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

Step-by-step 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 : $\hat{p}=\dfrac{43}{136}=0.316176470588\approx0.3162$

Test statistic for proportion :-

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

$\Rightarrow z=\dfrac{0.3162-0.23}{\sqrt{\dfrac{0.23(1-0.23)}{136}}}\\\\\Rightarrow\ z=\dfrac{0.0862}{\sqrt{0.00130220588235}}\\\\\Rightarrow\ z=\dfrac{0.0862}{0.0360860898734}\\\\=2.38873206552\approx2.39$

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