Question

The average time to run the 5K fun run is 22 minutes and the standard deviation is 2.5 minutes. 49 runners are randomly selected to run the 5K fun run. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. What is the distribution of ∑x∑x? ∑x∑x ~ N(,)
4. If one randomly selected runner is timed, find the probability that this runner's time will be between 22.0643 and 22.4643 minutes.
5. For the 49 runners, find the probability that their average time is between 22.0643 and 22.4643 minutes.
6. Find the probability that the randomly selected 49 person team will have a total time more than 1053.5.
7. For part e) and f), is the assumption of normal necessary? YesNo
8. The top 10% of all 49 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

(there are more than 4 parts, as per policy i am answering first 4 parts)

a.

for X :

mean = 22 min

SD = 2.5 min

therefore distribution is : N(22, 2.52)

b.

for x̄

mean = population mean = 22 min

SD = SDpopulation / (n^0.5)

= 2.5 / (49^0.5)

= 0.3571 min

therefore distribution is : N(22, 0.3572)

c.

Σx = 49*x̄

for ax :

mean (ax) = a*mean(x)

var(ax) = a^2 * var(x)

therefore, SD(ax) = a*SD(x)

here ax = Σx = 49x̄

therefore,

mean(Σx) = mean(49x̄) = 49*mean(x̄) = 49*22 = 1078 min

SD(Σx) = SD(49x̄) = 49*SD(x̄) = 49*0.357 = 17.493 min

therefore distribution is : N(1078, 17.4932)

d.

N(22, 0.3572)

P( 22.0643 <= x <= 22.4643 ) :

P( 22.0643 <= x <= 22.4643 ) = 0.3318

probability that this runner's time will be between 22.0643 and 22.4643 min is 0.3318

(PLEASE UPVOTE)