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?

a) Yes, the evidence supports the claim for Kentucky, and the results are applicable across the country.

b) Yes, the evidence supports the claim for Kentucky, but the results may not be applicable across the country.

c) No, the evidence does not support the claim for Kentucky, and the results are not applicable across the country.

d) No, the evidence does not support the claim for Kentucky, but the results may be applicable across the country.

Answer 1

The Kentucky sample does not provide enough evidence to refute the claim that 51.7% of homes are heated by natural gas. However, these findings from Kentucky cannot be generalized to infer natural gas usage patterns across the entire United States due to regional differences. None of the above option is correct.

Step-by-step explanation:

The question relates to hypothesis testing, and specifically whether a sample from Kentucky supports the claim that 51.7% of homes in the U.S. are heated by natural gas.

Based on a sample of 221 homes in Kentucky, where 115 were found to be heated by natural gas, a hypothesis test at the alpha level of 0.05 should be conducted to determine if these results support the national claim.

To analyze this, a proportion test is appropriate.

Given that the test statistic does not lead to the rejection of the null hypothesis, it is concluded that there is not enough evidence to suggest that the proportion of homes using natural gas in Kentucky differs from the national claim of 51.7%.

However, the results derived from the Kentucky sample cannot be assumed to be applicable across the country. Geographic variability in natural gas usage due to climate differences, and other socioeconomic factors suggest that different regions may have significantly different natural gas usage.

Therefore, a larger, more diverse sample would be required to generalize this claim nationally.

None of the above option is correct.