A study of Hub Furniture regarding the payment of invoices reveals the time from billing until payment is received follows the normal distribution. The mean time until payment is received is 20 days and the standard deviation is 5 days.
a. What percent of the invoices are paid within 15 days of receipt?
b. What percent of the invoices are paid in more than 28 days?
c. What percent of the invoices are paid in more than 15 days but less than 28 days?
d. The management of Hub Furniture wants to encourage their customers to pay their monthly invoices as soon as possible. Therefore, it announced that a 2 percent reduction in price would be in effect for customers who pay within 7 days of the receipt of the invoice. What percent of customers will earn this discount?
To answer these questions, we will use the Z-score formula and the standard normal distribution table.
First, let's define the Z-score formula:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Random variable (number of days)
μ = Mean (average number of days)
σ = Standard deviation
a. To find the percent of invoices paid within 15 days, we need to calculate the Z-score for 15 days.
Z = (15 - 20) / 5
Z = -1
Using the standard normal distribution table, we can find the corresponding percentile for a Z-score of -1. The table shows that the percentile is approximately 0.1587.
Therefore, approximately 15.87% of the invoices are paid within 15 days of receipt.
b. To find the percent of invoices paid in more than 28 days, we need to calculate the Z-score for 28 days.
Z = (28 - 20) / 5
Z = 1.6
Using the standard normal distribution table, we can find the percentile for a Z-score of 1.6. The table shows that the percentile is approximately 0.9452.
Therefore, approximately 94.52% of the invoices are paid in more than 28 days.
c. To find the percent of invoices paid in more than 15 days but less than 28 days, we need to calculate the Z-scores for both 15 days and 28 days and find the difference.
Z1 = (15 - 20) / 5
Z1 = -1
Z2 = (28 - 20) / 5
Z2 = 1.6
Now we find the difference in percentiles between these two Z-scores, which is the percent of invoices paid between 15 and 28 days.
Percent = Percentile(Z2) - Percentile(Z1)
Percent = 0.9452 - 0.1587
Percent = 0.7865
Therefore, approximately 78.65% of the invoices are paid in more than 15 days but less than 28 days.
d. To find the percent of customers who will earn the 2 percent discount for paying within 7 days, we need to calculate the Z-score for 7 days.
Z = (7 - 20) / 5
Z = -2.6
Using the standard normal distribution table, we can find the percentile for a Z-score of -2.6. The table shows that the percentile is approximately 0.0049.
Therefore, approximately 0.49% of customers will earn the 2 percent discount by paying within 7 days of receipt.