Question

in studies for amedication 15 percent of patients gained weight as a side effect. Suppose 533 patients are randomly selected. Use the normal approximation to the binomial to approximate the probabilty that

a) exactly 80 patients will gain weight as side effect

b) no more than 80 patients will gain weight as side effect

c) at least 91 patients will gain weight as side effect

since 91 is ......(fewer) (more) than 15% of patients, this suggest that the proportion of patiens that gain weight as a side effect is (greater than) (eqaul to) (less than) 0.15

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 533 * 0.15 ) = 79.95
Variance = n * P * Q = ( 533 * 0.15 * 0.85 ) = 67.9575
Standard deviation = √(variance) = √(67.9575) = 8.2436

Part a)

P ( X = 80 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 80 - 0.5 < X < 80 + 0.5 ) = P ( 79.5 < X < 80.5 )

X ~ N ( µ = 79.95 , σ = 8.2436 )
P ( 79.5 < X < 80.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 79.5 - 79.95 ) / 8.2436
Z = -0.05
Z = ( 80.5 - 79.95 ) / 8.2436
Z = 0.07
P ( -0.05 < Z < 0.07 )
P ( 79.5 < X < 80.5 ) = P ( Z < 0.07 ) - P ( Z < -0.05 )
P ( 79.5 < X < 80.5 ) = 0.5279 - 0.4801
P ( 79.5 < X < 80.5 ) = 0.0478

Part b)

P ( X <= 80 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 80 + 0.5 ) = P ( X < 80.5 )

X ~ N ( µ = 79.95 , σ = 8.2436 )
P ( X < 80.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 80.5 - 79.95 ) / 8.2436
Z = 0.07
P ( ( X - µ ) / σ ) < ( 80.5 - 79.95 ) / 8.2436 )
P ( X < 80.5 ) = P ( Z < 0.07 )
P ( X < 80.5 ) = 0.5279

Part c)

P ( X >= 91 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 91 - 0.5 ) =P ( X > 90.5 )

X ~ N ( µ = 79.95 , σ = 8.2436 )
P ( X > 90.5 ) = 1 - P ( X < 90.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 90.5 - 79.95 ) / 8.2436
Z = 1.28
P ( ( X - µ ) / σ ) > ( 90.5 - 79.95 ) / 8.2436 )
P ( Z > 1.28 )
P ( X > 90.5 ) = 1 - P ( Z < 1.28 )
P ( X > 90.5 ) = 1 - 0.8997
P ( X > 90.5 ) = 0.1003

since 91 is (fewer) than 15% of patients, this suggest that the proportion of patiens that gain weight as a side effect is (less than) 0.15.