Question

The US Department of Energy reported that $51.7\%$ of homes were heated by natural gas. A random sample of $221$homes in Kentucky found that $115$ were heated by natural gas. Does the evidence support the claim for Kentucky at the $\alpha =0.05$ level in Kentucky? Are the results applicable across the country? Why?

Short answer

The results support the claim of Kentucky.

The results are not applicable across the country because the sample was selected from Kentucky only and not the representative.

Step-by-step solution

Given information

$x=115,n=221,p_0=0.517,a=0.05$

Explanation

State the hypothesis:

The null hypothesis states that $51.7\%$of homes were heated by natural gas. In symbols:

$H_0:p=0.517$

The alternative hypothesis states that not $51.7\%$of homes were heated by natural gas. In symbols:

$H_a:p\ne 0.517$

The sample proportion is:

$$\begin{gathered} \stackrel{⏜}{P}=\frac{x}{n} \\ =\frac{115}{221} \\ =0.520 \end{gathered}$$

The test statistic is:

$$\begin{gathered} z=\frac{\stackrel{⏜}{p}-p_0}{\sqrt{{\displaystyle \frac{p_0\left(1-p_0\right)}{n}}}} \\ =\frac{0.520-0.517}{\sqrt{{\displaystyle \frac{0.517\left(1-0.517\right)}{221}}}} \\ =\frac{0.003}{0.0336} \\ =0.10 \end{gathered}$$

The p-value is:

$$\begin{gathered} p-value=P\left(z<-0.10\right)+P\left(z>0.10\right) \\ =0.4602+0.4602 \\ =0.9204 \end{gathered}$$

Since $p-value=0.9203>0.05$, do not reject the null hypothesis. There is sufficient evidence to conclude that $51.7\%$of homes were heated by natural gas. The results are not applicable across the country because the sample was selected from Kentucky only and may not be representative of the country.