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 4552 seconds, while Mary completed the race in 5798 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 4312 seconds with a standard deviation of 596 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5248 seconds with a standard deviation of 799 seconds.

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

Remember: a better performance corresponds to a faster finish.

a)What percent of the triathletes did Leo finish slower than in his group?

b) What percent of the triathletes did Mary finish faster than in her group?

c) What is the cutoff time for the fastest 24% of athletes in the men's group, i.e. those who took the shortest 24% of time to finish?

d)What is the cutoff time for the slowest 38% of athletes in the women's group?

Answer 1

a)

for normal distribution z score =(X-μ)/σx
here mean=       μ=	4312
std deviation   =σ=	596.0000

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

probability =

P(X<4552)

=

P(Z<0.4)=

65.54%

b)

percent of the triathletes did Mary finish faster than in her group :

probability =

P(X>5798)

=

P(Z>2.49)=99.36%

c) for 24th percentile ; critical z = -0.71

therefore corresponding value =mean+z*std deviation =4312-0.71*596=3884.84 ~3885 seconds

d)

for highest 38% ; critical z =0.31

therefore corresponding value =mean+z*std deviation =4312+0.31*596=4496.76 ~4497 seconds