Question

As reported in Runner’s World magazine, the times of the finishers in the New York City 10-km run are normally distributed with mean 61 minutes and standard deviation 9 minutes.
a.Determine the percentage of finishers who have times between 50 and 70 minutes.

b. Obtain and interpret the 40th percentile for the finishing times.

c. Find the middle 30% of the finishing times.

Answer 1

Solution :

Given that ,

mean = = 61

standard deviation = = 9

a.

P(50 < x < 70) = P[(50 - 61)/ 9) < (x - ) /  < (70 - 61) / 9) ]

= P(-1.22 < z < 1)

= P(z < 1) - P(z < -1.22)

= 0.8413 - 0.1112

= 0.7301

percentage = 73.01%

b.

Using standard normal table ,

P(Z < z) = 40%

P(Z < -0.25) = 0.4

z = -0.25

Using z-score formula,

x = z * +

x = -0.25 * 9 + 61 = 58.75

40th percentile is 58.75

c.

Middle 30% as the to z values are -0.385 and 0.385

Using z-score formula,

x = z * +

x = -0.385 * 9 + 61 = 57.54

and

x = 0.385 * 9 + 61 = 64.47

the middle 30% of the finishing times is 57.54 and 64.47