Constructing and Interpreting Confidence Intervals. In Exercises 13–16, use the given sample data and confidence level. In each case, (a) find the best point estimate of the population proportion p; (b) identify the value of the margin of error E; (c) construct the confidence interval; (d) write a statement that correctly interprets the confidence interval.

Medical Malpractice In a study of 1228 randomly selected medical malpractice lawsuits, it was found that 856 of them were dropped or dismissed (based on data from the Physicians Insurers Association of America). Construct a 95% confidence interval for the proportion of medical malpractice lawsuits that are dropped or dismissed.

In a sample of 1228 medical malpractice lawsuits, 856 were dropped or dismissed.

(a)

The best point estimate of the proportion of medical malpractice lawsuits that are dropped or dismissed is computed below:

$$\begin{gathered} \hat{p}=\frac{856}{1228} \\ =0.697 \end{gathered}$$

Thus, the sample proportion of medical malpractice lawsuits that are dropped or dismissed equal to 0.697 is the best point estimate of the proportion of medical malpractice lawsuits that are dropped or dismissed.

(b)

The confidence level is equal to 95%. Thus, the corresponding level of significance is equal to 0.05.

From the standard normal distribution table, the right-tailed value of $z_{\frac{\alpha }{2}}$ for$\alpha =0.05$ is equal to 1.96.

The margin of error is calculated below:

$$\begin{gathered} E=1.96\times \sqrt{\frac{0.697\times 0.303}{1228}} \\ =0.0257 \end{gathered}$$

Thus, the margin of error is equal to 0.0257.

(c)

The formula for computing the confidence interval estimate of the population proportion is written below:

$CI=\left(\hat{p}-E,\hat{p}+E\right)$

The 95% confidence interval becomes equal to:

$$\begin{gathered} CI=\left(0.697-0.0257,0.697+0.0257\right) \\ =\left(0.671,0.723\right) \end{gathered}$$

Therefore, the 95% confidence interval estimate of the population proportion of medical malpractice lawsuits that are dropped or dismissed is equal to (0.671, 0.723).

(d)

There is 95% confidence that the true proportion of medical malpractice lawsuits that are dropped or dismissed will lie between the values, 0.671 and 0.723.