Question

a) Sports shirts are frequently classified as S, M, L, XL for small, medium, large

and extra-large neck sizes. S fits a neck circumference of less than 37cm, M

fits between 37 and 40.5cm and L fits between 40.5cm and 44cm while XL fits

necks over 44cm in circumference. The neck circumference of adult males has

a normal distribution with µ = 40cm and σ = 2cm.

(i) What proportion of shirts should be manufactured in each category?

(ii) If you wanted to define categories S, M, L, XL so that the categories con-

tained 20%, 30%, 30% and 20% respectively, of the total population of

adult males, what neck sizes must you assign to each of the categories

Answer 1

(i)

1.We make standard normal distribution by the formula:

$X'=\frac{X-\mu}{]sigma}=\frac{X-40}{]2}$

2.Next we apply this formula to neck circumference  divisions:

$d_1=\frac{37-40}{2}=-1.5;$

$d_2=\frac{40.5-40}{2}=0.25;\\ d_3=\frac{44-40}{2}=2;\\$

3.Find p values of z-scors using tables in internet:

p(-1.5)=-.0668;

p(0.25)=0.5987;

p(2.0)=-.9772;

4.Do calculation of probabilities of interval between divisions:

P(X$\le$ 37)=p(-1.5)-p(-$\infty)=0.0668-0=0.0668;$

P(37<X$\le$ 40.5)=p(0.25)-p(-1.5)=0.5987-0.0668=  0.5319;

P(40.5<X$\le$ 44)=p(2.0)-p(0.25)=0.9772-0.5987=0.3785;

P(X>44)=p($\infty)-p(2)=$ 1-0.9772=0.022800;

5.Useful verification: 0.0668+0,5319+0.3785+-0.0228=1;

6,. Now we have desired proportions:

Less 37 sm- 6.68%;

Greate 37 but less 40.5 sm- 53.18%;

Greater 40.5 but less than 44 sm- 37.8%;

Greater 44 sm- 2.28%

(ii)

In this part we must find z scores by p values:

0.2+0.3=0.5;

z(0.5)=0;

0.5+0.3=0.8;

z(0.8)=0.84;

Now we transform z scores to real data by the formula

d_i=zi$\cdot \sigma+\mu=z_i\cdot 2+40;$

We have divisions&

d₁=-0.85$\cdot 2+40=$ 38.3 sm;

d₂=$0\cdot 2+40=40$ sm;

d₃=$-.84\cdot 2+40=41.68$ sm;

So desired divisions are 38.3, 40,41,68 sm.