Question

Review the Payment Time Case Study and Data Set. Develop a 700-word report including the following calculations and using the information to determine whether the new billing system has reduced the mean bill payment time: Assuming the standard deviation of the payment times for all payments is 4.2 days, construct a 95% confidence interval estimate to determine whether the new billing system was effective. State the interpretation of 95% confidence interval and state whether or not the billing system was effective. Using the 95% confidence interval, can we be 95% confident that µ ≤ 19.5 days? Using the 99% confidence interval, can we be 99% confident that µ ≤ 19.5 days? If the population mean payment time is 19.5 days, what is the probability of observing a sample mean payment time of 65 invoices less than or equal to 18.1077 days? The payment time case

Answer 1

Using Minitab:

Choose Stat > Basic Statistics > 1-Sample Z.

Manual Calsulation:

EQUATION

CI = mean ± Z _x α/√N

following equation:

Using this calculation for the data provided we can calculate it as 4.2/√ (65) = 0.52.

A 95% critical value provides a 1.96 confidence interval. The sample mean of the PayTime 65 data sets was 18.11. Using the data we can calculate the confidence interval as:

((18.11 – (1.96*0.52)), (18.11 + (1.96*0.52))) = (18.11 – 1.02, 18.11 + 1.02) = (17.09, 19.13)

Confidence Interval Estimate for the Mean
Population Standard Deviation	4.2
Sample Mean	18.11
Sample Size	65
Confidence Level	95%
Standard Error	0.52
z-Value	1.96
Confidence Interval Lower Limit	17.09
Confidence Interval Upper Limit	19.13

With the calculations it is certain with 95% confidence that payment time can be achieved in ≤ 19.5 days. This is due to the fact the lower and upper limits; 17.09 and 19.13 respectfully, are below 19.5 days.

Using the 95% confidence interval, can we be 95% confident that µ ≤ 19.5 days?

EQUATION

CI = X ± Z x α/√N

Confidence Interval Estimate for the Mean:

Population Standard Deviation           4.2

Sample Mean 18.1077

Sample Size    65

Confidence level         95%

Solution:

Standard error of the mean = 0.5209

Z Value           = 1.9600

Interval half width = 1.0210

Confidence Interval

Interval Lower Limit = 17.0867

Interval Upper Limit = 19.1287

Based on 95% confidence interval level, it is evident that both 17.0866 and 19.1287 are below 19.5 days

Utilizing the 95% confidence interval, it is evident that 95% confident that µ ≤ 19.5 days

Then 95% CI = (17.0867, 19.1287) less than 19.5

Therefore 95% confident that µ ≤ 19.5 days

Payment time 99% Confidence Estimate for Mean:

Formula Confidence interval :   

((18.11 – (2.58*0.52)), (18.11 + (2.58*0.52))) = (18.11 – 1.34, 18.11 + 1.34) = (16.77, 19.45)

If the population mean payment time is 19.5 days, what is the probability of observing a sample mean payment time of 65 invoices less than or equal to 18.1077 days?

Assuming the population mean payment time is 19.5 days, the probability of observing a sample mean payment time of the 65 invoices that’s less than or equal to 181.1077 days comes to 0.0037 or can be figured as 37percent. The sample distribution is used to determine that has reduced the mean payment at least 50 percent.

Z value= (18.1077-19.5)/0.5209=-2.67

P(<18.1077)= P(z-<-2.67) =0.0037

The 37.07 percent suggest that the population payment mean shows a reduction in the actual payment time by at least 50 percent. Whereas, the payment time of using the old payment system it shows a 37.07 chance that the sample mean would be 18.1077 days.

Hope this will be helpful Thanks and God Bless you:)