The proportion of homes in Illinois owned by investors rather than homeowners is \( 19 \% \). 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? Use the formula \( n=\frac{p(1-p)}{\sigma_{p}^{2}} \) 77 770 1539 2039

To find the minimum sample size \( n \) for the given proportion of homes in Illinois owned by investors, we can use the formula: \[ n = \frac{p(1-p)}{\sigma_{p}^{2}} \] Given that \( p = 0.19 \) and \( \sigma_{p} = 0.01 \), we first need to compute \( \sigma_{p}^{2} = (0.01)^{2} = 0.0001 \). Now, calculate \( p(1-p) = 0.19(1-0.19) = 0.19 \times 0.81 = 0.1539 \). Plugging this back into the formula gives: \[ n = \frac{0.1539}{0.0001} = 1539 \] Thus, the minimum sample size is \( 1539 \). The correct answer is **1539**.