Question

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 6.9-in and a standard deviation of 1.2-in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 1.9% or largest 1.9%.

What is the minimum head breadth that will fit the clientele?
min =

What is the maximum head breadth that will fit the clientele?
min =

Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

Answer 1

Solution:

Given: the population of potential clientele have head breadths that are normally distributed with a mean of 6.9-in and a standard deviation of 1.2-in.

That is: Mean =

Standard Deviation =

the helmets will be designed to fit all men except those with head breadths that are in the smallest 1.9% or largest 1.9%.

Part a) What is the minimum head breadth that will fit the clientele?

That is find x value such that:

P( X < x ) = 1.9%

P(X < x) = 0.0190

Thus find z value such that:

P( Z < z ) = 0.0190

Look in z table for Area = 0.0190 or its closest area and find area.

Area 0.0190 is between 0.0188 and 0.0192

Area 0.0188 corresponds to -2.08

Area 0.0192 corresponds to -2.07

Thus average of both z values is:

z = ( -2.08 + -2.07 ) / 2

z = -4.15 / 2

z = -2.075

Thus required z value is -2.075

Now use following formula to find x value.

Thus the minimum head breadth that will fit the clientele = 4.4 in.

Part b) What is the maximum head breadth that will fit the clientele?

That is find x value such that:

P( X > x ) = 1.9%

P( X > x ) = 0.0190

Thus find z value such that:

P( Z > z ) = 0.0190

Since Standard Normal distribution is symmetric,

P( Z < - z ) = P( Z > +z )

We have P( Z < -2.075) = 0.0190

thus

P( Z > 2.075 ) = 0.0190

Thus use following formula to find x :

Thus the maximum head breadth that will fit the clientele = 9.4 in