NEED HELP!!!!! PLEASE

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

The head diameters are normally distributed with a mean of 6-in and a standard deviation of 0.8-in. 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%.

Q1: What is the minimum head diameter that will fit the clientele?

For the smallest diameter,

We need d_min such that P( D <= d_min) = 0.015

From the Z-table or calculator, we get P( Z <= -2.17) = 0.015

Now, we will have to convert this Z number to our regular diameter.

Using standard normal conversion formula,

Z = (dmin - mean)/sd = (dmin - 6)/0.8 = -2.17

=> dmin -6 = -2.17 *0.8 = -1.736

=> dmin = 6 - 1.736 = 4.264

Q2: What is the maximum head diameter that will fit the clientele?

We need d_max such that P( D >= d_max) = 0.015

From the Z-table or calculator, we get P( Z >= 2.17) = 0.015

Now, we will have to convert this Z number to our regular diameter.

Using standard normal conversion formula,

Z = (dmin - mean)/sd = (dmin - 6)/0.8 = 2.17

=> dmin - 6 = 2.17 * 0.8

=> dmin = 6 + 1.736 = 7.736

So, the minimum diameter is 4.264 and the maximum diameter is 7.736