Question

7. Engineers develop two bicycle helments: one for adults and one for children. The engineers must consider head breadths when designing helmets in order to ensure they fit most people. Company researchers have determined that adults’ head breadths are normally distributed with a mean of 7.1 inches and a standard deviation of 1.3 inches. Children’s head breadths are normally distributed wtih a mean of 5.2 inches and a standard deviation of 0.9 inches. Due to financial constraints, the helmets will be designed to fit everyone except those with head breadths that are in the smallest 2.5% or largest 2.5% of their respective population.

a. What is the maximum head breadth that will fit an adult?

b. How large must an adult’s head be in order to be larger than 92% of all adults?

c. Consider the minimum head breadth of an adult that will fit in the helmet. If a child had this same head breadth, they would have a head larger than what percentage of the child population?

d. What percent of the adult population has a head breadth between 6.5 and 8.2 inches?

e. What percent of the child population has a head breadth greater than 4.9 inches?

Answer 1

let X is head breadth of adult hence X is normal with mean=7.1 SD=1.3

Y is head breadth of child hence Y is normal with mean=5.2 SD=0.9

Helmet does not fit for the smallest 2.5% and largest 2.5%

a)

let "a" is maximum head breadth of an adult that will fit in the helmet hence

P(X>a)=0.025

now

From Z table

P(Z>1.96)=0.025

Hence on comparing both

b)

let "b" is such adult head that larger than 92% of all adults

so

P(X<b)=0.92

now

from Z table

P(Z<1.405)

so on comparing both we get

c)

let minimum adult head breadth that fits in helmet is "c" then

P(X<c)=0.025 hence

from Z table

P(Z<-1.96)=0.025

hence

we have to find P(Y<4.552) Now

Hence required % is 23.58%

d)

we have to find P(6.5<X<8.2)

=P(-0.462<Z<0.846)

=P(Z<0.846) - P(Z<-0.462)

=0.801-0.322

=0.479 Hence required % is 47.9%

e)

we have to find P(Y>4.9)

Hence required % is 59.1%