Respuesta :
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
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.
Data Set as provided
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
[tex] X bar (Sample Mean)= \frac{\sum x}{n} [/tex]
A. Confidence Interval at 95%
The confidence level at 95% is:
[tex] CI = 17.08663837 < \mu < 19.12874624 [/tex]
The confidence level is used to make an inference about the population mean, μ, with the help of the sample mean (also known as a point estimate).
With the help of the confidence interval constructed above, we can say that the mean payment time was somewhere between 17.08663837 and 19.12874624 days.
Since the upper bound of the confidence interval, 19.12874624 days is less than the previous population mean at 19.5, we can conclude that the mean payment time was less than 19.5 days and therefore the new billing system was effective.
We begin by calculating the mean of all the 65 observations by using the formula:
[tex] X bar (Sample Mean)= \frac{\sum x}{n} [/tex]
[tex] X bar (Sample Mean)= \frac{1177}{65} [/tex]
X bar (Sample Mean) = 18.10769231
Since the population standard deviation is given, we can use the following formula to construct the Confidence Interval (CI).
[tex] CI = X bar \pm Z\frac{\sigma }{\sqrt{n}} [/tex]
where
CI = confidence interval
Z = two tailed Z score at 95% confidence level
σ = population standard deviation
n = sample size.
Substituting the values in the equation above, we get,
[tex] CI = 18.10769231 \pm 1.96\frac{4.2}{\sqrt{65}} [/tex]
[tex] CI = 18.10769231 \pm 1.96* 0.520945885} [/tex]
[tex] CI = 18.10769231 \pm 1.021053935 [/tex]
[tex] CI = 17.08663837 < \mu < 19.12874624 [/tex]
B. If the population mean payment time is 19.50, the probability of observing a sample mean payment time of 18.1077 days in a sample of 65 invoices is 0.3763%.
We use the following formula to calculate the Z score:
[tex] Z = \frac{X bar - \mu }{\frac{\sigma }{\sqrt{n}}} [/tex]
Substituting the values in the equation above, we get,
[tex] Z = \frac{18.1077 - 19.5 }{\frac{4.2 }{\sqrt{65}}} [/tex]
[tex] Z = \frac{-1.3923}{0.520945885}} [/tex]
[tex] Z = -2.672638444 [/tex]
By looking up the probabilities for standardized z score, we get
[tex] P(Z\leq -2.672638444) = 0.0037629 [/tex] or
P(Z ≤ -2.672638444) = 0.3763%.
