Question

A pharmaceutical development corporation tests a new pain reliever on 78 people with head colds. Assume that 45% of head colds would subside naturally without the drug.

1. Find the probability that exactly 20 people will be cured without a pain reliever

2. Find the probability that less than 40 people will be cured without a pain reliever.

3. Use the normal approximation to approximate the probability that of the 78 people chosen, between 27 and 47 (inclusive) will be cured without a pain reliever.

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 78 * 0.45 ) = 35.1
Variance = n * P * Q = ( 78 * 0.45 * 0.55 ) = 19.305
Standard deviation = √(variance) = √(19.305) = 4.3937

Part a)

P ( X = 20 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 20 - 0.5 < X < 20 + 0.5 ) = P ( 19.5 < X < 20.5 )

X ~ N ( µ = 35.1 , σ = 4.3937 )
P ( 19.5 < X < 20.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 19.5 - 35.1 ) / 4.3937
Z = -3.55
Z = ( 20.5 - 35.1 ) / 4.3937
Z = -3.32
P ( -3.55 < Z < -3.32 )
P ( 19.5 < X < 20.5 ) = P ( Z < -3.32 ) - P ( Z < -3.55 )
P ( 19.5 < X < 20.5 ) = 0.0005 - 0.0002
P ( 19.5 < X < 20.5 ) = 0.0003

part b)

P ( X < 40 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 40 - 0.5 ) = P ( X < 39.5 )

X ~ N ( µ = 35.1 , σ = 4.3937 )
P ( X < 39.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 39.5 - 35.1 ) / 4.3937
Z = 1
P ( ( X - µ ) / σ ) < ( 39.5 - 35.1 ) / 4.3937 )
P ( X < 39.5 ) = P ( Z < 1 )
P ( X < 39.5 ) = 0.8413

Part C)

P ( 27 <= X <= 47 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 27 - 0.5 < X < 47 + 0.5 ) = P ( 26.5 < X < 47.5 )

X ~ N ( µ = 35.1 , σ = 4.3937 )
P ( 26.5 < X < 47.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 26.5 - 35.1 ) / 4.3937
Z = -1.96
Z = ( 47.5 - 35.1 ) / 4.3937
Z = 2.82
P ( -1.96 < Z < 2.82 )
P ( 26.5 < X < 47.5 ) = P ( Z < 2.82 ) - P ( Z < -1.96 )
P ( 26.5 < X < 47.5 ) = 0.9976 - 0.025
P ( 26.5 < X < 47.5 ) = 0.9726