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.

Verified step by step guidance

1. Step 1: Identify the sample proportion (p̂). The sample proportion is calculated as the number of successes (lawsuits dropped or dismissed) divided by the total sample size. Use the formula: $\mathrm{p̂}=\frac{x}{n}$, where x = 856 and n = 1228.
2. Step 2: Calculate the standard error (SE) of the sample proportion. The standard error is given by the formula: $\mathrm{SE}=\sqrt{\frac{\mathrm{p̂}}{(}}$. Substitute the value of p̂ from Step 1 and n = 1228 into this formula.
3. Step 3: Determine the critical value (z) for a 95% confidence level. For a 95% confidence level, the critical value z is approximately 1.96. This value corresponds to the z-score that captures the middle 95% of the standard normal distribution.
4. Step 4: Calculate the margin of error (E). The margin of error is given by the formula: $E=z\times \mathrm{SE}$. Use the z value from Step 3 and the SE from Step 2 to compute E.
5. Step 5: Construct the confidence interval. The confidence interval is given by the formula: $\left[\mathrm{p̂}-E,\mathrm{p̂}+E\right]$. Substitute the values of p̂ from Step 1 and E from Step 4 to find the interval. Finally, interpret the confidence interval by stating that you are 95% confident that the true proportion of medical malpractice lawsuits that are dropped or dismissed lies within this interval.

Key Concepts

Point Estimate

A point estimate is a single value that serves as a best guess for a population parameter. In the context of proportions, it is calculated by dividing the number of successes (e.g., lawsuits dropped) by the total number of observations (e.g., total lawsuits). This estimate provides a quick snapshot of the population characteristic being studied.

Margin of Error

The margin of error quantifies the uncertainty associated with a point estimate. It reflects the range within which the true population parameter is expected to lie, given a certain confidence level. The margin of error is influenced by the sample size and the variability in the data, and it is crucial for constructing confidence intervals.

Confidence Interval

A confidence interval is a range of values derived from sample data that is likely to contain the true population parameter with a specified level of confidence, such as 95%. It is calculated using the point estimate and the margin of error, providing a more informative estimate than the point estimate alone. The interpretation of a confidence interval involves understanding that if the same sampling process were repeated, a certain percentage of intervals would contain the true parameter.