Question 6.53

For Exercises 6.51 and 6.52, see pages 320–321; for 6.53 to 6.55, see pages 323–324; for 6.56 to 6.59, see pages 325–326; for 6.60 to 6.62, see pages 328–329; for 6.63 and 6.64, see page 331; and for 6.65 and 6.66, see page 332.

Question 6.53

6.53 Spending on housing.

The Census Bureau reports that households spend an average of 31% of their total spending on housing. A homebuilders association in Cleveland wonders if the national finding applies in its area. It interviews a sample of 40 households in the Cleveland metropolitan area to learn what percent of their spending goes toward housing. Take $μ$ to be the mean percent of spending devoted to housing among all Cleveland households. We want to test the hypotheses

$\begin{array}{l} H_{0} : μ = 31 % \\ H_{a} : μ \neq 31 % \end{array}$

The population standard deviation is $σ = 9.6 %$ .

The study finds $\bar{x} = 28.6 %$ for the 40 households in the sample. What is the value of the test statistic $z$ ? Sketch a standard Normal curve, and mark $z$ on the axis. Shade the area under the curve that represents the $P$ -value.
Calculate the $P$ -value. Are you convinced that Cleveland differs from the national average?

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6.53

(a) $Z = - 1.58$ . (b) 0.1142. No.