Question

(1 point) 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 4988 seconds, while Mary completed the race in 5654 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 4331 seconds with a standard deviation of 578 seconds. The finishing times of the Women, Ages 25 - 29 group has a mean of 5233 seconds with a standard deviation of 834 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 NG The Women, Ages 25 - 29 group has a distribution of NG 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. Mary ranked better B. Leo 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 20% of athletes in the men's group, i.e. those who took the shortest 20% of time to finish? seconds 8. What is the cutoff time for the slowest 18% of athletes in the women's group? seconds

Answer 1

- of age group 30-34 Let X denotes finishing time of the men XN NCH, = 4331, **: 578%) of age group 25-29 - Let y denotes finishing time of the women YN N (M = 5233, O2: 834) . Leo completed in the men 30-34 group & Mary completed in the Hermosa Beach hathion women, age 25-29 group. Leo completed the save in 4988 seconds & Mary completed the race in 5654 seconds. 1) The men, age 30-40 group has a distribution of N (4,5 4331, 6, 5789) i) The Women, age g 25-29 group has a distribution of N ( Me: 5233, 0,*: 834%) 2) z scose for Leo's finishing time uj X-e, 4988 - 4331 - 578 1.1367 z : : ») z score for Mary's finishing time us 2:8. 5654 - 5238 0.5048 834 4) Leo ranked bellen in their group.

5) Probability rave slower leo finish the than in his group that Le Probability that EPCX > 4988) . 4988-4 P / X - Me :P (2 > 4988724331) 4988 - 433) 578 :P (2) 1.1367) VILL O 1.1367 - 0.1271 finish • slower than in his Percentage of traithletes did to group : 100 x P (x>4998). = 100 x 0.1271 = 12.71.). - 6) the race faster than in her Probability that many finish group = p (7 < 565%) :p ( C 554") :p ( 2 < 5059-5233) 834 = p (2< 0.5048) il-Prz>0.5048) 31 - 0.3085 = 0.6915 in Percentage of her group = traithletes did Mary finish the race faster than 100 x P (4 <5654) = 100x o. 6915 z eg. 18 %

7) P(x > 2.) = 0.20 PC Kits how ) = 0.20 - 4331 P (2) = 0.20 578 4381 x to probabicity 0.20 is 0.84 The pooo ordinate corresponding i.e.P(Z >0.84) - 0.20 :. 2 XI-9331.0.84 578 :: c = (0.84x578) + 4331 2, = 4816.52 for time cutoff the of athletes fastest 20% :. 4816.52 is the in he men's group. 8) PCY SY,) = 0.18 P (41 242 < Sorte) -0.16 iy, -5233 ) = 0.18 przi A L L y, 5233 834 . probability 0.10 es -0.91 The ordinate coresponding to Le P(Z <-0.91) = 0.18 :. 4. - 5233 -0.98 834 30.59 . 9, 5233 - (0.91 x 834) = 4479.06 the slowest time for 18% of athletes in :: 4414.06 is the cutoff hе штель дир.