
(e) To verify that F = t^2 for the slope, we need to compare the F-statistic in the ANOVA table with the square of the t-statistic for the slope. In this case, the F-statistic is 66.97 and the t-statistic is 8.183. Taking the square of the t-statistic (8.183^2) gives us approximately 66.97, which confirms that F = t^2 for the slope.
(f) The R^2 value of 0.519 indicates that approximately 51.9 percent of the variation in total revenue can be explained by the linear relationship with total assets. The standard error of 6.977 indicates the average distance between the actual total revenue values and the predicted values based on the regression equation. Overall, the fit of this regression model suggests a moderate degree of association between total assets and total revenue, with room for improvement.