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 4503 seconds, while Mary completed the race in 5565 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 four 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 4310 seconds with a standard deviation of 591 seconds. The finishing times of the Women, Ages 25 - 29 group has a mean of 5203 seconds with a standard deviation of 791 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. They ranked the same
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 8% of athletes in the men's group, i.e. those who took the shortest 8% of time to finish? (In seconds)

8. What is the cutoff time for the slowest 22% of athletes in the women's group? (In seconds)

Answer 1

he Men, Ages 30 - 34 group has a distribution of N(4310 ,591 )

The Women, Ages 25 - 29 group has a distribution of N (5203 ,791 )

2)

z score =(4503-4310)/591 =0.3266

3)

z score =0.4576

4)

A. Leo ranked better

5)

What percent of the triathletes did Leo finish slower than in his group :

probability =P(X<4503)=(Z<4503-4310)/591)=P(Z<(0.33)=0.628~ 62.80 %

6)

probability =P(X>5565)=P(Z>(5565-5203)/791)=P(Z>0.46)=1-P(Z<0.46)=1-0.6764=0.3236~ 32.36 %

7)

for 8th percentile critical value of z=		-1.405
therefore corresponding value=mean+z*std deviation=			3479.6027 seconds

8)

for 78th percentile critical value of z=		0.772
therefore corresponding value=mean+z*std deviation=			5813.8048