To qualify as a contestant in a​ race, a runner has to be in the fastest​ 16% of all applicants. The running times are normally​ distributed, with a mean of 59 min and a standard deviation of 6 min. To the nearest​ minute, what is the qualifying time for the​ race?

Answer 1

To find the qualifying time for the race, we need to determine the time that corresponds to the 84th percentile of the normal distribution. This represents the top 16% of applicants.

Using a standard normal probability table, the z-score corresponding to the 84th percentile is approximately 0.9945.

We can then use the formula for z-score:
z = (X - μ) / σ

Where:
X = Qualifying time
μ = Mean time (59 min)
σ = Standard deviation (6 min)

0.9945 = (X - 59) / 6
0.9945 * 6 = X - 59
5.967 = X - 59
X = 64.967

Rounded to the nearest minute, the qualifying time for the race is 65 minutes.