Example 33

EXAMPLE 33 Using confidence intervals for $σ$ to conduct two-tailed $χ^{2}$ tests for $σ$

A 95% confidence interval for the population mean sodium content of breakfast cereals, in milligrams (mg) per serving, is given by

$(44.53 mg, 81.50 mg)$

Assume that the data are normally distributed. Test, using level of significance $α = 0.05$ , whether $σ$ differs from the following.

Solution

For the hypothesis test $H_{0} : σ = 80$ versus $H_{a} : σ \neq 80, σ_{0} = 80$ lies between the lower bound 44.53 and the upper bound 81.50 of the confidence interval, and we therefore do not reject $H_{0}$ . There is insufficient evidence that the population standard deviation of sodium content differs from 80 mg.
For the hypothesis test $H_{0} : σ = 40 versus H_{a} : σ \neq 40, σ_{0} = 40$ lies outside the confidence interval, and we therefore reject $H_{0}$ . There is evidence that the population standard deviation of sodium content differs from 40 mg.

NOW YOU CAN DO

Exercises 19–22.