Question

Engineers must consider the diameters of heads when designing helmets. The company researchers have determined that the population of potential clientele have head diameters that are normally distributed with a mean of 6.6-in and a standard deviation of 1.1-in. Due to financial constraints, the helmets will be designed to fit all men except those with head diameters that are in the smallest 0.8% or largest 0.8%.1. What is the minimum head breadth that will fit the clientele?
2. What is the maximum head breadth that will fit the clientele?

Answer 1

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 $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 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 $\mu = 6.6, \sigma = 1.1$

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.

$Z = \frac{X - \mu}{\sigma}$

$-2.41 = \frac{X - 6.6}{1.1}$

$X - 6.6 = -2.41*1.1$

$X = 3.95$

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.

$Z = \frac{X - \mu}{\sigma}$

$2.41 = \frac{X - 6.6}{1.1}$

$X - 6.6 = 2.41*1.1$

$X = 9.25$

The maximum head breadth that will fit the clientele is of 9.25-in.