Question

4 it is claimed that less than half of households in the United States have carbon monoxide detectors. A survey of 1005 households resulted in 462 who have carbon monoxide detectors. Find the value of the test statistic Z

Answer 1

Answer:

The value of the test statistic is -2.54.

Step-by-step explanation:

A hypothesis test is conducted to determine whether the proportion of households in the United States having carbon monoxide detectors is less than 50% or not.

The hypothesis is:

$H_{o}$: The proportion of households in the United States having carbon monoxide detectors is less than 50%, i.e. p < 0.50.

$H_{a}$: The proportion of households in the United States having carbon monoxide detectors is not less than 50%, i.e. p ≥ 0.50.

The test statistic is:

$z=\frac{\hat p -p}{\sqrt{\frac{p(1-p)}{n} }}$

Given:

Sample size, n = 1005

Number of households that have carbon monoxide detectors is, X = 462.

The sample proportion is: $\hat p=\frac{X}{n}= \frac{462}{1005}= 0.4597\approx0.46$

Compute the value of the test statistic as follows:

$z=\frac{\hat p -p}{\sqrt{\frac{p(1-p)}{n} }}\\=\frac{0.46-0.50}{\sqrt{\frac{0.50(1-0.50)}{1005}}} \\=-2.54$

Thus, the value of the test statistic is -2.54.