EXAMPLE 14 Performing a set of tests for the significance of a set of individual variables

Using the results from Example 13, do the following, using level of significance :

  • Test 1: Test whether a significant linear relationship exists between rating and vitamins.
  • Test 2: Test whether a significant linear relationship exists between rating and sodium.

Solution

The regression assumptions were verified in Example 13.

  • Step 1 For each hypothesis test, state the hypotheses and the rejection rule. Vitamins is , and sodium is , so the hypotheses are

    • Test 1:
      • . No linear relationship exists between rating and vitamins.
      • . A linear relationship exists between rating and vitamins.
      • Reject if the .
    • Test 2:
      • . No linear relationship exists between rating and sodium.
      • . A linear relationship exists between rating and sodium.
      • Reject if the .
  • Step 2 For each hypothesis test, find the statistic and the -value. Figure 25 is an excerpt of the Minitab results from Figure 24, with the relevant statistics and -values highlighted. For Test 1, the -value represents , because it is a two-tailed test. For Test 2, the -value represents .
    • Test 1: , with -value 0.336.
    • Test 2: , with -value 0.002.
    image
    Figure 13.26: FIGURE 25 statistics and -values for the test for the significance of the variables vitamins and sodium.
  • Step 3 For each hypothesis test, state the conclusion and interpretation.
    • Test 1: The , which is not . Therefore, we do not reject . There is insufficient evidence of a linear relationship between rating and vitamins when sodium is held constant. Perhaps surprisingly, the variable vitamins is not significant, meaning that the amount of vitamins in the breakfast cereal is not helpful in predicting nutritional rating.
    • Test 2: The , which is . Therefore, we reject . There is evidence of a linear relationship between rating and sodium when vitamins is held constant. The variable sodium is significant, meaning that the amount of sodium in the breakfast cereal is helpful in predicting nutritional rating.

NOW YOU CAN DO

Exercises 23 and 30.