EXAMPLE 5 Hypothesis test for the slope using the -value method and technology

shortmemory

Table 13.5: Table 4
Time Score
1 9
1 10
2 11
3 12
3 13
4 14
5 19
6 17
7 21
8 24

In Section 4.3, we considered a study on short-term memory. Ten subjects were given a set of nonsense words to memorize within a certain amount of time and were later scored on the number of words they could remember. The results are repeated here in Table 4. Use the -value method and technology to test, using level of significance , whether a linear relationship exists between time and score.

Solution

We begin by verifying the regression assumptions. The scatterplot of the residuals versus the fitted values in Figure 8 shows no strong evidence that the independence assumption, the constant variance assumption, or the zero-mean assumption is violated. Also, the normal probability plot of the residuals in Figure 9 offers evidence of the normality of the results. Therefore, we conclude that the regression assumptions are verified, and proceed with the hypothesis test.

  • Step 1 State the hypotheses and the rejection rule.

    • No linear relationship exists between time and score.
    • A linear relationship exists between time and score.

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    Figure 13.8: FIGURE 8 Residuals versus fitted values plot.
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    Figure 13.9: FIGURE 9 Normal probability plot of the residuals.

    The rejection rule is: reject if the .

  • Step 2 Calculate .

    From page 226 in Section 4.3, we have . From Example 13 in Chapter 4 on page 228, we have

    From the TI-83/84 summary statistics, we have the standard deviation of the (time) data to be . Thus, using the relationship we learned in Example 3:

    Therefore,

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    TI-83/84 summary statistics for (time) data.

  • Step 3 Find the -value. For instructions, see the Step-by-Step Technology Guide on page 730. The regression results (including the -value) for the TI-83/84, Excel, Minitab, and CrunchIt! are shown in Figures 10, 11, 12, and 13. (Differing results are due to rounding.)

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    Figure 13.10: FIGURE 10 TI-83/84 regression results.

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    Figure 13.11: FIGURE 11 Excel regression result.
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    Figure 13.12: FIGURE 12 Minitab regression results.
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    Figure 13.13: FIGURE 13 Crunchit! regression results.

  • Step 4 The -value of about , so we reject . Evidence exists, at level of significance , for a linear relationship between time and score.

NOW YOU CAN DO

Exercises 19–22.