In triathlons, it is common for racers to be placed into age and gender groups.

Question

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.

Here is some information on the performance of their groups:

• The finishing times of the Men, Ages 30–34, have a mean of 4313 seconds with a standard deviation of 583 seconds.

• The finishing times of the Women, Ages 25–29, have 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) Did Leo or Mary rank better in their respective groups?

(b) What proportion of athletes did Leo finish faster than in his group?

(c) What is the distribution of the difference in times between the two groups (finishing times of the women minus finishing times of the men?)

Explanation / Answer

a) z - score for Mary = (5513 - 5261)/ 807 = 0.31

z - score for Leo = (4948 - 4313)/ 583 = 1.09

Since z - score of Leo is greater than z - score for Mary, Leo performed better in the respective group as compared to Mary.

b) P(z < 1.09) = 0.8621

Hence,

Leo finished faster than 86.21% of athletes in his group.

c) Distribution of the difference in time will also follow a normal distribution since distribution of finishing time of both groups are approximately normal.

Mean of this distribution = 5261 - 4313 = 948 seconds

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