Question

4. NFL Combine

The 40-yard dash time (i.e. the amount of time it takes to run 40 yards of the prospective NFL running backs is normally distributed with a mean of 4.53 seconds and a standard deviation of 0.2 seconds. For the corner backs, it is also normally distributed with a mean of 4.46 seconds and a standard deviation of 0.3 seconds.

(a) Find the probability that a running back finishes the 40 yard dash between 4.33 and 4.73 seconds.

(b) Find the probability that a randomly selected running back outruns a randomly selected corner back (i.e. the probability that the running back completes a 40 yard dash more quickly than a corner back).

(c) Find the probability that a random sample of n=25 corner backs will have an average dash time of less than 4.40 seconds.

(d) Suppose that a single line backer (not a corner back and not a running back- so you don't know the population mean or variance) ran 41 40 yard dashes, over the course of one week. His average time over these 41 trials was 4.50 seconds and the standard deviation was 0.3 seconds. Find a 95% confidence interval for the mean of his true 40 yard dash time.

Answer 1

Solution:

Given that
x- the time to take to complete NFL running backs μ_x = 4.53, σ_x = 0.2
Z_x = x-μ_x/σ_x = x- 4.53/0.2

y- the time to take to complete NFL running backs μ_y = 4.46, σ_y = 0.3
Z_y = y-μ_y/σ_y = y - 4.46/0.3

a) The probability that a running back finishes the 40 yard dash between 4.33 and 4.73 seconds.
P(4.33 ≤ x ≤ 4.73) = P ( 4.33−4.53/0.2< z < 4.73−4.53/0.2)
= P(-1 < Z < 1)
= 0.6826
b) The probability that a randomly selected running back outruns a randomly selected corner back
P(x < 4.46) = P(z < 4.46−4.53/0.2) = P (z< -0.35) = 0.3632

c) If sample size n = 25 (Corner back)
μy = 4.46, σy = σ/√n = 0.3/√25 = 0.06
The probability that a random sample of n=25 corner backs will have an average dash time of less than 4.40 seconds
P(y < 4.40) = P(z < 4.40−4.46/0.3) = P (z< -0.2) = 0.4207

d) The 95% confidence interval for the mean of his true 40 yard dash time for the selected runner
CI = x̅ ± Zα/2 * s/√n
= 4.50 ± (2.021 * 0.3/√41)
= (4.41, 4.59)