Question

16. Let W be a random variable modeled as a binomial with p = 0.42 and n = 35. a. Find the exact value of P(W = 15) by using the binomial probability formula. b. Find the approximate value of P(14 < W< 16) by using a normal curve approximation. c. Round the probabilities in parts a. and b. to two decimal places and compare.

Answer 1

a)

X ~ bin ( n , p)

Where n = 35, p = 0.42

binomial probability distribution is

P(X) = nCx * p^x * ( 1 - p)^n-x

P(X = 15 ) = ³⁵C₁₅ * 0.42¹⁵ * ( 1 - 0.42)²⁰

= 0.1346

b)

Using Normal Approximation to Binomial
Mean = n * P = ( 35 * 0.42 ) = 14.7
Variance = n * P * Q = ( 35 * 0.42 * 0.58 ) = 8.526
Standard deviation = √(variance) = √(8.526) = 2.9199

P ( 14 < X < 16 ) = ?
Using continuity correction
P ( n + 0.5 < X < n - 0.5 ) = P ( 14 + 0.5 < X < 16 - 0.5 )

= P ( 14.5 < X < 15.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 14.5 - 14.7 ) / 2.9199
Z = -0.0685
Z = ( 15.5 - 14.7 ) / 2.9199
Z = 0.274
P ( -0.07 < Z < 0.27 )
P ( 14.5 < X < 15.5 ) = P ( Z < 0.27 ) - P ( Z < -0.07 )
P ( 14.5 < X < 15.5 ) = 0.608 - 0.4727
P ( 14.5 < X < 15.5 ) = 0.1353

c)

Rounded to 2 decimal places,

probability in part a = 0.13 , probability in part b = 0.14

both probabilities are not approximately same. (Since difference between probabilities are greater than 0.05)