Answer:
1. The minimum head breadth that will fit the clientele is of 4.41-in.
2. The maximum head breadth that will fit the clientele is of 7.19-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 zscore 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 pvalue, we get the probability that the value of the measure is greater than X.
Normally distributed with a mean of 5.8-in and a standard deviation of 0.8-in.
This means that ![\mu = 5.8, \sigma = 0.8](https://tex.z-dn.net/?f=%5Cmu%20%3D%205.8%2C%20%5Csigma%20%3D%200.8)
1. What is the minimum head breadth that will fit the clientele?
The 4.1st percentile, that is, X when Z has a pvalue of 0.041, so X when Z = -1.74.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-1.74 = \frac{X - 5.8}{0.8}](https://tex.z-dn.net/?f=-1.74%20%3D%20%5Cfrac%7BX%20-%205.8%7D%7B0.8%7D)
![X - 5.8 = -1.74*0.8](https://tex.z-dn.net/?f=X%20-%205.8%20%3D%20-1.74%2A0.8)
![X = 4.41](https://tex.z-dn.net/?f=X%20%3D%204.41)
The minimum head breadth that will fit the clientele is of 4.41-in.
2. What is the maximum head breadth that will fit the clientele?
100 - 4.1 = 95.9th percentile, that is, X when Z has a pvalue of 0.959, so X when Z = 1.74.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.74 = \frac{X - 5.8}{0.8}](https://tex.z-dn.net/?f=1.74%20%3D%20%5Cfrac%7BX%20-%205.8%7D%7B0.8%7D)
![X - 5.8 = 1.74*0.8](https://tex.z-dn.net/?f=X%20-%205.8%20%3D%201.74%2A0.8)
![X = 7.19](https://tex.z-dn.net/?f=X%20%3D%207.19)
The maximum head breadth that will fit the clientele is of 7.19-in.