Question

The average time to run the 5K fun run is 20 minutes and the standard deviation is 2.3 minutes. 7 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution. What is the distribution of X X ? X X ~ N(,) What is the distribution of ¯ x x¯ ? ¯ x x¯ ~ N(,) What is the distribution of ∑ x ∑x ? ∑ x ∑x ~ N(,) If one randomly selected runner is timed, find the probability that this runner's time will be between 19.696 and 20.996 minutes. For the 7 runners, find the probability that their average time is between 19.696 and 20.996 minutes. Find the probability that the randomly selected 7 person team will have a total time more than 137.9. For part e) and f), is the assumption of normal necessary? NoYes The top 10% of all 7 person team relay races will compete in the championship round. These are the 10% lowest times. What is the longest total time that a relay team can have and still make it to the championship round? minutes

Answer 1

a)

X ~ N(20,2.3)

b) x¯ ~ N(20 ,0.8693)

c)

∑x ~ N(140 , 6.0852)

d) If one randomly selected runner is timed, find the probability that this runner's time will be between 19.696 and 20.996 minutes.

probability =P(19.696<X<20.996)=P((19.696-20)/2.3)<Z<(20.996-20)/2.3)=P(-0.13<Z<0.43)=0.6675-0.4474=0.2201

e)

probability =P(19.696<X<20.996)=P((19.696-20)/0.869)<Z<(20.996-20)/0.869)=P(-0.35<Z<1.15)=0.874-0.3633=0.5108

f)

probability =P(X>137.9)=P(Z>(137.9-140)/6.085)=P(Z>-0.35)=1-P(Z<-0.35)=1-0.365=0.6350

h)

yes

i)

for 10th percentile critical value of z=		-1.28
therefore corresponding value=mean+z*std deviation=			132.2015