Question

A study by a wholesaler regarding the payment of invoices reveals the time from billing until payment is received is approximately normally distributed. The mean time until payment is received is 18 days and the standard deviation is 5 days.

1. What percentage of the invoices is paid within 14 days?
2. What percentage of invoices is paid in more than 14 days but less than 25?
3. The management want to encourage customers to pay their monthly invoices as soon as possible. Therefore, it announced that a 2% reduction in price would be in effect for customers who pay within 7 days of the billing date. A study conducted a few months after the announcement showed that the percentage of customers who paid within 7 days was 4.8%. What was the change in the percentage of customers who paid within 7 days?

Answer 1

Consider the random variable X as

X: Time until payment is received

X ~ N ( Mean = 18 , variance = 5²)

E(X) = 18 and SD(X) = 5

a) Required probability = P ( X < 14)

= P (( X- E(X) ) / SD(X) < ( 14-18) / 5)

= P ( Z < - 0.8) since Z = ( X- E(X) ) / SD(X) ~ N(0,1)

From normal probability table

P(Z< -0.8) = 0.2119

Percentage of invoices paid within 14 days = 21.19%

b) Required probability = P( 14 < X < 25)

= P ( (14-18)/5 < ( X- E(X) ) / SD(X) < (25-18) /5)

= P ( -0.8 < z < 1.4)

= P (Z < 1.4) - P(Z< -0.8)

From normal probability table

P(Z< 1.4) = 0.9192 , P(Z< -0.8) = 0.2119

P( 14 < X < 25) = 0.9192 - 0.2119 = 0.7073

Percentage of invoices paid in more than 14 days but less than 25 days = 70.73%

c) P ( invoices paid within 7 days) = P( X< 7)

= P ( ( X- E(X) ) / SD(X) < (7-18) /5)

=P ( Z < -2.2)

From normal probability table

P(Z< -2.2) = 0.0139

Percentage of invoices paid within 7 days = 1.39%

But After encouraging the customers the percentage of invoices paid within 7 days =4.8%

Hence Change in the percentage of customers who paid within 7 days = 4.8 % - 1.39% = 3.41%.