If the heights of 300 students are normally distributed with

If the heights of 300 students are normally distributed with mean 68.0 inches and standard deviation 3.0 inches, how many students have heights (a) greater than 72 inches, (b) less than or equal to 64 inches, (c) between 65 and 71 inches inclusive, (d) equal to 68 inches? Assume the measurements to be recorded to the nearest inch.

Solution

X~N(68,9)

Y=(X-68)/3 ~N(0,1)

The measurements are recorded in nearest inch.

(a)

P(X>72.5)=P(Y>1.5)=0.0668072

Number of students=0.0668072*300=20

(b)

P(X<64.5)=P(Y<-3.5/3)=0.1216725

Number of students=0.1216725*300= 37

(c)

P(64.5<=X<71.5)=P(-3.5/3<=Y<3.5/3)=P(- 1.166667<=Y< 1.166667)=0.756655

Number of students=0.756655*300=227

(d)

P(67.5<=X<68.5)=P(-0.5/3<=Y<0.5/3)=0.1323677

Number of students=0.1323677*300=40