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 α = 0.05 level in Kentucky? Are the results applicable across the country? Why?

Short answer

We would need to expand our sample base to include these possibilities if we wanted to generalize this claim to the entire nation.

Step-by-step solution

State the null and alternate hypothesis. we have to conduct hypothesis test if homes of Kentucky take 51.7% of the natural gas according to US Department of Energy or not, on average. 

$H_0:p=0.517;H_a:p\ne 0.517$

Calculate the p-value using the normal distribution for proportions: 

$p-value=0.9203$

In one to two complete sentences, explain what the p-value means for this problem. If the null hypothesis is true (the proportion is 0.517), then there is a 0.9203 probability that the sample (estimated) proportion is 0.520 or more.

Compare α and the p-value:Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.

alpha	decision	reason for decision
0.05	Do not reject the null hypothesis.	$p-value>0.05$

Conclusion: At the 5% significance level, there is not enough evidence to conclude that the proportion of homes in Kentucky that are heated by natural gas is 0.517.

Are the results applicable across the country

However, we cannot generalize this result to the entire nation. First, the sample’s population is only the state of Kentucky. Second, it is reasonable to assume that homes in the extreme north and south will have extreme high usage and low usage, respectively. We would need to expand our sample base to include these possibilities if we wanted to generalize this claim to the entire nation.