Question

according to statistics reported on cnbc, a surprising number of motor vehicles are not covered by insurance. sample results, consistent with the cnbc report, showed 23 of 174 vehicles were not covered by insurance.
a. what is the point estimate of the proportion of vehicles not covered by insurance (to 4 decimals)?
0.1322
b. develop a 95% confidence interval for the population proportion (to 4 decimals).
()

Explanation:

Step1: Calculate the standard error

The formula for the standard error of a proportion is $SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion and $n$ is the sample size.
Given $\hat{p}=0.1322$ and $n = 174$.
$SE=\sqrt{\frac{0.1322\times(1 - 0.1322)}{174}}=\sqrt{\frac{0.1322\times0.8678}{174}}\approx\sqrt{\frac{0.1147}{174}}\approx\sqrt{0.000659}\approx0.0257$.

Step2: Find the z - value

For a 95% confidence interval, the z - value $z_{\alpha/2}$ is 1.96.

Step3: Calculate the margin of error

The margin of error $ME=z_{\alpha/2}\times SE$.
$ME = 1.96\times0.0257\approx0.0504$.

Step4: Calculate the confidence interval

The confidence interval for the population proportion $p$ is given by $\hat{p}-MELower limit: $0.1322-0.0504 = 0.0818$.
Upper limit: $0.1322 + 0.0504=0.1826$.

Answer:

$(0.0818,0.1826)$