Question

Compute PIX) using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If so, approximate PX) using the normal distribution and compare the result with the exact probability n=47, p=0.5, and X = 21 For n = 47. p=0.5, and X = 21, use the binomial probability formula to find PC 0.0892 (Round to four decimal places as needed) Can the normal distribution be used to approximate this probability? O A. Yes, because np(1 - p) 2 10 OB. Yes, because np(1-) 210 OC. No, because np(1 - p) 5 10 OD. No, because np(1-P) 10

Answer 1

Using binomial probability formula,

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

So,

P(X = 21) = 47C21 * 0.521 * 0.526

= 0.0892

np ( 1 - p) = 47 * 0.5 * 0.5 = 11.75 >= 10

Normal approximation can be used .

Yes, because np ( 1 - p) >= 10

Using Normal Approximation to Binomial
Mean = n * P = ( 47 * 0.5 ) = 23.5
Variance = n * P * Q = ( 47 * 0.5 * 0.5 ) = 11.75
Standard deviation = √(variance) = √(11.75) = 3.4278

With continuity correction

P(X = 21) = P(20.5 < X < 21.5)

P ( 20.5 < X < 21.5 ) = P ( Z < ( 21.5 - 23.5 ) / 3.4278 ) - P ( Z < ( 20.5 - 23.5 ) / 3.4278 )
= P ( Z < -0.58) - P ( Z < -0.88 )
= 0.281 - 0.1894
= 0.0915