Answer:
1. The minimum head breadth that will fit the clientele is of 3.95-in.
2. The maximum head breadth that will fit the clientele is of 9.25-in.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean
and standard deviation
, the z-score of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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 p-value, we get the probability that the value of the measure is greater than X.
Mean of 6.6-in and a standard deviation of 1.1-in.
This means that ![\mu = 6.6, \sigma = 1.1](https://tex.z-dn.net/?f=%5Cmu%20%3D%206.6%2C%20%5Csigma%20%3D%201.1)
1. What is the minimum head breadth that will fit the clientele?
The 0.8th percentile, which is X when Z has a p-value of 0.008, so X when Z = -2.41.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-2.41 = \frac{X - 6.6}{1.1}](https://tex.z-dn.net/?f=-2.41%20%3D%20%5Cfrac%7BX%20-%206.6%7D%7B1.1%7D)
![X - 6.6 = -2.41*1.1](https://tex.z-dn.net/?f=X%20-%206.6%20%3D%20-2.41%2A1.1)
![X = 3.95](https://tex.z-dn.net/?f=X%20%3D%203.95)
The minimum head breadth that will fit the clientele is of 3.95-in.
2. What is the maximum head breadth that will fit the clientele?
The 100 - 0.8 = 99.2nd percentile, which is X when Z has a p-value of 0.992, so X when Z = 2.41.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![2.41 = \frac{X - 6.6}{1.1}](https://tex.z-dn.net/?f=2.41%20%3D%20%5Cfrac%7BX%20-%206.6%7D%7B1.1%7D)
![X - 6.6 = 2.41*1.1](https://tex.z-dn.net/?f=X%20-%206.6%20%3D%202.41%2A1.1)
![X = 9.25](https://tex.z-dn.net/?f=X%20%3D%209.25)
The maximum head breadth that will fit the clientele is of 9.25-in.