Q:

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 Ξ± = 0.05. What is the value of the test statistic for this scenario? Round your answer to two decimal places.

Accepted Solution

A:
Answer: z =2.39Step-by-step explanation:Given : [tex]H_0: p \leq0.23\\\\ H_a: p > 0.23[/tex]A random sample of 136 households found that 43 consisted of one person.i.e. Sample size : n= 136Sample proportion : [tex]\hat{p}=\dfrac{43}{136}=0.316176470588\approx0.3162[/tex]Test statistic for proportion :-[tex]z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}[/tex][tex]\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[/tex]Hence, the value of the test statistic: z =2.39