A banker finds that the number of times people use automated-teller machines in a year are normally distributed with a mean of 40.0 and a standard deviation of 11.4. What is the percentage of customers who use them less than 25 times?

1 Answer
Jason K. · Geoff K.
Nov 12, 2017

9.34%.

Explanation:

Since we are told this distribution is normal, we can convert our provided values to a standardized z-score value, and then consult a z-score table to find the requested percentage.

First, note that we have #mu = 40.0# and #sigma = 11.4#. This describes a normal distribution of #N(40.0,11.4^2)#. We can use the formula for the z-score to convert the value of 25 times into its representative z-score:

#z = (x-mu)/sigma = (25 - 40.0)/11.4 = (-15)/11.4 = -1.32#

To find the percentage of customers who use the ATMs less than 25 times, we simply refer to a z-score table and find the z-score we just determined (-1.32) and use the table to find the left-tail area under the normal curve.

Using one table, we see that this corresponds to an area of 0.0934, or roughly 9.34% of all customers.

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