An insurance company wants to study the relationship between

An insurance company wants to study the relationship between the amount of fire damage and the distance from the burning house to the responding fire station. This information will be used in setting rates for insurance coverage. An analyst randomly selects 30 claims filed last year and determines the distance from each house to the responding fire station. The analyst constructs a simple regression model with the distance from the fire station X measured in miles and the amount of fire damage Y measured in thousands of dollars. The following table displays some information for regression.

Regression Output

ANOVA Table

Variables

Coefficients

Source

SS

x(bar) = 7.8

Intercept

12.3601 (point estimate of the y-intercept)

Regression

1864.5728 (Explained variation)

SS_xx= 81.07469

Distance-X

4.7956 (point estimate of the slope)

Residual

1344.4934 (Unexplained variation)

SS_xx= 388.8018

1) Write down the least square line

2) What does the slope indicate?

3) How much damage would you estimate for a fire 5 miles from the fire station?

4) Determine and interpret the coefficient of determination.

5) Determine the correlation coefficient. Interpret its value.

6) Calculate the mean square error s²and the standard error s.

7) Calculate s_b1 and the t statistic for testing the significance of the slope

8) In order to test H₀ : ₁= 0 versus H_a : ₁ 0 on, determine whether we can reject H₀ by setting equal to 0.10, 0.05, and 0.01. What do you conclude?

9) Calculate a 95 percent confidence interval for ₁

10) Calculate a 99 percent confidence interval for x=10

11) Calculate a 90 percent prediction interval for x=10

12) Use the explained variation and the unexplained variation as given on the output to calculate the F (model) statistic.

13) Using F-Table, test the significance of the regression model at the 0.10, 0.05, and 0.01 levels of significance. What do you conclude?

Solution

1. Least square line is

Y = 12.3601 + 4.7956 X

2. The slope = 4.7956 indicates that, for a unit increase in X, there is 4.7956 units inccrease in Y, intercept held constant.

3. For a fire 5 miles from the fire station, the estimate of the damage is

hat Y = 12.3601 + 4.7956 x 5 = 36.3381

4. The coefficient of determination is given as

R² = SS Residual / SS Regression = 1344.4934 / 1864.5728 = 0.7211

5. The correlation coefficient is calculated as

r = (0.7211)^1/2 = 0.8492

This shows a relatively high degree of positive correlation between the distance X and the damage Y, i.e., as the distance increases, the damage increases.