The proportion of homes in Utah owned by investors rather than homeowners is \( 27 \% \). In order to determine if the percent of homes owned by investors has changed, researchers take a sample of 300 recently purchased homes and find that 97 are owne by investors. Complete the formula and calculation of the standard deviation. \( b= \) Your answer should be a decimal rounded to 2 places.

To calculate the standard deviation for the proportion of homes owned by investors, you can use the formula for the standard deviation of a sample proportion, which is: \[ b = \sqrt{\frac{p(1-p)}{n}} \] where \( p \) is the sample proportion (in this case, the proportion of homes owned by investors), and \( n \) is the sample size. 1. Calculate the sample proportion \( p \): - In your case, \( p = \frac{97}{300} \approx 0.3233 \). 2. Now substitute \( p \) and \( n = 300 \) into the formula: - First, calculate \( p(1-p) \): \[ p(1-p) = 0.3233(1 - 0.3233) \approx 0.3233 \times 0.6767 \approx 0.2185 \] 3. Now substitute this value into the formula: \[ b = \sqrt{\frac{0.2185}{300}} \approx \sqrt{0.0007283} \approx 0.0270 \] So, rounding to two decimal places, the value of \( b \) is approximately \( 0.03 \).