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?

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:

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

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.

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.

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

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