Question

The manager of a night club in Boston stated that 85% of the customers are between the ages of 21 and 27 years. If the age of customers is normally distributed with a mean of 24 years, calculate its standard deviation.

Answer 1

The standard deviation of the age of the customers was of 2.08 years.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question:

`\mu = 24`

85% of the customers are between the ages of 21 and 27 years.

Since the normal distribution is symmetric, this means, for example, that 27 is the 50 + (85/2) = 92.5th percentile, that is, when
`X = 27`, Z has a pvalue of 0.925. So when
`X = 27, Z = 1.44`. We use this to find
`\sigma`

`Z = (X - \mu)/(\sigma)`

`1.44 = (27 - 24)/(\sigma)`

`1.44\sigma = 3`

`\sigma = (3)/(1.44)`

`\sigma = 2.08`

The standard deviation of the age of the customers was of 2.08 years.