In the previous sections, we presented the basic elements necessary to fit and use a straight-line regression model. In this section, we will assemble these elements by applying them in an example with the aid of computer software.

Suppose a fire insurance company wants to relate the amount of fire damage in major residential fires to the distance between the burning house and the nearest fire station. The study is to be conducted in a large suburb of a major city; a sample of 15 recent fires in this suburb is selected. The amount of damage, y, and the distance between the fire and the nearest fire station, x, are recorded for each fire. The results are given in Table 9.5 and saved in the FIREDAM file.

1. Step 1 First, we hypothesize a model to relate fire damage, y, to the distance from the nearest fire station, x. We hypothesize a straight-line probabilistic model:

   $$y={\beta }_0+{\beta }_{1}x+\epsilon$$

2. Step 2 Next, we open the FIREDAM file and use statistical software to estimate the unknown parameters in the deterministic component of the hypothesized model. The SAS printout for the simple linear regression analysis is shown in Figure 9.25. The least squares estimate of the slope ${\beta }_1$ and intercept ${\beta }_0,$ highlighted on the printout, are

   $$\begin{array}{r}{\widehat{\beta }}_1=4.91933\end{array}$$
   $$\begin{array}{r}{\widehat{\beta }}_0=10.27793\end{array}$$

   and the least squares equation is (rounded)

   $$\widehat{y}=10.28+4.92x$$

   This prediction equation is graphed by MINITAB in Figure 9.26, along with a plot of the data points.

   

   Figure 9.25

   The least squares estimate of the slope, ${\widehat{\beta }}_1=4.92,$ implies that the estimated mean damage increases by $4,920 for each additional mile from the fire station. This interpretation is valid over the range of x, or from .7 to 6.1 miles from the station. The estimated y-intercept, ${\widehat{\beta }}_0=10.28,$ has the interpretation that a fire 0 miles from the fire station has an estimated mean damage of $10,280. Although this would seem to apply to the fire station itself, remember that the y-intercept is meaningfully interpretable only if $x=0$ is within the sampled range of the independent variable. Since $x=0$ is outside the range, ${\widehat{\beta }}_0$ has no practical interpretation.

   

   Figure 9.26

3. Step 3 Now we specify the probability distribution of the random-error component $\epsilon .$ The assumptions about the distribution are identical to those listed in Section 9.3. Although we know that these assumptions are not completely satisfied (they rarely are for practical problems), we are willing to assume that they are approximately satisfied for this example. The estimate of the standard deviation $\sigma$ of $\epsilon ,$ highlighted on the SAS printout, is

   $$s=2.31635$$

   This implies that most of the observed fire damage (y) values will fall within approximately $2s=4.64$ thousand dollars of their respective predicted values when the least squares line is used.

4. Step 4 We can now check the usefulness of the hypothesized model—in other words, whether x really contributes information for the prediction of y by the straight-line model. First, test the null hypothesis that the slope ${\beta }_1$ is 0—that is, that there is no linear relationship between fire damage and the distance from the nearest fire station—against the alternative hypothesis that fire damage increases as the distance increases. We test

   $$H_0:{\beta }_1=0$$
   $$H_{\mathrm{a}}:{\beta }_1>0$$

   The two-tailed observed significance level for testing $H_{\mathrm{a}}:{\beta }_1\ne 0,$ highlighted on the SAS printout, is less than .0001. Thus, the p-value for our one-tailed test is less than half of this value (.00005). This small p-value leaves little doubt that mean fire damage and distance between the fire and the fire station are at least linearly related, with mean fire damage increasing as the distance increases.

   We gain additional information about the relationship by forming a confidence interval for the slope ${\beta }_1.$ A 95% confidence interval, highlighted on the SAS printout, is (4.071, 5.768). Thus, with 95% confidence, we estimate that the interval from $4,071 to $5,768 encloses the mean increase $({\beta }_1)$ in fire damage per additional mile in distance from the fire station.

   Another measure of the utility of the model is the coefficient of determination, $r^2.$ The value (also highlighted on the printout) is $r^2=.9235,$ which implies that about 92% of the sample variation in fire damage (y) is explained by the distance (x) between the fire and the fire station.

   The coefficient of correlation, r, that measures the strength of the linear relationship between y and x is not shown on the SAS printout and must be calculated. Using the facts that $r=\sqrt{r^2}$ in simple linear regression and that r and ${\widehat{\beta }}_1$ have the same sign, we calculate

   $$r=+\sqrt{r^2}=\sqrt{.9235}=.96$$

   The high correlation confirms our conclusion that ${\beta }_1$ is greater than 0; it appears that fire damage and distance from the fire station are positively correlated. All signs point to a strong linear relationship between y and x.

5. Step 5 We are now prepared to use the least squares model. Suppose the insurance company wants to predict the fire damage if a major residential fire were to occur 3.5 miles from the nearest fire station. The predicted value (highlighted at the bottom of the SAS printout) is $\widehat{y}=27.496,$ while the 95% prediction interval (also highlighted) is (22.324, 32.667). Therefore, with 95% confidence, we predict fire damage in a major residential fire 3.5 miles from the nearest station to be between $22,324 and $32,667.