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 4609 seconds, while Mary completed the race in 5783 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 4333 seconds with a standard deviation of 592 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5249 seconds with a standard deviation of 781 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. They ranked the same
B. Mary ranked better
C. Leo 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 18% of athletes in the men's group, i.e. those who took the shortest 18% of time to finish?

seconds

8. What is the cutoff time for the slowest 34% of athletes in the women's group?

seconds

Answer 1

Note: Round all calculated answers to four decimal places. This is possible only if you use technology (calculator, excel etc). While using tables for normal distribution, all the z values and percentages are rounded to 2 decimals.

The finishing times of the Men, Ages 30 - 34 group has a mean of 4333 seconds with a standard deviation of 592 seconds.

The finishing times of the Women, Ages 25 - 29 group has a mean of 5249 seconds with a standard deviation of 781 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.

Let X be the finishing times of a randomly selected man in , Ages 30 - 34 group. X has a normal distribution with mean and standard deviation

ans: The Men, Ages 30 - 34 group has a distribution of .

Note: standard format to designate a normal distribution is N(mean,variance). Here the variance is

Let Y be the finishing times of a randomly selected wonman in , Ages 25 - 29 group. Y has a normal distribution with mean and standard deviation

​​​​​​​ans: The Women, Ages 25 - 29 group has a distribution of .

Note: standard format to designate a normal distribution is N(mean,variance). Here the variance is

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

Leo completed the race in 4609 seconds

the z score is

ans: the Z score for Leo's finishing time is z=0.4662

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

Mary completed the race in 5783 seconds.

the z score is

ans: the Z score for Mary's finishing time is z=0.6837

4. Did Leo or Mary rank better in their respective groups?
In this case, the lower score means you have finished faster and hence have a better rank.

Leo ranks better in his group compared to Mary as he has a lower z score
ans: C. Leo ranked better

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

Leo has finished slower than all the trialthetes who have completed the race in less than 4609 seconds

The probability that a randomly selected man in Ages 30 - 34 group completes the race in less than 4609 seconds is

ans: The percent of the triathletes did Leo finish slower than in his group is 68.08%

Note: If we use technology to calculate the probability (calculator or excel function =NORM.DIST(4609,4333,592,TRUE)) we get the % as 67.9470% (to 4 decimal places)

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

Mary completed the race in 5783 seconds. Mary would finish faster than all the althetes who have completed the race is more than 5783 seconds

The probability that a randomly selected woman in Ages 25-29 group completes the race in more than 5783 seconds is

ans: the percent of the triathletes did Mary finish faster than in her group is 24.83%

to get answer rounded to 4 decimals, we need to use technology (calculator or excel function =1-NORM.DIST(5783,5249,781,TRUE)) and get the answer as 24.7070%

7. What is the cutoff time for the fastest 18% of athletes in the men's group, i.e. those who took the shortest 18% of time to finish?

Let q be the cutoff time for the fastest 18% of athletes in the men's group. This is same as the probability that a randomly selected man in men's group finishes in less than q seconds is 0.18

P(X<q)=0.18

The z scoe corresponding to probability 0.18 would be

P(Z<z)=0.18.

However, we know that the area under a standard normal curve to the left of mean 0, is 0.5. Sine th erequired probability left of z is less than 0.5, we know that z must be negative

Hence we need

Using the standard normal tables, for z=0.92, we get P(Z<0.92)=0.82

Hence

We need

P(X<q) =0.18

We can equate the z score of q to -0.92 and get

ans: the cutoff time for the fastest 18% of athletes in the men's group is 3788.36 seconds

Note: to get an answer accurate to 4 decimals use the excel function =NORM.INV(0.18,4333,592) and get 3791.1039 seconds

8. What is the cutoff time for the slowest 34% of athletes in the women's group?

Let q seconds be the cutoff time for the slowest 34% of athletes in the women's group. This is same as the probability that a randomly selected woman in the women's group takes more than q seconds is 0.34

P(Y>q)=0.34

The z scoe corresponding to probability 0.34 would be

Using the standard normal tables, for z=0.41, we get P(Z<0.41)=0.66

Hence

We need

P(Y>q) =0.34

We can equate the z score of q to 0.41 and get

ans: the cutoff time for the slowest 34% of athletes in the women's group is 5559.21 seconds

Note: to get an answer accurate to 4 decimals use excel function =NORM.INV(0.66,5249,781) and get 5571.1337 seconds