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 6.5-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 1.9% or largest 1.9%. a) What is the minimum head breadth that will fit the clientele? b) What is the maximum head breadth that will fit the clientele?

Answer 1

a) min = 4.4251

b) max = 8.5743

Explanation:

Given:

Mean, u = 6.5

Standard deviation = 1

To get our values, we are to use the inverse of the standard normal table.

a) The minimum head breadth that will fit.

P(Z < z) = 1.9%

P(Z < z) = 0.019

P(Z < -2.0749) = 0.019

z = -2.0749

From the z-score formula, we have:

`(x - \mu)/(\sigma) = Z_0_._0_1_9`

Let x be the breath of heads.

Making x the subject of the formula, we have:

`x = z * \sigma + \mu`

We already have:

z = -2.0749

u = 6.5

s.d = 1

Substituting figures, we have:

x = (-2.0749 * 1) + 6.5

x = 4.4251

The minimum head breadth that will fit the clientele is 4.4251

b) The maximum head breadth that will fit.

P(Z > z) = 1.9%

1 - P(Z < z) = 0.019

P(Z < z) = 1 - 0.019 = 0.981

P(Z < 2.0749) = 0.981

From the z-score formula, we have:

`(x - \mu)/(\sigma) = Z_0_._0_1_9`

Let x be the breath of heads.

Making x the subject of the formula, we have:

`x = z * \sigma + \mu`

We already have:

z = 2.0749

u = 6.5

s.d = 1

Substituting figures, we have:

x = (2.0749 * 1) + 6.5

x = 4.4251

x = 8.5743

The maximum head breadth that will fit the clientele is 8.5743