The essay seeks to analyze whether the new billing system introduced at the trucking company, Stockton, has cut down the bill payment time. The analysis will determine whether the consulting firms can promote the system to other trucking companies. From the information given the new system has achieved a mean payment time of 18.1 days down from 19.5 days in the old system. The results were collected from 65 sample invoice. The analysis from the new system used a standard deviation of payment to be 4.2 days. This analysis will calculate whether one can be say the mean payment time, μ, will be lower than 19.5 days with a 95 percent confidence. The analysis will also calculate whether one can be 95 percent confident that the μ is less than 19.5 days. The analysis will also calculate the probability that the mean payment time from a sample of 65 invoices can be less or equivalent to 18.10 days.
Using the data sets above, the analysis is as follows:
95% confident μ≤ 19.5 days
Data: Population standard deviation, α = 4.2,
Sample mean, μ = 18.10
Confidence level = 95 percent
Sample size, N = 65
To estimate the confidence, the equation CI = X ± Z c is used. Where X is the sample mean and Z c is the z value for confidence level. A 95 percent confidence interval yields an interval estimate of 17.0867 and 19.128. Since both 17.0867 and 19.128 are below 19.5 day we can conclude that the mean payment time will be less or equal to 19.5 days with a 95 percent confidence.
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99% confident μ≤ 19.5 days
Using the same data as above, but a different confidence level of 99 percent, the results yield CI of 16.7658 and 19.4496 both of which are less 19.5. We can therefore be 99 percent confident that the billing system will have an average bill payment time of less or equal to 19.5 days.
The probability of sample mean payment time being less or equal to 18.10 days.
To determine the probability, one needs to determine the Z score also referred to as the standard score ( Gulay, 2018 ). The score is a valuable statistic as it permits one to determine the probability of a score arising within a normal distribution ( Gulay, 2018 ). One can determine the Z score using the formula Z= . Where σ is the standard error of the mean. σ is given by the formula σ = and z is representative of the standardized random variable and probabilities that are linked to ranges of Z. The probability is determined using the Z distribution table.
σ = = 0.5209
In this case Z = = -2.607
The probability, P (mean, μ < 18.10) = P (Z < -2.67) stands at 0.0038. This figure is very low and shows that it is no likely that from 65 invoices, the mean payment time can be less or equivalent to 18.10 days.
Hypothesis Testing
To conclusive make a decision a hypothesis testing of the results would be necessary. Hypothesis testing helps one to weigh whether it is statistically worth it to market the billing system. The technique is helpful in that it evaluates two opposing arguments about the system (Ramsey, & Schafer, 2012). In this case, one would test the null and alternative hypothesis.
Let the null hypothesis denote a mean greater than 19.5 days, μ > 19.5 and the alternative hypothesis indicate a mean payment time less than 19.5 days, μ≤ 19.5.
Since a 95 percent confidence level is required, one can set the set the α at 0.05 and conduct the z test. The z test is also informed by the fact that the sample size is larger than 30. Ogunjimi (2017) observed that the false discovery rate (FDR) rejects the null is the value of “z” is lower than the critical value (CRV) of 1.654. This value is for a one-tail test. To determine the value of z, the formula z= is used.
z=
z= = -2.67265
The value realized, -2.67, is lower than the critical value of 1.645; hence, the null is rejected. The hypothesis that the billing system makes a significant statistical difference in decreasing the payment time is valid. The decision is, therefore, to carry on with promoting of the system to other trucking companies.
Conclusion
The consulting company had expressed doubt on whether the system could indeed lessen the billing payment time of a trucking company. It thus follows that a statistical analysis of the data they obtained was necessary in order to go ahead and promote the billing system to other trucking firms. The analysis successful used the 65 samples provided to determine that the new billing system will make a significant statistical reduction in payment time. Trucking firms wishing to adopt the new system can do so with confidence that it is beneficial to their operations.
References
Gulay, E. (2018). Comparing Simple Forecasting Methods and Complex Methods: A Frame of Forecasting Competition. Scientific Annals of Economics and Business , 65 (2), 159-169. doi:10.2478/saeb-2018-0010
Ogunjimi, A. (2017, March 12). How Is a Hypothesis Important in Business? Retrieved from https://smallbusiness.chron.com/hypothesis-important-business-34382.html
Ramsey, F., & Schafer, D. (2012). The Statistical Sleuth: A Course in Methods of Data Analysis . Boston, MA: Cengage Learning.
