Answer to Question #233879 in Statistics and Probability for samuel

Question #233879

 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? [6marks] (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


1
Expert's answer
2021-09-07T03:03:44-0400

(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.



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