If the height of 500 students is normally distributed with the mean 68 inches and S.D. 4 inches , estimate the number of students having heights (i) greater than 72 inches,(ii)less than 62 inches ,(iii) between 65 and 71 inches.

Subject: Applied Mathematics 2

Topic: Probability Distribution

Difficulty: High

Standard normal variate ,S.N.V; Z = $\frac{(X-m)}{σ}$ = $\frac{(X-68)}{4}$

(i) When X= 72 , Z = $\frac{(72-68)}{4}$ = 1

∴ P ( X> 72) = Area (Z > 1) = 0.5 – (area between Z=0 to Z=1) = 0.5 – 0.3413 = 0.1587

∴ The number of students having heights greater than 72 = 500 × 0.1587 = 79

(ii) When X= 62 , Z = $\frac{(62-68)}{4}$ = - 1.5

∴ P ( X<62) =Area (Z < - 1.5) = Area (Z > 1.5) = 0.5 – (area between Z=0 to Z=1.5 = 0.5 – 0.4332 = 0.0668

∴ The number of students having heights less than 62= 500 ×0 .0668= 33

(iii) When X= 65 , Z = $\frac{(65-68)}{4}$ = - 0.75 & When X= 71 , Z = $\frac{(71-68)}{4}$ = 0.75

∴ P( 65< X< 71) = P( - 0.75 < Z< 0.75) = 2( 0.5 – area from Z= 0 to Z 0.75) = 2(0.2734) = 0.5468

∴ The number of students having heights between 65 and 71 inches = 500 × 0.5468 = 273.