Respuesta :
Answer:
The standard deviation of the age of the customers was of 2.08 years.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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:
[tex]\mu = 24[/tex]
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 [tex]X = 27[/tex], Z has a pvalue of 0.925. So when [tex]X = 27, Z = 1.44[/tex]. We use this to find [tex]\sigma[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.44 = \frac{27 - 24}{\sigma}[/tex]
[tex]1.44\sigma = 3[/tex]
[tex]\sigma = \frac{3}{1.44}[/tex]
[tex]\sigma = 2.08[/tex]
The standard deviation of the age of the customers was of 2.08 years.
