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 6.8-in and a standard deviation
of 0.8-in. Due to financial constraints, the helmets will be
designed to fit all men except those with head breadths that are in
the smallest 3% or largest 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.
Solution:-
Given that,
mean = = 6.8
standard deviation = = 0.8
Using standard normal table,
P(Z < z) = 3%
= P(Z < z ) = 0.03
= P(Z < -1.881 ) = 0.03
z = -1.881
Using z-score formula,
x = z * +
x = -1.881 * 0.8 + 6.8
x = 5.3
min. = 5.3 in.
Using standard normal table,
P(Z > z) = 3%
= 1 - P(Z < z) = 0.03
= P(Z < z) = 1 - 0.03
= P(Z < z ) = 0.97
= P(Z < 1.881 ) = 0.97
z = 1.881
Using z-score formula,
x = z * +
x = 1.881 * 0.8 + 6.8
x = 8.3
max. = 8.3 in.
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