Problem

Automobile Collisions. The number of collision claims (COLLISION) reported for 1984–1986 c...

Automobile Collisions. The number of collision claims (COLLISION) reported for 1984–1986 cars are listed in the same categories as described for Example 9.1 and are available in a file named CRASH9 on the CD. For MINITAB, the file will have two columns, one with the data and one with a classification variable. The Excel spreadsheet will have nine columns containing the data for each type of car. (See Using tire Computer for more on setting up data for analysis with ANOVA).

Using the classification variable (CARCLAS) described in that example, an ANOVA was run on the number of collisions. Figure 9.7 summarizes the results. Determine whether there is a difference in the average number of collisions for different types of cars. Use a 5% level of significance. State the hypotheses to be tested, the decision rule, the test statistic, and your decision. (source: Copyright, September 28, 1987. U.S. News and World Report. Used with permission.)

FIGURE 9.7: ANOVA Results for Automobile Collisions Example.

Example 9.1:

Automobile Injuries Injury claims (INJURY) for 1984–1986 cars are listed in Table 9.1 (see the file INJURY9 on the CD) for cars in the following categories:

small two-door
midsized two-door
large two-door
small four-door
midsized four-door
large four-door
small station wagons and vans
midsized station wagons and vans
large station wagons and vans

The variable CARCLAS is used to indicate into which category each car falls. The categories are coded from 1 to 9 in the order shown.

TABLE 9.1: Data for Automobile Injuries Example

Source: Copyright. September 28, 1987. U.S. News and World Report. Reprinted with permission.

In ANOVA terminology, the dependent variable is INJURY. Our concern is whether the average number of injuries differs for the nine different categories or populations. Our data comprise nine samples from each of these nine populations. The explanatory variable or factor is CARCLAS. Note that this is a qualitative variable. ANOVA is designed to investigate the relationship between a quantitative dependent variable and a qualitative independent variable or factor. There are nine levels of this factor because there are nine types of cars coded. The averages of the injuries for each type of car are the factor level means.

We want to test whether the factor level means are equal:

H0: μ1 = μ2 = ⋯ = μ9

Ha: Not all of the means are equal

Figure 9.1 shows the MINITAB ANOVA output for this problem. Figure 9.2 shows the Excel output and Figure 9.3 shows the SAS output. Figure 9.4 shows the general form of the ANOVA table for each of the software packages.

In Figure 9.4(a). the general form of the MINITAB output is shown. The sums of squares due to the treatment variable, error, and the total sum of squares are printed along with their degrees of freedom. MSTR, MSE, the F statistic, and a p value associated with the observed F statistic are computed. Below the ANOVA table, MINITAB provides the standard error, S (se in this text), which is the square root of MSE, the R2 and . These values have interpretations similar to regression. This makes sense when you think of ANOVA as a special case of regression on indicator variables. MINITAB also lists summary information for the factor levels. The sample size, mean, and standard deviation for the observations in each factor-level are given. Individual confidence intervals for each factor-level (population) mean are shown graphically. (Note that these intervals should be used with caution for comparing two factor-level means. Better approaches are discussed later.)