The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean µ = 3.2 minutes and a standard deviation σ = 1.6 minutes. if a random sample of 64 customers is observed, find the probability that their mean time at the teller's window is:

a. at most 2.7 minutes.

b. more than 3.5 minutes.

c. at least 3.2 minutes but less than 3.4 minutes

The general approach to answering this item is to determine first the z-score of the given data and convert the z-scores to percentile. The equation for z-score determination is,

z-score = (X - μ) / σ

where X is the data, μ is the mean or average, and σ is the standard deviation.

(A) at most 2.7 minutes

z-score = (2.7 - 3.2) / 1.6 = - 0.3125

This is equivalent to 37.73%

(B) more than 3.5 minutes

z-score = (3.5 - 3.2) / 1.6 = 0.1875

This is equivalent to 57.44%. We are asked for more than so we take,

100 - 57.44% = 42.56%

(C) z-score of 3.2 minutes

z-score = (3.2 - 3.2) / 1.6 = 0

This is equivalent to 50%.

z-score of 3.4 minutes

z-score = (3.4 - 3.2) / 1.6 = 0.125

This is equivalent to 54.97%

The difference of the two percentiles is 4.97%.