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 6.7-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 4.2% or largest 4.2%.

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.

Please explain steps and how z score was found.

Answer 1

Solution

Let X = Head breadth (in inches). Then, we are given X ~ N(µ, σ²), where µ = 6.7, and σ = 1…..........................… (1)

Back-up Theory

If a random variable X ~ N(µ, σ²), i.e., X has Normal Distribution with mean µ and variance σ², then,

Z = (X - µ)/σ ~ N(0, 1), i.e., Standard Normal Distribution ………………........................................................………..(2)

P(X ≤ or ≥ t) = P[{(X - µ)/σ} ≤ or ≥ {(t - µ)/σ}] = P[Z ≤ or ≥ {(t - µ)/σ}] .……...............................................................…(3)

Probability values for the Standard Normal Variable, Z, can be directly read off from Standard Normal Tables ..... (4a)

or can be found using Excel Function: Statistical, NORMSDIST(z) which gives P(Z ≤ z) ……..............................…(4b)

Percentage points of N(0, 1) can be found using Excel Function: Statistical, NORMSINV(Probability) which gives values of t for which P(Z ≤ t) = given probability……………….................................................... ………(4c)

Now to work out the solution,

Let t₁ and t₂ be respectively the minimum and maximum head breadth that will fit the clientele…........................... (5)

Part (a)

Given, ‘the helmets will be designed to fit all men except those with head breadths that are in the smallest 4.2%’

=> P(X < t₁) = 0.042 [i.e., 4.2%] [vide (5)]

=> P[Z < {(t₁ – 6.7)/1}] = 0.042 [vide (3) and (1)]

=> {(t₁ – 6.7)/1} = - 1.7279 [vide (4c)]

=> t₁ = 4.9721

Thus, the minimum head breadth that will fit the clientele is: 5.0 inches Answer

Part (b)

Given, ‘the helmets will be designed to fit all men except those with head breadths that are in the largest 4.2%’

=> P(X > t₂) = 0.042 [i.e., 4.2%] [vide (5)]

=> P[Z < {(t₂ – 6.7)/1}] = 0.042 [vide (3) and (1)]

=> {(t₂ – 6.7)/1} = 1.7279 [vide (4c)]

=> t2 = 8.4279

Thus, the maximum head breadth that will fit the clientele is: 8.4 inches Answer

DONE