Question

Triathlon times. 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 4981 seconds, while Mary completed the race in 5737 seconds. Obviously Leo finished faster, but they are curious about how they did within their respective groups. Can you help them? Round all calculated answers to two decimal places.

Here is some information on the performance of their groups:

The finishing times of the Men, Ages 30 - 34 group has a mean of 4345 seconds with a standard deviation of 586 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5236 seconds with a standard deviation of 830 seconds.

The distributions of finishing times for both groups are approximately Normal.

Remember: a better performance corresponds to a faster finish.

1. Write the short-hand for these two normal distributions.

The Men, Ages 30 - 34 group has a distribution of N( , ).

The Women, Ages 25 - 29 group has a distribution of N( , ).

2. What is the Z score for Leo's finishing time? z =

3. What is the Z score for Mary's finishing time? z =

4. Did Leo or Mary rank better in their respective groups?

A. Leo ranked better

B. Mary ranked better

C. They ranked the same

5. What percent of the triathletes did Leo finish slower than in his group? %.

6. What percent of the triathletes did Mary finish faster than in her group? %.

7. What is the cutoff time for the fastest 21% of athletes in the men's group, i.e. those who took the shortest 21% of time to finish?

seconds

8. What is the cutoff time for the slowest 24% of athletes in the women's group?

Answer 1

Hello Sir/ Mam

(1) Men, Ages 30 - 34

Group has a mean of 4345 seconds with a standard deviation of 586 seconds.

Group has a distribution of

Women, Ages 25 - 29

Group has a mean of 5236 seconds with a standard deviation of 830 seconds

Group has a distribution of

(2) Leo's finishing time = 4981 seconds

Hence,

(3) Mary's finishing time = 5737 seconds

Hence,

(4) THE CORRECT ANSWER IS OPTION B : MARY RANKED BETTER

As the z-score of Mary is less than that of Leo, Mary has performed better.

(5) THE REQUIRED ANSWER IS 86.11%

The p-value of z score 1.08532 is 0.861111

(6) THE REQUIRED ANSWER IS 27.305%

The p-value of z score 0.6036 is 0.72695

Hence, the required p value = 1 - 0.72695 = 0.27305

(7) THE REQUIRED ANSWER IS 3873 seconds.

For calculating the cut-off time for the fastest 21%, we have to calculate X using z-value from p-value = 0.21.

Hence,

(8) THE REQUIRED ANSWER IS 5822 seconds.

For calculating the cut-off time for the slowest 24%, we have to calculate X using z-value from p-value = 0.76.

Hence,

I hope this solves your doubt.

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