Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


Tennis Challenges In a recent U. S. Open tennis tournament, women playing singles matches used challenges on 137 calls made by the line judges. Among those challenges, 33 were found to be successful with the call overturned.


b. Compare the result from part (a) to this 99% confidence interval for the percentage of successful challenges made by the men playing singles matches: . Does it appear that either gender is more successful than the other?

Verified step by step guidance

1. Step 1: Identify the given data for the problem. The total number of challenges made by women is 137, and the number of successful challenges is 33. The confidence level is 99%.
2. Step 2: Calculate the sample proportion (p̂) of successful challenges for women. Use the formula: p̂ = x / n, where x is the number of successful challenges (33) and n is the total number of challenges (137).
3. Step 3: Determine the standard error (SE) for the sample proportion. Use the formula: SE = sqrt((p̂ * (1 - p̂)) / n), where p̂ is the sample proportion and n is the sample size.
4. Step 4: Find the critical value (z*) for a 99% confidence level. This value corresponds to the z-score that leaves 0.5% in each tail of the standard normal distribution. Commonly, z* for 99% confidence is approximately 2.576.
5. Step 5: Construct the confidence interval for the population proportion (p) using the formula: CI = p̂ ± z* * SE. Compare this confidence interval to the given 99% confidence interval for men’s successful challenges to determine if there is a significant difference between genders.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. For example, a 99% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 99% of those intervals would contain the true proportion of successful challenges.

Proportion

Proportion refers to the fraction of the total that possesses a certain characteristic. In this context, it is the ratio of successful challenges to the total number of challenges made. Understanding proportions is essential for comparing success rates between different groups, such as male and female players in the tennis tournament.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, it involves comparing the proportions of successful challenges between genders to determine if there is a statistically significant difference in their success rates. This process typically includes formulating null and alternative hypotheses and calculating p-values.