91Ó°ÊÓ

Triathlon times, Part I. In triathlons, it is common for racers to be placed into age and gender groups. Friends Leo and Mary both completed the Hermosa Beach Triathlon, where Leo competed in the Men, Ages 30 - 34 group while Mary competed in the Women, Ages 25 - 29 group. Leo completed the race in 1:22:28 (4948 seconds), while Mary completed the race in 1:31:53 (5513 seconds). Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Here is some information on the performance of their groups: \- The finishing times of the Men, Ages 30 - 34 group has a mean of 4313 seconds with a standard deviation of 583 seconds. \- The finishing times of the Women, Ages \(25-29\) group has a mean of 5261 seconds with a standard deviation of 807 seconds. \- The distributions of finishing times for both groups are approximately Normal. Remember: a better performance corresponds to a faster finish. (a) Write down the short-hand for these two normal distributions. (b) What are the Z-scores for Leo's and Mary's finishing times? What do these Z-scores tell you? (c) Did Leo or Mary rank better in their respective groups? Explain your reasoning. (d) What percent of the triathletes did Leo finish faster than in his group? (e) What percent of the triathletes did Mary finish faster than in her group? (f) If the distributions of finishing times are not nearly normal, would your answers to parts (b) - (e) change? Explain your reasoning.

Step by step solution

01

Define Normal Distributions

For part (a), define the normal distribution for each group using the mean and standard deviation. For Leo's group (Men, Ages 30-34):The distribution is Normal with a mean of 4313 seconds and a standard deviation of 583 seconds, written as \(N(4313, 583)\).For Mary's group (Women, Ages 25-29):The distribution is Normal with a mean of 5261 seconds and a standard deviation of 807 seconds, written as \(N(5261, 807)\).

02

Calculate Z-scores

For part (b), calculate the Z-scores using the formula:\[ Z = \frac{X - \mu}{\sigma} \]For Leo:\[ Z_{\text{Leo}} = \frac{4948 - 4313}{583} \approx 1.09 \]For Mary:\[ Z_{\text{Mary}} = \frac{5513 - 5261}{807} \approx 0.31 \]The Z-scores indicate how many standard deviations the individual's time is above the mean. Higher Z-scores indicate slower performance relative to the group's average.

03

Compare Z-scores

For part (c), compare the Z-scores to determine who ranked better in their group. Leo's Z-score of 1.09 is higher than Mary's Z-score of 0.31, which means Leo performed worse relative to his group compared to Mary. Therefore, Mary ranked better in her group than Leo did in his.

04

Calculate Percentages for Leo

For part (d), determine what percentage of triathletes Leo finished faster than by using a standard normal distribution table to find the cumulative probability for Leo's Z-score of 1.09. A Z-score of 1.09 corresponds to a cumulative probability of approximately 0.8621, meaning Leo finished faster than about 86.21% of the triathletes in his group.

05

Calculate Percentages for Mary

For part (e), determine what percentage of triathletes Mary finished faster than by using a standard normal distribution table to find the cumulative probability for Mary's Z-score of 0.31. A Z-score of 0.31 corresponds to a cumulative probability of approximately 0.6217, meaning Mary finished faster than about 62.17% of the triathletes in her group.

06

Consider Non-Normal Distribution

For part (f), if the distributions were not normal, the Z-scores would be less informative because they rely on normality to relate the relative standing of finish times. Percent calculations using the normal distribution table would not be accurate, possibly affecting conclusions about relative performance strengths.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

When analyzing triathlon performances, like those of Leo and Mary, we often rely on normal distribution models. A normal distribution is a bell-shaped curve that describes how variables are spread out. It is fully defined by two parameters: the mean and the standard deviation.
The mean indicates the center of the distribution, and the standard deviation shows the level of variance from this mean.

• In Leo's group (Men, Ages 30-34), the performances are modeled as a normal distribution with a mean ( abla) of 4313 seconds and a standard deviation ( abla) of 583 seconds.
• For Mary's group (Women, Ages 25-29), the times follow a normal distribution with a mean of 5261 seconds and a standard deviation of 807 seconds.

The beauty of normal distributions is that they are symmetrical, which allows for easy comparative analysis through tools like Z-scores and percentile calculations.

Z-score Calculation

Z-scores are a way to express how far away a point is from the mean of a data set in terms of standard deviations. They are invaluable for comparing individual performances to the overall group.
To calculate a Z-score, we use the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where:

• \(X\) is the individual's score or result.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation.

For Leo, his Z-score calculation was:
\[ Z_{\text{Leo}} = \frac{4948 - 4313}{583} \approx 1.09 \]
For Mary, her Z-score was calculated as:
\[ Z_{\text{Mary}} = \frac{5513 - 5261}{807} \approx 0.31 \]
These Z-scores show how many standard deviations Leo and Mary's times lie above their group's average. Higher Z-scores suggest poorer relative performance.

Statistical Comparison

After calculating Z-scores, they can be used for statistical comparison. This involves assessing each individual's performance relative to their group.
In this context, Leo's Z-score of 1.09 indicates he is 1.09 standard deviations slower than the group mean, while Mary's Z-score of 0.31 signifies she is 0.31 standard deviations slower than her mean.

• By comparing these Z-scores, we see that Leo performed worse than Mary when assessed relative to their respective groups.

The smaller Z-score means an individual is closer to what is considered 'average' or 'better' within their group. Therefore, mathematically, Mary achieved a better ranking.

Percentile Ranking

Percentile ranks show how an individual's performance compares to others in the dataset. In a normal distribution, Z-scores can be converted to percentile ranks using standard normal distribution tables.
For instance:

• Leo's Z-score of 1.09 corresponds to a percentile rank of approximately 86.21%, meaning he performed better than about 86% of his peers.
• Mary's Z-score of 0.31 gives her a percentile rank of about 62.17%, indicating she outran about 62% of her group.

Percentile ranking is a powerful tool that provides insight beyond mere Z-scores by showing the position and relevance of the performance in the group context.

Statistical Assumptions

Statistical analyses often depend on assumptions made about the data distribution. In this exercise, we assumed a normal distribution.

• The normality assumption allows the straightforward application of Z-scores and percentile rankings.
• If data were not normally distributed, these methods might not accurately reflect the relative standings of Leo and Mary’s performances.

Non-normal distributions may skew or distort typical performance metrics, resulting in potentially misleading interpretations. Therefore, understanding these statistical assumptions is crucial before delving into comparisons and analyses.