Question 8 $7 / 8$ answered The proportion of homes in Vermont owned by investors rather than homeowners is $17 \%$. Consider a sampling distribution where $n=140$. (a) A news headline claims that more than a third of all homes sold are being sold to investors rather than homeowners. Determine the $P(X>0.33)$ for Vermont to determine if the claim is reasonable. \[ P(X>0.33)= \] Is the claim reasonable? The claim is unreasonable. If $17 \%$ of homes are being sold to investors, there is less than a $5 \%$ chance that a sample of homes sold had more than $33 \%$ sold to investors by chance alone The claim is reasonable. If 0.17 of homes are being sold to investors, there is more than a $5 \%$ chance that a sample of homes sold had more than $33 \%$ sold to investors by chance alone (b) What is the probability that between $14 \%$ and $20 \%$ of homes sold in Vermont are sold to investors?

Step 1 ：First, we need to calculate the mean and standard deviation of the binomial distribution. The mean is np and the standard deviation is sqrt(np(1-p)).

Step 2 ：Given that n = 140, p = 0.17, we can calculate the mean as \(140 \times 0.17 = 23.8\)

Step 3 ：The standard deviation is calculated as \(\sqrt{140 \times 0.17 \times (1 - 0.17)} = 4.444547221033882\)

Step 4 ：Next, we calculate the z-score for 0.33. The z-score is calculated as \((0.33 \times 140 - 23.8) / 4.444547221033882 = 5.0398834540426725\)

Step 5 ：Finally, we find the probability that the z-score is greater than the calculated value. This is the probability that more than 33% of homes are sold to investors.

Step 6 ：The probability that more than 33% of homes are sold to investors is approximately \(2.33 \times 10^{-7}\)

Step 7 ：This means that it is very unlikely that more than 33% of homes are sold to investors if the true proportion is 17%. Therefore, the claim that more than a third of all homes sold are being sold to investors rather than homeowners is not reasonable.

Step 8 ：Final Answer: The probability that more than 33% of homes are sold to investors is approximately \(\boxed{2.33 \times 10^{-7}}\). The claim that more than a third of all homes sold are being sold to investors rather than homeowners is not reasonable.