The Payment Time Case
The purpose of the electronic billing system is to make the processing easier and organizations. The new billing system must reduce the payment time process. Considering the given scenario, the old billing system takes more than 39 days which is against the standard payment time of 30 days. The new billing system is expected to take 50% time less than the old billing system, which should be less than 19.5 days. To analyze the effectiveness of the system, the organization has selected random samples of 65 invoices out of 7,823 invoices processed during the first three months of the installation of the new electronic billing system. The confidence interval will give the range of approval of the probability of the sample. Most of the times, 95% confidence interval is utilized due to its high acceptance.
Effectiveness Of The New Billing System
Considering the standard deviation of all payments to be 4.2 days, the confidence interval of the new billing system will be constructed for 95%. The formula that will help to determine the population mean is x̅ + – z * (σ/n). The formula can be expressed as x̅ + z * (σ/n) and x̅ – z * (σ/n)
Sample size 65
Conf. coeff. 1.96
Margin of error 1.021053935
Upper bound 19.12874624
Lower bound 17.08663837
The calculated value comes out to be 17.09 days and 19.19 days. The margin of error is obtained by deducting lower bound from the upper bound value which is 1.02. The critical value is 1.96. The calculated value is less than the critical value which shows that the system is effective because the occurrence of error is less than the estimated chances of error. (Cleaves, Hobbs, & Noble, 2017)
95% confidence interval
At a 95% confidence interval, the computed mean is obtained as 18.11, the median is 17, the mode is 16, the standard deviation is 3.96, and variance is 15.69. The critical value is 1.96. If the critical value is more than the calculated value then we should accept the null hypothesis. The calculation is to be made to check if the probability of mean payment time is less than equal to 19.5 days. (Black, 2017) The formula used to calculate the confidence interval is x ± z (α/2) σ/√n
Standard Error 0.491330159
Standard Deviation 3.9612
Sample Variance 15.6913
Confidence Level(95.0%) 0.981544819
The 95% confidence interval results in the lower bound to be 17.1261 and upper bound to be 19.0892. The confidence interval is 0.98154, which is less than the critical value. Hence, we will accept the null hypothesis, that we are 95% confident that means is less than equal to 19.5 days.
99% confidence interval
At 99% confidence interval, the computed mean value is 18.1077 and the standard deviation is 3.96. The critical value is 2.56. If the critical value is more than the calculated value then we will accept the null hypothesis. The calculation will be made to check if we are 99% confident that the mean will be less than equal to 19.5 days. The formula used to check the confidence interval is x ± z (α/2) σ/√n
Standard Error 0.491330159
Standard Deviation 3.961230384
Sample Variance 15.69134615
Confidence Level(99.0%) 1.304409996
The lower bound value is 16.8033 and upper bound value is 19.4121. The confidence interval is 1.3044. The confidence interval is less than the critical value. This means we will accept the null hypothesis that we are 99% confident that mean is less than or equal to 19.5 days.
Probability to observe a sample mean and payments time of 65 invoices
The population mean is given as 19.5 days and the standard deviation is 4.2 days. The formula applicable in this case is the Z test, the formula is (x – μ) / (σ / √n). The x is 18.1077, σ is 4.2, and n is 65.
z test = (18.1077-19.5) / (4.2/√65)
= (18.1077-19.5) / 0.521
P (mean < 18.1077) = P (z < -2.67)
Therefore, the probability is 0.0038.
We conclude that the firm should implement the new system. In order to maximize profits and receive the payments on a timely basis, the new billing system must be marketed to the other trucking companies in the country.
Black, K. (2017). Business Statistics: For Contemporary Decision Making (9th ed.). Danvers, MA: Wiley
Cleaves, C., Hobbs, M., & Noble, J. (2017). Business Math (11th ed.). New York City, NY: Pearson Education.