EXAMPLE 13 test for the overall significance of the multiple regression

For the breakfast cereal data, we are interested in determining whether a linear relationship exists between , and and .

  1. Determine whether the regression assumptions have been violated.
  2. Perform the test for the overall significance of the multiple regression of rating on vitamins and sodium, using level of significance .

Solution

  1. Figure 22, a scatterplot of the residuals versus fitted values, contains no strong evidence of unhealthy patterns. Although the two cereals, All-Bran with Extra Fiber and Product 19, are unusual, the vast majority of the data suggest that the independence, constant variance, and zero-mean assumptions are satisfied. Figure 23 indicates that the normality assumption is satisfied.
  2. The Minitab multiple regression results are provided in Figure 24.

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    Figure 13.23: FIGURE 22 Scatterplots of residuals versus fitted values.
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    Figure 13.24: FIGURE 23 Normal probability plot of the residuals.
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    Figure 13.25: FIGURE 24 Minitab results for regression of rating on vitamins and sodium.

  • Step 1 State the hypotheses and the rejection rule. We have variables, so the hypotheses are

    • . No linear relationship exists between rating and either vitamins and sodium. The overall multiple regression is not significant.
    • and . A linear relationship exists between rating and at least one of vitamins and sodium. The overall multiple regression is significant.

    Reject if the .

  • Step 2 Find the statistic and the -value. These are located in the ANOVA table portion of the printout, denoted Analysis of Variance. From Figure 24, we have and . The -value represents .
  • Step 3 Conclusion and interpretation. The , so we reject . There is evidence, at level of significance , for a linear relationship between rating and at least one of vitamins and sodium. The overall multiple regression is significant.

The ANOVA table is a convenient way to organize a set of statistics, which can be used to perform multiple regression as well as ANOVA.

NOW YOU CAN DO

Exercises 21, 22, 27, and 28.