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 5.8-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 0.9% or largest 0.9%.
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.)
Solution :
Given that,
mean = = 5.8
standard deviation = = 1
- Using standard normal table ,
P(Z < z) = 0.9%
P(Z < z) = 0.009
P(Z < -2.366) = 0.009
z = -2.366
Using z-score formula,
x = z * +
x = -2.366 * 1 + 5.8 = 3.4
The minimum head breadth that will fit the clientele is min = 3.4
- Using standard normal table ,
P(Z > z) = 0.9%
1 - P(Z < z) = 0.009
P(Z < z) = 1 - 0.009 = 0.991
P(Z < 2.366) = 0.991
z = 2.366
Using z-score formula,
x = z * +
x = 2.366 * 1 + 5.8 = 8.2
The maximum head breadth that will fit the clientele is max = 8.2
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