in a beach town, 13% of the residents own boats. a random sample of 100 residents was selected. what is the…

Explanation:

Step1: Identify population proportion and sample size

Let $p = 0.13$ (population proportion of residents who own boats) and $n=100$ (sample size).

Step2: Calculate mean and standard deviation of sampling - distribution of sample proportion

The mean of the sampling - distribution of the sample proportion $\hat{p}$ is $\mu_{\hat{p}}=p = 0.13$. The standard deviation is $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.13\times(1 - 0.13)}{100}}=\sqrt{\frac{0.13\times0.87}{100}}=\sqrt{\frac{0.1131}{100}}=\sqrt{0.001131}\approx0.0336$.

Step3: Calculate the z - score

We want to find $P(\hat{p}<0.11)$. The z - score is calculated as $z=\frac{\hat{p}-\mu_{\hat{p}}}{\sigma_{\hat{p}}}=\frac{0.11 - 0.13}{0.0336}=\frac{- 0.02}{0.0336}\approx - 0.595$.

Step4: Find the probability using the z - table

Looking up the z - score of $-0.595$ in the z - table, we find the corresponding probability. The closest z - score in the table is $z=-0.60$, and the probability $P(Z < - 0.60)=0.2743\approx0.276$.

Answer:

A. 0.276