Normal Distribution/Standard Deviation – Shoe Sizes

Suppose the shoe size of workers is normally distributed, with a meanof 10.0 inches, and a standard deviationof 0.5 inch. A clueless shoe manufacturer is going to introduce a new line of shoes specifically for these workers. Assume that if the shoe size falls between two shoe sizes, you purchase the next larger shoe size. How many pairs of each of the following sizes should be included in a batch of 1,000 pair of shoes?
a. 8.5 b. 9.0 c. 9.5 d. 10.0
e. 10.5 f. 11.0 g. 11.5

a) P(x<=8.5) = P(z<(8.5-10)/0.5)=P(z<-3) = 0.50-0.4987=0.0013
Number of pairs of size 8.5 = 0.0013*1000=1.3 roundoff we get 1.

b) P(8.5<x<9.0) = P((8.5-10)/0.5<z<(9.0-10)/0.5)=P(-3<z<-2) = 0.4987-0.4772=0.0215
Number of pairs of size 9.0 = 0.0215*1000=21.5 …

The solution uses normal distribution to determine the size of shoes to be produced.