Question

Over the past several years, the proportion of one-person households has been increasing. The Census Bureau would like to test the hypothesis that the proportion of one-person households exceeds 0.23. The hypotheses of interest are given below: H0: p ≤ 0.23 Ha: p > 0.23 A random sample of 136 households found that 43 consisted of one person. The Census Bureau would like to set α = 0.05. What is the value of the test statistic for this scenario? Round your answer to two decimal places.

Answer 1

Answer: z =2.39

Explanation:

Given :

`H_0: p \leq0.23\\\\ H_a: p > 0.23`

A random sample of 136 households found that 43 consisted of one person.

i.e. Sample size : n= 136

Sample proportion :
`\hat{p}=(43)/(136)=0.316176470588\approx0.3162`

Test statistic for proportion :-

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

`\Rightarrow z=\frac{0.3162-0.23}{\sqrt{(0.23(1-0.23))/(136)}}\\\\\Rightarrow\ z=(0.0862)/(√(0.00130220588235))\\\\\Rightarrow\ z=(0.0862)/(0.0360860898734)\\\\=2.38873206552\approx2.39`

Hence, the value of the test statistic: z =2.39