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 4762 seconds, while Mary completed the race in 5531 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 4349 seconds with a standard deviation of 593 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5218 seconds with a standard deviation of 821 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? seconds.

3. What is the Z score for Mary's finishing time? seconds. 4. Did Leo or Mary rank better in their respective groups?

A. They ranked the same B. Leo ranked better C. Mary ranked better

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 11% of athletes in the men's group, i.e. those who took the shortest 11% of time to finish? seconds

8. What is the cutoff time for the slowest 31% of athletes in the women's group?

Answer 1

1)The Men, Ages 30 - 34 group has a distribution of M ~ N(4349, 593)

The Women, Ages 25 - 29 group has a distribution of W ~N(5218,821)

2) The Z-score for Leo is

Leo was 0.70 SD’s above the mean for his group

3 ) The Z - score for MAry is

Mary was 0.38 SD’s above the mean for his group

4)

Since Mary was only 0.38 SD’s above the mean for her group, and Leo was 0.70 SD’s above the mean for his group, then Mary ranked better in here group than Leo did in his group

Answer: Option C) Mary ranked better

5) For Leo

Answer: 75.8% of the triathletes finish slower than Leo in his group

6) For Mary

Answer: 35.2%

Q 7 The critical value of Z for fastest 11% of theletes in the men's group is 1.227

Therefore the cutoff is

Q 8) The critical value of Z for slowest 31% of atheletes in the women's group is -0.495

Therefore the cutoff is