Example 16

EXAMPLE 16 Equivalence of two-tailed tests and confidence intervals

Recall Example 4 from Section 8.1 (page 432), where we were 90% confident using a $Z$ interval that the population mean score on the 2014 SAT Math test lies between 471.2 and 548.8. Test, using level of significance $α = 0.10$ , 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 $μ_{0}$ as necessary, as long as we use level of significance $α = 0.10$ (see Table 7).

If any values of $μ_{0}$ lie inside the confidence interval, that is, between 471.2 and 548.8, we will not reject $H_{0}$ for this value of $μ_{0}$ .
If any values of $μ_{0}$ lie outside the confidence interval, that is, either to the left of 471.2 or to the right of 548.8, we will reject $H_{0}$ , as shown in Figure 16.

FIGURE 16 Reject $H_{0}$ for values of $μ_{0}$ that lie outside (471.2, 548.8).

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

$H_{0} : μ = 470 versus H_{a} : μ \neq 470$
$H_{0} : μ = 510 versus H_{a} : μ \neq 510$
$H_{0} : μ = 550 versus H_{a} : μ \neq 550$

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To perform each hypothesis test, simply observe where each value of $μ_{0}$ falls on the number line shown in Figure 16. For example, in the first hypothesis test, the hypothesized value $μ_{0} = 470$ lies outside the interval (471.2, 548.8). Thus, we reject $H_{0}$ . The three hypothesis tests are summarized here.

Value of $μ_{0}$	Form of hypothesis test, with $α = 0.10$	Where $μ_{0}$ lies in relation to 90% confidence interval	Conclusion of hypothesis test
a. 470	$H_{0} : μ = 470 vs . H_{a} : μ \neq 470$	Outside	Reject $H_{0}$
b. 510	$H_{0} : μ = 510 vs . H_{a} : μ \neq 510$	Inside	Do not reject $H_{0}$
c. 550	$H_{0} : μ = 550 vs . H_{a} : μ \neq 550$	Outside	Reject $H_{0}$

NOW YOU CAN DO

Exercises 41–46.