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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. $$ \begin{array}{|lrrrr|} \hline \text { ANOVA table } & & & & & & \\ \text { Source } & & \text { SS } & \text { df } & \text { MS } & \text { F } \\\ \text { Regression } & 1,864.5782 & 1 & 1,864.5782 & 38.83 \\ \text { Residual } & 1,344.4934 & 28 & 48.0176 & \\ \text { Total } & 3,209.0716 & 29 & & & \\ \text { Regression } & \text { output } & & & & \\ \text { Variables } & \text { Coefficients } & \text { Std. Error } t(\mathrm{df} & =28) \\ \text { Intercept } & 12.3601 & & 3.2915 & 3.755 \\ \text { Distance-X } & 4.7956 & & 0.7696 & 6.231 \\ \hline \end{array} $$ Answer the following questions. a. Write out the regression equation. Is there a direct or indirect relationship between the distance from the fire station and the amount of fire damage? b. How much damage would you estimate for a fire 5 miles from the nearest fire station? c. Determine and interpret the coefficient of determination. d. Determine the correlation coefficient. Interpret its value. How did you determine the sign of the correlation coefficient? e. Conduct a test of hypothesis to determine if there is a significant relationship between the distance from the fire station and the amount of damage. Use the .01 significance level and a two-tailed test.

Step by step solution

01

Write the Regression Equation

The regression equation is derived from the coefficients given in the output table. The general form of a simple linear regression is: \[ y = \beta_0 + \beta_1 x \]Given the coefficients, where the intercept is 12.3601 and the slope (Distance-X) is 4.7956, the regression equation becomes:\[ y = 12.3601 + 4.7956x \]Since the coefficient of Distance-X is positive, there is a direct relationship. This means as the distance from the fire station increases, the amount of fire damage increases.

02

Estimate Damage for a Fire 5 Miles Away

To estimate the fire damage for a distance of 5 miles, substitute \( x = 5 \) into the regression equation:\[ y = 12.3601 + 4.7956(5) \]Calculate the expression:\[ y = 12.3601 + 23.978 \]\[ y = 36.3381 \]The estimated fire damage is approximately $36,338 (since damage is in thousands of dollars).

03

Determine the Coefficient of Determination

The coefficient of determination, \( R^2 \), is found by dividing the regression sum of squares (SS) by the total sum of squares:\[ R^2 = \frac{SS_{Regression}}{SS_{Total}} = \frac{1864.5782}{3209.0716} \]Calculate the value:\[ R^2 = 0.581 \]The coefficient of determination is 0.581, indicating that approximately 58.1% of the variation in fire damage can be explained by the distance from the fire station.

04

Determine and Interpret the Correlation Coefficient

The correlation coefficient, \( r \), is the square root of the coefficient of determination. Since the relationship is direct (positive slope), \( r \) is positive:\[ r = \sqrt{R^2} = \sqrt{0.581} \]Calculate the value:\[ r \approx 0.762 \]The correlation coefficient, \( r = 0.762 \), indicates a strong positive linear correlation between the distance and fire damage.

05

Hypothesis Testing for Significant Relationship

To test if the relationship is significant, use the provided t-statistic for \( \beta_1 \) (slope):- Null Hypothesis \( H_0: \beta_1 = 0 \) (no relationship)- Alternative Hypothesis \( H_a: \beta_1 eq 0 \) (relationship exists)Given: \( t = 6.231 \), df = 28. At \( \alpha = 0.01 \) for a two-tailed test, the critical \( t \)-value is approximately 2.763 (from the \( t \)-distribution table). Since 6.231 > 2.763, reject \( H_0 \).Conclusion: There is a significant relationship between distance and fire damage at the 0.01 significance level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coefficient of Determination

The coefficient of determination, often represented as \( R^2 \), is a key concept in linear regression analysis. It tells us how well our regression model explains the variability of the dependent variable. In the context of our exercise, \( R^2 \) for the relationship between distance from the fire station and fire damage amount is calculated to be 0.581.
This indicates that 58.1% of the variation in fire damage can be explained by the distance from the nearest fire station. Another way of understanding this figure is that our regression line or model captures more than half of the data variability about fire damage.
Having a \( R^2 \) value closer to 1 would signify a model that better fits the data, while a value closer to 0 would indicate a poor fit. Therefore, a value like 0.581 suggests a moderate to strong relationship, implying that distance is quite an important factor influencing fire damage.

Hypothesis Testing

Hypothesis testing in the context of linear regression is used to assess the significance of the predictors in the model. This involves testing whether the slope of the regression line is significantly different from zero. In simpler terms, it helps us determine if there is a meaningful relationship between our independent variable (distance) and the dependent variable (fire damage).
In this exercise, we conduct a two-tailed test with a null hypothesis \( H_0: \beta_1 = 0 \) that suggests no relationship, and an alternative hypothesis \( H_a: \beta_1 eq 0 \) that indicates a significant relationship exists. The calculated t-value is 6.231, with degrees of freedom 28.

• Significance Level: 0.01
• Critical Value: 2.763
• Result: Since 6.231 is greater than 2.763, we reject the null hypothesis.

A large t-value in comparison to the critical value confirms that there is indeed a significant relationship between distance and fire damage.
This conclusion implies that the distance to the nearest fire station does have a substantial impact on the amount of fire damage.

Correlation Coefficient

The correlation coefficient, denoted by \( r \), measures the strength and direction of the linear relationship between two variables. In this scenario, we are investigating how distance affects fire damage. The value of \( r \) is derived from the square root of \( R^2 \), and here it turns out to be approximately 0.762.
This positive correlation coefficient signifies a strong positive correlation, meaning as the distance increases, the amount of fire damage also tends to increase. The sign of \( r \) (positive in this case) corresponds to the direction of the relationship indicated by the slope of our regression equation.
Such a correlation, being close to 1, suggests that distance from the fire station is a significant predictor of fire damage, justifying the decision-making in setting insurance rates based on this variable. A correlation of this magnitude is a clear indicator of the importance of location with respect to the nearest fire station.