Example 23

EXAMPLE 23 Using a confidence interval to perform two-tailed $t$ tests

In Example 14 of Chapter 8 (page 452), we found the 99% $t$ -confidence interval for $μ$ , the population mean sodium content per serving of all breakfast cereals, to be the following:

$\begin{array}{l} Lower bound = 144.1 grams & Upper bound = 227.7 grams \end{array}$

Test, using level of significance $α = 0.01$ , whether the population mean amount of sodium differs from the following values: (a) 100 grams, (b) 170 grams, (c) 250 grams.

Solution

The key words “differs from” mean that we are using two-tailed tests. Then, for each hypothesized value of $μ_{0}$ , we determine whether it falls inside or outside the given confidence interval.

$H_{0} : μ = 100 versus H_{a} : μ \neq 100$

The confidence interval is (144.1, 227.7), and because $μ_{0} = 100$ lies outside the interval (see Figure 33), we reject $H_{0}$ .

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$\begin{array}{l} H_{0} : μ = 170 & versus & H_{a} : μ \neq 170 \end{array}$
$μ_{0} = 170$ lies inside the interval, so we do not reject $H_{0}$ .
$\begin{array}{l} H_{0} : μ = 250 & versus & H_{a} : μ \neq 250 \end{array}$
$μ_{0} = 250$ lies outside the interval, so we reject $H_{0}$ .

FIGURE 33 Reject $H_{0}$ for values of $μ_{0}$ that lie outside (144.1, 227.7).

NOW YOU CAN DO

Exercises 31–36.