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. 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.

1. The cutoff time for the fastest 5% of athletes in the men's group, i.e. those who took the shortest 5% of time to finish.

2. The cutoff time for the slowest 10% of athletes in the women's group.

Answer 1

SOLUTION:

LET M DENOTE THE FINISHING TIME FOR MEN, AGED 30-34. THEN NORMAL DISTRIBUTION FOR FINISHING TIME IS M~N(4313,538). ALSO W DENOTE THE FINISHING TIME FOR WOMEN, AGED 25-29. THEN NORMAL DISTRIBUTION FOR FINISHING TIME IS W~N(5261,807).

LEO'S Z-SCORE IS

MARY'S Z-SCORE IS

THESE TWO Z-SCORES IMPLY THAT LEO WAS 1.09 SD'S ABOVE THE MEAN OF HIS GROUP AND MARY WAS 0.31 SD'S ABOVE THE MEAN OF HER GROUP. SO MARY RANKED BETTER IN HER GROUP AS COMPARED TO LEO'S PERFORMANCE IN HIS GROUP.

ALSO FOR LEO, , SO LEO FINISHED FASTER THAN 13.8% OF RUNNERS IN HIS GROUP

ALSO FOR MARY, , SO MARY FINISHED FASTER THAN 37.8% OF THE RUNNERS IN HER GROUP.

REMARK- ALL THE NECESSARY EXPLAINATIONS HAVE BEEN GIVEN ABOVE AND WORKOUTS PERFORMED. IN CASE OF DOUBTS, COMMENT BELOW. AND DO LIKE IF POSSIBLE.