Question

question 5
juries should have the same racial distribution as the surrounding communities. according to the u.s. census bureau, 18% of residents in minneapolis, minnesota, are african americans. suppose a local court will randomly sample 100 state residents and will then observe the proportion in the sample who are african american. how likely is the resulting sample proportion to be between 0.066 and 0.294 (i.e., 6.6% to 29.4% african american)?
a. there is roughly a 68% chance that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.
b. there is roughly a 95% chance that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.
c. it is certain that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.
d. there is roughly a 99.7% chance that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.

Explanation:

Step1: Identify the mean and standard deviation of the sampling distribution

The population proportion $p = 0.18$ and the sample size $n=100$. The mean of the sampling - distribution of the sample proportion $\hat{p}$ is $\mu_{\hat{p}}=p = 0.18$, and the standard deviation is $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.18\times(1 - 0.18)}{100}}=\sqrt{\frac{0.18\times0.82}{100}}=\sqrt{\frac{0.1476}{100}} = 0.0384$.

Step2: Calculate the z - scores

For $\hat{p}_1 = 0.066$, the z - score is $z_1=\frac{\hat{p}_1-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.066 - 0.18}{0.0384}=\frac{- 0.114}{0.0384}\approx - 3$.
For $\hat{p}_2 = 0.294$, the z - score is $z_2=\frac{\hat{p}_2-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.294 - 0.18}{0.0384}=\frac{0.114}{0.0384}\approx3$.

Step3: Use the empirical rule

The empirical rule for a normal distribution states that approximately 99.7% of the data lies within 3 standard deviations of the mean. Since $z_1\approx - 3$ and $z_2\approx3$, there is roughly a 99.7% chance that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.

Answer:

D. There is roughly a 99.7% chance that the resulting sample proportion will be between 0.066 and 0.294 of the true proportion.