Answer to Question #228135 in Statistics and Probability for chisenga

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

Answer 1

If we were given mean as 40 and standard deviation as 2 and asked the proportion of shirts manufactured in each category, the following solution will apply

Small

$0<X<37$

Since $\mu = 40$ and $\sigma = 2$

$= P(0<X<37)\\ = P(0-40<X-\mu<37-40)\\ = P(\frac{0-40}{2}<\frac{X-\mu}{\sigma}<\frac{37-40}{2})\\ = P(-20<Z<-1.5)\\ =0.0668\\ \implies P_{small}= 0.0668*100=7 \%$

Medium

$37<X<40.5$

Since $\mu = 40$ and $\sigma = 2$

$= P(37<X<40.5)\\ = P(37-40<X-\mu<40.5-40)\\ = P(\frac{37-40}{2}<\frac{X-\mu}{\sigma}<\frac{40.5-40}{2})\\ = P(-1.5<Z<0.25)\\ =0.5319\\ \implies P_{medium}= 0.5319*100=53 \%$

Large

$40.5<X<44$

Since $\mu = 40$ and $\sigma = 2$

$= P(40.5<X<44)\\ = P(40.5-40<X-\mu<44-40)\\ = P(\frac{40.5-40}{2}<\frac{X-\mu}{\sigma}<\frac{44-40}{2})\\ = P(0.25<Z<2)\\ =0.3785\\ \implies P_{large}= 0.3785*100=38 \%$

Extra-Large

$X>44$

Since $\mu = 40$ and $\sigma = 2$

$= P(X>44)\\ = P(X-\mu>44-40)\\ = P(\frac{X-\mu}{\sigma}>\frac{44-40}{2})\\ = P(Z>2)\\ =0.0228\\ \implies P_{extra-large}= 0.0228*100=2 \%$