Question

Shami buys 20 items, their prices are iid random variables with mean 75 and standard deviation 10. Let X be the total amount spent for all 20 items a) Estimate probability of ? ≥ 1700. b) Estimate probability of 1400 ≤ ? ≤ 1600

Answer 1

µ = 20 * 75 = 1500

σ = 10 * √20 = 44.7214

Part a)

X ~ N ( µ = 1500 , σ = 44.7214 )
P ( X > 1700 ) = 1 - P ( X < 1700 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1700 - 1500 ) / 44.7214
Z = 4.4721
P ( ( X - µ ) / σ ) > ( 1700 - 1500 ) / 44.7214 )
P ( Z > 4.4721 )
P ( X > 1700 ) = 1 - P ( Z < 4.4721 )
P ( X > 1700 ) = 1 - 1
P ( X > 1700 ) = 0

Part b)

X ~ N ( µ = 1500 , σ = 44.7214 )
P ( 1400 < X < 1600 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 1400 - 1500 ) / 44.7214
Z = -2.2361
Z = ( 1600 - 1500 ) / 44.7214
Z = 2.2361
P ( -2.24 < Z < 2.24 )
P ( 1400 < X < 1600 ) = P ( Z < 2.24 ) - P ( Z < -2.24 )
P ( 1400 < X < 1600 ) = 0.9873 - 0.0127
P ( 1400 < X < 1600 ) = 0.9747