Question

1. As reported in Runners World magazine, the times of the finishers in the New York City 10 km run follow the normal model with mean=61 minutes and standard deviation= 9 minutes.

a) Using the 68-95-99.7% rule, 68% of runners will finsh between which two times? (In minutes)

b) Using the 68-95-99.7% rule, 99.7% of runners will finish between which two times? (In minutes)

Use z-scores and your z-table for the following. Round z-scores to two decimal places but give the full decimal answer from the z-table.

c) What proportion of runners will finish in less than 49.75 minutes?

d) What proportion of runners will take more than 65 minutes to finish?

e) For a random runner of the 10 km run, what is the probability they will finish between 55 and 70 minutes?

f) What time in (in minutes) will 30% of all runners finish below?

g) What time (in minutes) will 10% of the runners finish above?

Answer 1

Solution:
Given in the question
mean = 61 minutes
Standard deviation = 9 minutes
Solution(a)
From empirical rule 68% of runners will finish b/w +/- 1 from the mean
So Lower bound = - 1 = 61 - 9 = 52
Upper bound = + 1 = 61 + 9 = 70
So 68% of runners will finish between 52 minutes and 70 minutes.
Solution(b)
From empirical rule 99.7% of runners will finish b/w +/- 3 from the mean
So Lower bound = - 3* = 61 - 3*9 = 34
Upper bound = + 3* = 61 + 3*9 = 88
So 99.7% of runners will finish between 34 minutes and 88 minutes.
Solution(c)
We need to calcualte P(X<49.75)=?
Z-score = (X-)/ = (49.75-61)/9 = -1.25
From Z table we found p-value
P(X<49.75) = 0.1057
So there is 10.57% proportion of runners will finish in less than 49.75 minutes.
solution(d)
P(X>65) = 1-P(X<=65)
Z = (65-61)/9 = 0.44
From Z table we found p-value
P(X>65) = 1-P(X<=65) = 1 - 0.6716 = 0.3284
So there is 32.84 proportion of runners will take more than 65 minutes to finish
Solution(e)
P(55<X<70) = P(X<70) - P(X<55)
Z = (55-61)/9 = -6/9= -0.67
Z = (70-61)/9 = 1
From Z table, we found p-value
P(55<X<70) = P(X<70) - P(X<55) = 0.8413 - 0.2514 = 0.5899
So there is 58.99% that they will finish between 55 and 70 minutes.
Solution(f)
P-value = 0.3
So from Z table Z-score = -0.5244
Estimated time can be calculated as
Time = mean + Z-score*Standard deviation = 61 - 0.5244*9 = 61 - 4.72 = 56.28 minutes
So Time 56.28 minutes will 30% of all runners finish below.
Solution(g)
P-value = 0.9
Z-score from Z table is 1.28
Estimated time can be calculated as
Time = mean + Z-score*Standard deviation = 61 + 1.28*9 = 61 + 11.52 = 72.52 minutes
So time 72.52 minutes will 10% of the runners finish above.