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.


Tennis Challenges In a recent U.S. Open tennis tournament, men playing singles matches used challenges on 240 calls made by the line judges. Among those challenges, 88 were found to be successful with the call overturned. Construct a 95% confidence interval for the proportion of successful challenges.

Verified step by step guidance

1. Step 1: Calculate the best point estimate of the population proportion p. The point estimate for p is given by the formula p̂ = x / n, where x is the number of successful challenges (88) and n is the total number of challenges (240). Substitute the values into the formula to find p̂.
2. Step 2: Compute the margin of error E. The formula for the margin of error in a confidence interval for a proportion is E = z * sqrt((p̂ * (1 - p̂)) / n), where z is the critical value corresponding to the confidence level (95% confidence level corresponds to z ≈ 1.96), p̂ is the point estimate, and n is the sample size. Substitute the values into the formula to calculate E.
3. Step 3: Construct the confidence interval. The confidence interval is given by the formula: CI = [p̂ - E, p̂ + E]. Use the values of p̂ and E calculated in the previous steps to construct the interval.
4. Step 4: Write the confidence interval in interval notation. Express the confidence interval as [lower bound, upper bound], where the lower bound is p̂ - E and the upper bound is p̂ + E.
5. Step 5: Interpret the confidence interval. Write a statement explaining that we are 95% confident that the true proportion of successful challenges lies within the calculated interval. Ensure the interpretation is clear and relates to the context of the problem.

Key Concepts

Point Estimate

A point estimate is a single value that serves as a best guess or approximation of a population parameter. In the context of proportions, it is calculated as the ratio of successful outcomes to the total number of trials. For example, in the tennis challenges scenario, the point estimate for the proportion of successful challenges is 88 successful calls out of 240 total challenges, which gives a point estimate of 0.3667.

Margin of Error

The margin of error quantifies the uncertainty associated with a point estimate. It reflects how much the estimate might vary from the true population parameter and is influenced by the sample size and the confidence level. For a proportion, the margin of error can be calculated using the formula E = z * sqrt[(p(1-p)/n)], where z is the z-score corresponding to the desired confidence level, p is the point estimate, and n is the sample size.

Confidence Interval

A confidence interval is a range of values, derived from the sample data, that is likely to contain the true population parameter with a specified level of confidence. For example, a 95% confidence interval for the proportion of successful challenges would provide a lower and upper bound around the point estimate, indicating that we can be 95% confident that the true proportion of successful challenges lies within this interval.