Triathlon Roberto finishes a triathlon (750-meter swim, 5-kilometer run, and 20-kilometer bicycle) in 63.2 minutes. Among all men in the race, the mean finishing time was 69.4 minutes with a standard deviation of 8.9 minutes. Zandra finishes the same triathlon in 79.3 minutes. Among all women in the race, the mean finishing time was 84.7 minutes with a standard deviation of 7.4 minutes. Who did better in relation to their gender?

Verified step by step guidance

1. Identify the relevant statistics for each athlete's gender group: for Roberto (men), the mean finishing time \(\mu_m = 69.4\) minutes and standard deviation \(\sigma_m = 8.9\) minutes; for Zandra (women), the mean finishing time \(\mu_w = 84.7\) minutes and standard deviation \(\sigma_w = 7.4\) minutes.
2. Calculate the z-score for Roberto's finishing time using the formula: \[Z = \frac{X - \mu}{\sigma}\] where \(X\) is Roberto's time (63.2 minutes), \(\mu\) is the men's mean, and \(\sigma\) is the men's standard deviation.
3. Calculate the z-score for Zandra's finishing time using the same formula, but with the women's statistics: \[Z = \frac{X - \mu}{\sigma}\] where \(X\) is Zandra's time (79.3 minutes), \(\mu\) is the women's mean, and \(\sigma\) is the women's standard deviation.
4. Interpret the z-scores: since finishing time is a measure where lower is better, a more negative z-score indicates a better performance relative to the group. Compare Roberto's and Zandra's z-scores to see who performed better relative to their gender group.
5. Conclude who did better by identifying whose z-score is lower (more negative), indicating a better relative performance in the triathlon.

Key Concepts

Z-Score

A z-score measures how many standard deviations a data point is from the mean of its distribution. It standardizes different data points, allowing comparison across different groups or scales by showing relative performance.

Mean and Standard Deviation

The mean is the average value of a dataset, while the standard deviation quantifies the amount of variation or dispersion from the mean. Together, they describe the central tendency and spread of finishing times in the race.

Comparing Performance Across Groups

To compare individuals from different groups with distinct averages and variability, we use standardized scores like z-scores. This approach accounts for group differences, enabling fair comparison of relative performance.