$0 / 8$ answered The proportion of homes in Vermont owned by investors rather than homeowners is $17 \%$. Consider a sampling distribution of $\hat{p}$. If the standard deviation of the sampling distribution can be no more than 0.01 , what is the minimum sample size for the sampling distribution? \[ n= \] Round to the nearest whole number Submit Next $\rightarrow$
Solution
Step 1 ：We are given that the proportion of homes in Vermont owned by investors rather than homeowners is 17%, which is 0.17 in decimal form. This is represented as p in our calculations.
Step 2 ：We are also given that the standard deviation of the sampling distribution can be no more than 0.01. This is represented as \(\sigma\) in our calculations.
Step 3 ：The standard deviation of the sampling distribution (also known as the standard error) can be calculated using the formula: \[\sigma = \sqrt{\frac{p(1-p)}{n}}\] where p is the proportion of success (in this case, the proportion of homes owned by investors), and n is the sample size.
Step 4 ：We need to find the minimum sample size n, so we can rearrange the formula to solve for n: \[n = \frac{p(1-p)}{\sigma^2}\]
Step 5 ：Substituting the given values into the formula, we get: \[n = \frac{0.17(1-0.17)}{0.01^2}\]
Step 6 ：Solving the above expression, we find that the minimum sample size for the sampling distribution is \(\boxed{1411}\).
