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 6.2-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 breadths that are in the smallest 2.6% or largest 2.6%. What is the minimum head breadth that will fit the clientele? min = What is the maximum head breadth that will fit the clientele? max = 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

Given that the distribution of head breadths are Normal with . We know that is a standard Normal distribution.

The smallest 2.6% means P(Z<?)=0.026 and the largest 2.6% means P(Z>?)=0.026.

We find the ? value from tables as -1.9431 ie P(Z<-1.9431)=0.026.

Let us find out the X.

min=4.1 inches.

Since, Normal distribution of symmetric, we can easily deduce the upper 2.6% is 1.9431.

Max=8.3 inches.

What is the minimum head breadth that will fit the clientele? min = 4.1 inches

What is the maximum head breadth that will fit the clientele? max = 8.3 inches