Question

The distribution of average wait times in drive-through restaurant lines in one town was approximately normal with mean μ=242 seconds and standard deviation σ=13 seconds.
Amelia only likes to use the drive-through for restaurants where the average wait time is in the bottom 15% for that town.
What is the maximum average wait time for restaurants where Amelia likes to use the drive-through? Round to the nearest whole second.

Answer 1

You're looking for the 15th percentile of a normal distribution with µ = 242 and σ = 13, which is to say you want to find x * such that

P(X ≤ x *) = 0.15

Transform the wait-time random variable X to Z, which follows the standard normal distribution with mean 0 and s.d. 1 :

Z = (X - µ) / σ   →   X = µ + σ Z = 242 + 13 Z

Now,

P(X ≤ x *) = P(242 + 13 Z ≤ x *) = P(Z ≤ (x * - 242)/13) = 0.15

Use a calculator (something with an inverse CDF function) or Z-table to look up the z-score, z = (x * - 242)/13, associated with a probability of 0.15. You would find that

z = (x * - 242)/13 ≈ -1.03643

Solve for x * :

x * ≈ 242 + 13 (-1.03643) ≈ 228.526 ≈ 229