A company is said to be out of compliance if more than 8% of all invoices contai

Question

A company is said to be out of compliance if more than 8% of all invoices contain errors, and it is said to be seriously out of compliance if more than 12% of all invoices contain errors. Suppose an auditor randomly selects a sample of 800 invoices and finds that 104 contained errors.

a) Construct a 90% confidence interval for this company's error rate.

b) How should the company be rated if statements about being out of compliance or seriously out of compliance require 5% level of significance?

c) What is the probability a company would be rated as seriously out of compliance by this test if 15% of all invoices at that company contain errors?

d) What sample size should the auditor use to estimate the error rate to within 2% with 95% confidence if it is assumed that the error rate will be no more than 15%?

e) Suppose the 104 erroneous invoices can be treated as a random sample from the population of all erroneous invoices. The error amounts are contained in the file
http://www.UTDallas.edu/~ammann/stat3355scripts/InvoiceErr.txt
Note: since this file just contains a single set of numeric values, you can use the scan() function in R to read this data. For example,

Construct a 95% confidence interval for the mean error amount. Also obtain and interpret a quantile-quantile plot of these invoice errors compared to the normal distribution.

Explanation / Answer

1. company error proportion p^ = 104/800 = 0.13

so for 95% confidence interval z - value = + - 1.96

so 95% CI = p^ +- 1.96 * sqrt [ p(1-p)/n]

= 0.13 +- 1.96 * sqrt [ 0.13 * 0.87/800]

= 0.13 +- 0.0119

= (0.118, 0.142)

(b) So, Here as confidence interval has the proportion of error rate > 0.08, so company must be rated out of compliances. For,COmpany must be rated seriously out of compliances, error rate shall be greater than 0.12 which is not the case here, so it is not seriously out of compliances,

(c) so here we have to calculate the probability if the real error rate = 0.15

so standard error of proportion = sqrt (0.15 * 0.85/800) = 0.0126

Z - value = (0.12 - 0.15)/ 0.0126 = - 0.03/0.01262 = -2.3764

so P( p> 0.12; 0.15; 0.01262) = 1 - 0.0087 = 0.9913

(d) Let the sample size = n

for 95% confidence Z - value = +- 1.96

here error rate will not be more than 15%, so we can say that upper limit of error rate confidence level is 15% , so p can be assumed 0.13 in this case

so 0.02 = 1.96* sqrt [0.13 * 0.87/n]

n = 1086.2124 that means n =1087 data records must be evaluated.

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