Engineers must consider the diameters of heads when designing helmets.

Question

Engineers must consider the diameters of heads when designing helmets. The company researchers have determined that the population of potential clientele have head diameters that are normally distributed with a mean of 6-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 diameters that are in the smallest 1.5% or largest 1.5%.

What is the minimum head diameter that will fit the clientele?

min = 

What is the maximum head diameter 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.

Answer 1

Find z -scores for bottom 1.5 percent or 1.5th percentile (0.015 probability). that z-score is -2.17

Since normal distribution is symmetric, z-score for top 1.5 percent or 98.5th percentile (0.985 probability) is 2.17

use standardized score z = (x - mean)/SD with mean given as 6 and SD given as 0.8 to find x for z = -2.17 and for 2.17. those will be the values for the 1.5th and 98.5th percentiles:

-2.17 = (x - 6)/0.8

x = -2.17 * 0.8 + 6 = 4.264 ~ 4.3 min = 4.3

2.17 = (x - 6)/0.8

x = 2.17 * 0.8 + 6 = 7.736 ~ 7.7 max = 7.7