Question

Purpose of Assignment

The purpose of the assignment is to develop students' abilities in using data sets to apply the concepts of sampling distributions and confidence intervals to make management decisions.

Assignment Steps

Resources: Microsoft Excel®, The Payment Time Case Study, The Payment Time Case Data Set

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?

Case Study – Payment Time Case Study

Major consulting firms such as Accenture, Ernst & Young Consulting, and Deloitte & Touche Consulting employ statistical analysis to assess the effectiveness of the systems they design for their customers. In this case, a consulting firm has developed an electronic billing system for a Stockton, CA, trucking company. The system sends invoices electronically to each customer’s computer and allows customers to easily check and correct errors. It is hoped the new billing system will substantially reduce the amount of time it takes customers to make payments. Typical payment times—measured from the date on an invoice to the date payment is received—using the trucking company’s old billing system had been 39 days or more. This exceeded the industry standard payment time of 30 days.

The new billing system does not automatically compute the payment time for each invoice because there is no continuing need for this information. The management consulting firm believes the new system will reduce the mean bill payment time by more than 50 percent. The mean payment time using the old billing system was approximately equal to, but no less than, 39 days. Therefore, if µ denotes the new mean payment time, the consulting firm believes that µ will be less than 19.5 days. Therefore, to assess the system’s effectiveness (whether µ < 19.5 days), the consulting firm selects a random sample of 65 invoices from the 7,823 invoices processed during the first three months of the new system’s operation. Whereas this is the first time the consulting company has installed an electronic billing system in a trucking company, the firm has installed electronic billing systems in other types of companies.

Analysis of results from these other companies show, although the population mean payment time varies from company to company, the population standard deviation of payment times is the same for different companies and equals 4.2 days. The payment times for the 65 sample invoices are manually determined and are given in the Excel^® spreadsheet named “The Payment Time Case”. If this sample can be used to establish that new billing system substantially reduces payment times, the consulting firm plans to market the system to other trucking firms.

PayTime
22
19
16
18
13
16
29
17
15
23
18
21
16
10
16
22
17
25
15
21
20
16
15
19
18
15
22
16
24
20
17
14
14
19
15
27
12
17
25
13
17
16
13
18
19
18
14
17
17
12
23
24
18
16
16
20
15
24
17
21
15
14
19
26
21

I do not need help with the paper. I need to understand the math.

Answer 1

Result:

· 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.

Descriptive statistics
	PayTime
n	65
mean	18.1077

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

Confidence Interval Estimate for the Mean
Data
Population Standard Deviation	4.2
Sample Mean	18.1077
Sample Size	65
Confidence Level	95%
Intermediate Calculations
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

95% CI = (17.0867, 19.1287)

Since the upper limit of the 95% CI is lower than 19.5, we conclude that µ ≤ 19.5 days.

Using the 99% confidence interval, can we be 99% confident that µ ≤ 19.5 days?

Confidence Interval Estimate for the Mean
Data
Population Standard Deviation	4.2
Sample Mean	18.1077
Sample Size	65
Confidence Level	99%
Intermediate Calculations
Standard Error of the Mean	0.5209
Z Value	2.5760
Interval Half Width	1.3419
Confidence Interval
Interval Lower Limit	16.7658
Interval Upper Limit	19.4496

99% CI = (16.7658, 19.4496)

Since the upper limit of the 99% CI is lower than 19.5, we conclude 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?

P( mean x < 18.1077)

= -2.6726

P( mean x < 18.1077) = P( z < -2.6726) = 0.0038

Z Test of Hypothesis for the Mean
Data
Null Hypothesis                       m=	19.5
Level of Significance	0.05
Population Standard Deviation	4.2
Sample Size	65
Sample Mean	18.1077
Intermediate Calculations
Standard Error of the Mean	0.5209
Z Test Statistic	-2.6726
Lower-Tail Test
Lower Critical Value	1.645
p-Value	0.0038
Reject the null hypothesis

The required probability = 0.0038