Insurance company Nero believes that the amount of damage to a house in the even

Question

Insurance company Nero believes that the amount of damage to a house in the event of a fire might depend crucially on the distance of the house from the closest fire station. Hence, it would like to take this into consideration in pcemium calculations. In order to investigate the relationship between damage and distance, the company Nero selects at random 15 recent claims, reading from each: , damage, the fire damage (in thousands of S): distance, the distance from the closest fire station( (in miles). Below you can find the R output (with some part removed) for a simple regression model of damage on distance. > summary(fire) distance damage Min. :0.70 Min. :18.30 1st Qu. :2.20 1st Qu':23.00 Median :3.10 Median :27.00 Mean :3.28 Mean 28.39 3rd Qu .:4.45 3rd Qu. :33.25 Max. :6.10 Max. 44.90 > cor distance, damage) [1] 0.9464122 > reg fire-1m(damage-distance) summary (reg fire) Call: 1m(formula damage distance) Residuals: Min 1Q Median 3Q Max 3.954-1.747-0.284 1.527 3.939 Coefficients: Estimate Std. Error t value pr( lt) 0.422 10.566 9.47e-08 , 0.001 ‘**, 0.01 .*' 0.05 (Intercept) 13.761 1.526 9.016 5.90e-07 distance 4.459 signif. codes: o " 0.1 ‘ , 1 Residual standard error: ???? on ???? degrees of freedom Multiple R-squared: ??77, Adjusted R-squared: ???? F-statistic: 111.64 on ???? and ???? DF, p-value: 9.466e-08 > reg.red-1m(damage-1) anova(regred, reg fire) Analysis of variance Table Model 1: damage 1 Model 2: damage distance Res.Df RSS of Sum of sq F ProF) 1 ?7?? 772.18 2 ?7?7 80.54 777? 691.64 111.64 9.466e-08 signif. codes: 0 ‘ , 0.001"** ' 0.01"*, 0.05 ‘." 0.1 . ' 1

Explanation / Answer

1. The regression equation is damage=intercept + slope*distance + error

The assumption on errors is that they are distributed with zero mean and constant variance and their correlation with the independent variable and the dependent variable is zero.

The least square estimate for the intercept is 13.761 and for slope is 4.459

2. Assuming normality of errors, the 95% CI for the slope is (slope - stderr(slope)*t0.025,slope+stderr(slope)*t0.025)

=(4.459-1.96*0.422,4.459+1.96*0.422) = (3.632,5.286)

The interpretation is that 95% of the times the mean slope shall lie in the above range.

3. Linear relationship is concluded as the slope coefficient is significant at 0.05 level.

Null hypothesis = slope=0

Alternate hypo: slope not equal to 0

p value is <0.05, so we reject the null hypothesis and conclude that the slope is significantly different from 0.

Another way to check at 0.95 confidence level is to find the confidence interval for the slope, here as found above. If 0 does not lie within the CI, then we reject the null hypothesis.

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