Final answer:
This is a hypothesis test to determine if the proportion of homes heated by natural gas in Kentucky is different from the national proportion. The results indicate that there is not enough evidence to support this claim at the 5 percent significance level.
Step-by-step explanation:
This question requires conducting a hypothesis test to determine if the proportion of homes heated by natural gas in Kentucky is different from the national proportion.
A) One-sample z-test for a proportion.
B) The test distribution to use is the standard normal distribution.
C) The random variable is the proportion of homes heated by natural gas in Kentucky.
To proceed with the hypothesis test, we can calculate the test statistic by first finding the standard error of the proportion and then using it to calculate the z-score. Finally, we can compare the z-score to the critical value to make a decision.
Using the given numbers, we have:
P-hat = 115/221 = 0.520 (sample proportion)
q-hat = 1 - p-hat = 0.480 (complement of the sample proportion)
Standard error = sqrt( ( p-hat * q-hat ) / n ) = sqrt( ( 0.520 * 0.480 ) / 221 ) ≈ 0.032
Z-score = ( p - P-hat ) / SE = ( 0.517 - 0.520 ) / 0.032 ≈ -0.094
Looking up the critical value for a one-tailed test with α = 0.05, we find the z-value to be approximately 1.645. Since -0.094 < -1.645, we fail to reject the null hypothesis.
Therefore, at the 5 percent significance level, there is not enough evidence to conclude that the proportion of homes in Kentucky heated by natural gas is significantly different from the national proportion of 0.517.