Question

4. According to statistics reported on IN-CORP a surprising number of motor vehicles are not covered by insurance. Sample results, consistent with the IN-CORP report, showed 46 of 200 vehicles were not covered by insurance. a. What is the point estimate of the proportion of vehicles not covered by insurance? b. Develop a 95% confidence interval for the population proportion.

Answer 1

Answer:

a) 0.23

b) The 95% confidence interval for the population proportion is (0.1717, 0.2883).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of $\pi$, and a confidence level of $1-\alpha$, we have the following confidence interval of proportions.

$\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}$

In which

z is the zscore that has a pvalue of $1 - \frac{\alpha}{2}$.

Point estimate

The point estimate is:

$\pi = \frac{46}{200} = 0.23$

95% confidence level

So $\alpha = 0.05$, z is the value of Z that has a pvalue of $1 - \frac{0.05}{2} = 0.975$, so $Z = 1.96$.

The lower limit of this interval is:

$\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.23 - 1.96\sqrt{\frac{0.23*0.77}{200}} = 0.1717$

The upper limit of this interval is:

$\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.23 + 1.96\sqrt{\frac{0.23*0.77}{200}} = 0.2883$

The 95% confidence interval for the population proportion is (0.1717, 0.2883).