EXAMPLE 16 Equivalence of two-tailed tests and confidence intervals

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Recall Example 4 from Section 8.1 (page 432), where we were 90% confident using a interval that the population mean score on the 2014 SAT Math test lies between 471.2 and 548.8. Test, using level of significance , whether the population mean SAT Math test score differs from these values: (a) 470, (b) 510, (c) 550.

Solution

Once we have the 90% confidence interval, we may test as many possible values for as necessary, as long as we use level of significance (see Table 7).

  • If any values of lie inside the confidence interval, that is, between 471.2 and 548.8, we will not reject for this value of .
  • If any values of lie outside the confidence interval, that is, either to the left of 471.2 or to the right of 548.8, we will reject , as shown in Figure 16.
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    Figure 9.19: FIGURE 16 Reject for values of that lie outside (471.2, 548.8).

We set up the three two-tailed hypothesis tests as follows:

518

To perform each hypothesis test, simply observe where each value of falls on the number line shown in Figure 16. For example, in the first hypothesis test, the hypothesized value lies outside the interval (471.2, 548.8). Thus, we reject . The three hypothesis tests are summarized here.

Value of Form of hypothesis test,
with 		Where lies in
relation to 90%
confidence interval		 Conclusion of
hypothesis test
a. 470 Outside Reject
b. 510 Inside Do not reject
c. 550 Outside Reject

NOW YOU CAN DO

Exercises 41–46.