Question

Problem 4. Using samples of 200 credit card statements, an auditor found the following: Number with errors 4 25 9 a. Determine the fraction defective in each sample. b. If the true fraction defective for this process is unknown, what is your estimate of it? e. What is your estimate of the mean and standard deviation of the sampling distribution of fractions defective for samples of this size? d. What control limits would give an alpha risk of 03 for this process? e. What alpha risk would control limits of 047 and.003 provide? f. Using control limits of .047 and .003, is the process in control? Problem 5. Given the following data for the number of defects per spool of cable, using three-sigma limits, is the process in control? 1 2 3 4 5 6 79 10 11 12 13 14 Number of defects Draw the c-hart using excel (preferred) or by hand while determining if the process is in control Hint: This is a c-chart type of a question. 名0 44 F1 3 5

Answer 1

4.

(a)

Sample	1	2	3	4
# with errors (np)	4	2	5	9
n	200	200	200	200
Fraction defective, p = np / n	0.02	0.010	0.025	0.045

(b)

The estimate for average fraction defective in population = p_bar = Average (4,2,3,9) / 200 = 0.025

(c)

p_bar = 0.025
Sp = [p_bar * (1 - p_bar)/n]^1/2 = sqrt(0.025*(1 - 0.025)/200) = 0.011

(d)

For alpha risk = 0.03, z = normsinv(1 - 0.03/2) = 2.17

UCL = p_bar + z * Sp = 0.025 + 2.17*0.011 = 0.04887
LCL = Max(0, p_bar - z * Sp) = Max(0, 0.025 - 2.17*0.011) = 0.00113

(e)

If UCL = 0.047

then, 0.025 + z*0.011 = 0.047
or, z = 2

Normxdist(2) = 0.9772

So, the alpha risk is (1 - 0.9772)*2 = 0.0456

(f)

Yes, in-control because all the p values as derived in part-a are within the [LCL, UCL] limit.

5.

UCL 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 5.174 LCL SampleDefects CL 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 c-chart 4 4 5.174 10 12 13 14 15 16 17 18 19 20 21 10 12 13 14 2 1.5 1.5 2 10 12 16 DefectsCLUCLLCL 23

2 Sample UCL LCL -$C$17-D3+3*SORT(D3) MAX(O,D3-3*SQRT(D3) -$C$17D4+3* SQRT(D4) MAX(0,D4-3*SQRT (D4)) -$C$17 D5+3*SQRT (D5) MAX(0,D5-3*SQRT(D5)) =$C$17 =D6+3*SQRT(D6) =MAX(0,D6-3*SQRT(D6)) $C$17 D7+3*SQRT(D7)MAX(O,D7-3*SQRT(D7)) -$C$17-D8+3*SQRT(D8) MAX(O,D8-3*SQRT(D8)) 12 10 13 11 14 12 151-13 16 14 -$C$17 D10+3*SQRT(D10) MAX(0,D10-3*SQRT(D10)) -$C$17D11+3 SQRT(D11) MAX(0,D11-3*SQRT(D11)) -$C$17D12+3*SQRT(D12) MAX(0,D12-3*SQRT (D12)) -$C$17D13+3 SQRT(D13) MAX(0,D13-3*SQRT(D13)) -$C$17D14+3 SQRT(D14) MAX(0,D14-3*SQRT(D14)) -$C$17D15+3 SQRT(D15) MAX(0,D15-3*SQRT(D15)) -$C$17 D16+3*SQRT(D16) MAX(O,D16-3*SQRT(D16) Avg AVERAGE(C3:C16)