Records show that 29 of all payments to a mailorder company

Records show that 29% of all payments to a mail-order company are submitted after the due date. Suppose that 50 payments are submitted this week. Let r be a random variable that represents the number of payments that are late. Use the normal approximation to the binomial to estimate:

a. Find the mean and standard deviation

b. Estimate P(r 20)

P(r 20) = P( ) = P( ) = _______________

c. Estimate P(20 r 25)

P(20 r 25) = P( ) = P( ) = ____________

Solution

Normal Approximation to Binomial Distribution
a)
Mean ( np ) =50 * 0.29 = 14.5
Standard Deviation ( npq )= 50*0.29*0.71 = 3.2086
Normal Distribution = Z= X- u / sd                   
b)
P(X >= 20) = (20-14.5)/3.2086
= 5.5/3.2086 = 1.7141
= P ( Z >1.714) From Standard Normal Table
= 0.0433                  

c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 20) = (20-14.5)/3.2086
= 5.5/3.2086 = 1.7141
= P ( Z <1.7141) From Standard Normal Table
= 0.95675
P(X < 25) = (25-14.5)/3.2086
= 10.5/3.2086 = 3.2725
= P ( Z <3.2725) From Standard Normal Table
= 0.99947
P(20 <= X <= 25) = 0.99947-0.95675 = 0.0427