Question

Use the normal approximation to find the indicated probability. The sample size is n, the population proportion of successes is p, and X is the number of successes in the sample.
n = 87, p = 0.72: P(X > 65)

Group of answer choices

0.7734

0.2266

0.2483

0.2643

Answer 1

Using Normal Approximation to Binomial
Mean = n * P = ( 87 * 0.72 ) = 62.64
Variance = n * P * Q = ( 87 * 0.72 * 0.28 ) = 17.5392
Standard deviation = √(variance) = √(17.5392) = 4.188

P ( X > 65 )
Using continuity correction
P ( X > n + 0.5 ) = P ( X > 65 + 0.5 ) = P ( X > 65.5 )

X ~ N ( µ = 62.64 , σ = 4.188 )
P ( X > 65.5 ) = 1 - P ( X < 65.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 65.5 - 62.64 ) / 4.188
Z = 0.68
P ( ( X - µ ) / σ ) > ( 65.5 - 62.64 ) / 4.188 )
P ( Z > 0.68 )
P ( X > 65.5 ) = 1 - P ( Z < 0.68 )
P ( X > 65.5 ) = 1 - 0.7517
P ( X > 65.5 ) = 0.2483