Question

Waterbury Insurance Company wants to study the relationship between the amount of fire damage and the distance between the burning house and the nearest fire station. This information will be used in setting rates for insurance coverage. For a sample of 30 claims for the last year, the director of the actuarial department determined the distance from the fire station (X) and the amount of fire damage, in thousands of dollars (Y). The MegaStat output is reported below.

ANOVA table
Source	SS			df			MS			F
Regression		1,865.5782			1			1,865.5782			39.71
Residual		1,315.4934			28			46.9819
Total		3,181.0716			29

Regression output
Variables	Coefficients			Std. Error			t(df=28)
Intercept		12.5538			3.0301			3.678
Distance-X		3.4664			8.63			6.3

(a-1)

Write out the regression equation. (Round your answers to 3 decimal places.)

Y =  +  X.

(a-2)

Is there a direct or indirect relationship between the distance from the fire station and the amount of fire damage?

The relationship between distance and damage is (Click to select)directinverse.

(b)	How much damage would you estimate (in dollars) for a fire 10 miles from the nearest fire station? (Round your answer to the nearest dollar amount.)

Estimated damage

$

(c-1)

Determine the coefficient of determination. (Round your answer to 3 decimal places.)

Coefficient of determination

(c-2)

Fill in the blank below. (Round your answer to 1 decimal place.)

% of the variation in damage is explained by variation in distance.

(d-1)

Determine the coefficient of correlation. (Round your answer to 3 decimal places.)

Coefficient of correlation

(d-2)

Choose the right option.

There is a fairly solid (Click to select)directinverse link between the variables.

(d-3)

How did you determine the sign of the correlation coefficient?

It is (Click to select)zeronegativepositive because the slope is (Click to select)negativepositivezero.

(e-1)

State the decision rule for 0.01 significance level: H0 : ρ = 0; H1 : ρ ≠ 0. (Negative amounts should be indicated by a minus sign. Round your answers to 3 decimal places.)

Reject H0 if t <  or t >

(e-2)

Compute the value of the test statistic. (Round your answer to 2 decimal places.)

Value of the test statistic

(e-3)

Is there any significant relationship between the distance from the fire station and the amount of damage? Use the 0.01 significance level.

(Click to select)Do not rejectReject H0. There is (Click to select)no significantsignificant relationship between distance and fire damage.

Answer 1

a-1.
The regression equation is,
Y = 12.5538 + 3.4664 X.

a-2.
Since the slope in the regression equation is positive,
The relationship between distance and damage is direct.

(b)
For x = 10,
Y = 12.5538 + 3.4664 * 10 = 47.2178
Estimated damage = $47

(c-1)
From the output,
Coefficient of determination = SS Regression / SS Total = 1,865.5782 / 3,181.0716 = 0.586
  
(c-2)
58.6 % of the variation in damage is explained by variation in distance.

(d-1)
Coefficient of correlation = = 0.765

(d-2)
Since the correlation is above 0.75, (Taking the positive sign as the slope of the regression equation is positive).
There is a fairly solid direct link between the variables.

(d-3)
It is positive because the slope is positive.

(e-1)
Degree of freedom of residual = 28
Critical value of t at 0.01 significance level and df = 28 is 2.76. Thus,
Reject H0 if t < -2.76 or t > 2.76

(e-2)
Value of the test statistic =

(e-3)

Since the test statistic is greater than 2.76,

Reject H₀. There is significant relationship between distance and fire damage.