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
(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;\\\\\nd_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
di=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.
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