Question

Yuki and Zana are on a swimming team. They often compete against each other in the 100 meter freestyle race. Yuki's times in this race are normally distributed with a mean of 80 seconds and a standard deviation of 4.2 seconds. Zana's times are also normally distributed with a mean of 85 seconds and a standard deviation of 5.6 seconds. We can assume that their times are independent.

Suppose we choose a random 100 meter freestyle race and calculate the difference between their times.

Find the probability that Yuki's time is faster than Zana's.
You may round your answer to two decimal places.

P(Yuki faster)≈

Answer 1

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N(u= 80, G=4+2) → E(Y)-30 , Var(^)(4.2)² Z- Zona's tine ~N (u.85, < «5.6) > E(±)= 85, Van (2) - (5:4)“ Y= Yubi's time %3D %3D * E(- z)- E()-E(E)> 80-85 - -5 Var(Y-z)= Var(Y)+ Varlz) [:Y,Z ane independent I =17:64 + 31-36 = 49 By Paopestiy of Normed disttion, Y-Z ~N (-5, 44) = N(u=-5, <s7). We have tofind, P(Yyz) > P(x-z 70) > P( Y-z+•5 : , 5) > |- P(NS5) [whune Nc Y=Z+5 7 1- E (0.7143) [tdffNCO,D=()- derined from Standard wornel ~N(0,1) ] table or oftuare , I will evse R Stadio . > |- 0-76 [R Code : Pwoum (0-7143) -0.24