Question

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  = 40cm and  = 2cm.
(i) What proportion of shirts should be manufactured in each category?
[6 marks]
(ii) If you wanted to define categories S, M, L, XL so that the categories contained
20%, 30%, 30% and 20% respectively, of the total population of
adult males, what neck sizes must you assign to each of the categories?
[6 marks]

Answer 1

(i)

$S:$

$$P(0<X<37)$$

$$=P(Z<\dfrac{37-40}{2})-P(Z\leq\dfrac{0-40}{2})$$

$$=P(Z<-1.5)-P(Z\leq-20)$$

$$\approx0.066807-0\approx0.0668$$

$6.68\%$

$M:$

$$P(37<X<40.5)$$

$$=P(Z<\dfrac{40.5-40}{2})-P(Z\leq\dfrac{37-40}{2})$$

$$=P(Z<0.25)-P(Z\leq-1.5)$$

$$\approx0.598706-0.066807\approx0.5319$$

$53.19\%$

$L:$

$$P(40.5<X<44)$$

$$=P(Z<\dfrac{44-40}{2})-P(Z\leq\dfrac{40.5-40}{2})$$

$$=P(Z<2)-P(Z\leq0.25)$$

$$\approx0.977250-0.598706\approx0.3785$$

$37.85\%$

$XL:$

$$P(X>44)$$

$$=1-P(Z\leq\dfrac{44-40}{2})$$

$$=1-P(Z\leq2)$$

$$\approx1-0.977250\approx0.0228$$

$2.28\%$

$$6.68+53.19+37.85+2.28=100$$

(ii)

$S:$

$$P(0<X<x_1)$$

$$=P(Z<\dfrac{x_1-40}{2})-P(Z\leq\dfrac{0-40}{2})$$

$$\approx P(Z<\dfrac{x_1-40}{2})-0\approx0.2$$

$$\dfrac{x_1-40}{2}\approx-0.8416$$ $$x_1\approx38.3$$

$M:$

$$P(38.3<X<x_2)$$

$$=P(Z<\dfrac{x_2-40}{2})-0.3=0.2$$

$$P(Z<\dfrac{x_2-40}{2})=0.5$$

$$\dfrac{x_2-40}{2}=0$$ $$x_2=40$$

$L:$

$$P(40<X<x_3)$$

$$=P(Z<\dfrac{x_3-40}{2})-0.5=0.3$$

$$P(Z<\dfrac{x_3-40}{2})=0.8$$

$$\dfrac{x_3-40}{2}=0.8416$$

$$x_3=41.7$$

S fits a neck circumference of less than 38.3 cm, M fits between 38.3 cm and 40 cm and L fits between 40 cm and 41.7 cm while XL fits necks over 41.7 cm in circumference.