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 5.6-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 0.5% or largest 0.5%.
What is the minimum head breadth that will fit the clientele? min =
What is the maximum head breadth that will fit the clientele? min
Solution:-
Given that,
mean =
= 5.6 in.
standard deviation =
= 1 in.
Using standard normal table,
P(Z < z) = 0.5%
= P(Z < z ) = 0.005
= P(Z < -2.58 ) = 0.005
z = -2.58
Using z-score formula,
x = z *
+
x = -2.58 * 1 + 5.6
x = 3.02 in.
min. 3.02 in.
Using standard normal table,
P(Z > z) = 0.5%
= 1 - P(Z < z) = 0.005
= P(Z < z) = 1 - 0.005
= P(Z < z ) = 0.995
= P(Z < 2.58 ) = 0.995
z = 2.58
Using z-score formula,
x = z *
+
x = 2.58 * 1 + 5.6
x = 8.18 in.
max. 8.18 in.
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