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 6.1-in and a standard deviation of 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 2.3% or largest 2.3%.

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

Solution :

mean = = 6.1

standard deviation = = 1

Using standard normal table,

(a)

P(Z < z) = 0.023

P(Z < -1.995) = 0.023

z = -1.995

Using z-score formula,

x = z * +

x = -1.995 * 1 + 6.1 = 8.095 = 4.1

The maximum head breadth that will fit the clientele = 4.1 - in

(b)

P(Z > z) = 2.3%

1 - P(Z < z) = 0.023

P(Z < z) = 1 - 0.023 = 0.977

P(Z < 1.995) = 0.977

z = 1.995

Using z-score formula,

x = z * +

x = 1.995 * 1 + 6.1 = 8.095 = 8.1

The minimum head breadth that will fit the clientele is =8.1 - in