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 6.6-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 2.1% or largest 2.1%. 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 = = 6.6
standard deviation = = 1
Using standard normal table,
P(Z < z) = 2.1%
= P(Z < -2.034 ) = 0.021
z = -2.034
Using z-score formula,
x = z * +
x = -2.034 * 1 + 6.6
x = 4.56
min. = 4.6 in.
Using standard normal table,
P(Z > z) = 2.1%
= 1 - P(Z < z) = 0.021
= P(Z < z) = 1 - 0.021
= P(Z < 2.034 ) = 0.979
z = 2.034
Using z-score formula,
x = z * +
x = 2.034 * 1 + 6.6
x = 8.63
max. = 8.6 in.
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