Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are Normally distributed with a mean of 5.8 inches and a standard deviation of 1.2 inches. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 2% or largest 2%. A. What is the minimum head breadth that will fit the clientele?
B. What is the maximum head breadth that will fit the clientele?

a) 3.3352 inches.

b) 8.2648 inches.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

In this question:

A. What is the minimum head breadth that will fit the clientele?

This is the 2nd percentile, which is X when Z has a pvalue of 0.02. So X when Z = -2.054.

So the minimum head breadth that will fit the clientele is 3.3352 inches.

B. What is the maximum head breadth that will fit the clientele?

The 100-2 = 98th percentile, which is X when Z has a pvalue of 0.98. So X when Z = 2.054.

So the maximum head breadth that will fit the clientele is 8.2648 inches.

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step-by-step explanation:

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