Question

A wholesaler has recently developed a computerized sales invoicing system. Prior to implementing this system, a manual system was used. The distribution of the number of errors per invoice for the manual system is as follows:

Errors per Invoice	0	1	2	3	More Than 3
Percentage of Invoices	80%	8%	5%	4%	3%

After implementation of the computerized system, a random sample of 500 invoices gives the following error distribution:

Errors per Invoice	0	1	2	3	More Than 3
Number of Invoices	462	15	10	8	5

pi	Ei	fi	(f – E) ^ 2/E
0.80	400	462	9.6100
0.08	40	15	15.6250
0.05	25	10	9.0000
0.04	20	8	7.2000
0.03	15	5	6.6667
	Chi-Square		48.10170		p-value 0.0000001096

(a)	Show that it is appropriate to carry out a chi-square test using these data.

Each Ei ≥

(b)	Use the Excel output shown above to determine whether the error percentages for the computerized system differ from those for the manual system at the .05 level of significance. What do you conclude?

Answer 1

Solution

Part (a)

It is appropriate to carry out a chi-square test using these data since

Each E_i ≥ 5 Answer 1

Part (b)

p-value is extremely low, smaller than any conceivable significance level. This implies that the Chi-square statistic is highly significant. This, in turn means that the error rate for the computerized system differ from those for the manual system at the .05 level of significance.

Going further, a closer look at the error rates reveals that for all the rates of 1, 2, 3 and more than 3, fi < Ei indicating that in each of these rates computerized system performs much better.

For the rate 0, fi > Ei, further confirming that computerized system is far superior.

In quantitative terms, average error rate for computerized system is just 0.15 per invoice whereas that for the manual system is 0.42 per invoice, almost three-fold. Answer 2

DONE