The Energy Information Administration reported that $51.5 \%$ of homes in the United States were heated by natural gas. A random sample of 200 homes found that 111 heated by natural gas. Does the evidence support the claim or has the percentage changed? Use $\alpha = 0.05$ and the $P$-value method.
A) Yes. There is not enough evidence to reject the claim that the percentage of homes that are heated by natural gas is 51.5%.
B) No. There is enough evidence to reject the claim that the percentage of homes that are heated by natural gas is 51.5%.
C) There is not enough information to draw a conclusion.

Question

Quizplus · Accepted Answer

To determine if the evidence supports the claim or if the percentage has changed, we perform a hypothesis test for a proportion. The null hypothesis $H_0$ is that the true proportion $p = 0.515$, and the alternative hypothesis $H_a$ is that $p \neq 0.515$. The sample proportion is $\hat{p} = \frac{111}{200} = 0.555$. The test statistic is calculated using the formula $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$, where $p_0 = 0.515$, $\hat{p} = 0.555$, and $n = 200$. Plugging in the values, we calculate the z-value and then find the corresponding P-value. If the P-value is greater than $\alpha = 0.05$, we do not have enough evidence to reject the null hypothesis, meaning the claim that 51.5% of homes are heated by natural gas cannot be rejected based on this sample.

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The Energy Information Administration Reported That $51.5 \%$ Of Homes in the United States Were Heated by Natural