so the largest small head that gets a custom helmet is 22.9 – 1.645 * 1.6 = 20.268 inches. in other words anyone with a head smaller than 20.268 will get a custom helmet

the smallest big head that gets a custom helmet is 22.9 + 1.645 * 1.6 = 25.532. in other words anyone with a head larger than 25.532 inches will get a custom helmet.

The rule states that approximately 95% of the observations

lie within two standard deviations of the mean.

The mean (µ) = 22.9 inches.

The standard deviation (σ) = 1.6 inches

That means the range of helmet sizes are:

= 22.9 inches ± 2 * (σ)

= 22.9 inches ± 2 * (1.6 inches)

= 22.9 inches ± 3.2 inches

= 19.7 to 26.1 inches

If you do a table lookup on 95%, it gives a z-value of 1.96 instead of 2 standard deviations. Don’t know if you’ve gotten that far along in stats, but the more accurate answer is:

= 22.9 inches ± 1.96 * (σ)

= 22.9 inches ± 1.96 * (1.6 inches)

= 22.9 inches ± 3.136 inches

= 19.764 to 26.036 inches

A short tutorial on standard deviation and the z-table is at:

http://www.mathsisfun.com/standard-normal-distribution-table.html

Good luck in your studies,

~ Mitch ~