Question

Understanding Variation, Question 1-10, (pg 114-116) Last page (60) shows formula. Must be answered in excel.

Understanding Variation /Managing Chaos WHAT SHOULD YOu DO NOW? Statistics has never been a spectator sport. Process behavior charts are no exception. You should see if you can compute the limits and construct an XmR chart 1. The U.S. Trade Deficits for the last half of 1988 are shown in Figure 6.7. Use the data of Figure 6.7, and the blank form in Figure 6.8 to plot the time series graph for the U.S. Trade Deficits for the last half of 1988 Jul Aug Sep Oct Nov Dec 1988 10.5 . 9.2 10.1 10.4 10.5 Figure 6.7: Monthly U.S. Trade Deficits, 1988 ($ billions) 2. Use the data in Figure 6.7 to compute the monthly moving ranges. Note that moving ranges are defined to be positive values, and are found by computing the differences between successive values. Write these values in the space provided in Figure 6.8, and plot the running record of the moving ranges. 3. Compute the average trade deficit for the last half of 1988 4. Compute the Average Moving Range 5. Compute the Natural Process Limits for the X-chart using the for- mulas on page 60 or on page 137. 6. Compute the Upper Range Limit using the formula on page 60 or on page 137. 114 Six Look What You've Been Missing ul Aug Sep Oct No Dec mR 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 3.0 2.0 E 1.0 2 Figure 6.8: Form for XmR Chart Plot the limits from Figure 6.8 on Figure 6.10. 8. The U.S. Trade Deficits for the first half of 1989 are shown on the next page in Figure 6.9. Plot these values and their moving ranges on rm given in Figure 6.10. Understanding Variation/Managing Chaos Jan Feb M Apr May Jun 8.7 9.6 9.0 7.0 6.8 1989 8.7 Figure 6.9: Monthly U.S. Trade Deficits, 1989 (S billions) Jan Feb Mar Apr May Jun mR 13.0 12.0 11.0 10.0 9.0 8.0 7.0 6.0 3.0 2.0 E 1.0 Figure 6.10: Form for XmR Chart There are two indications of shifts in Figure 6.10. The first shift was favorable, and the second was unfavorable 9. When is this favorable shift detected? 10. When might this favorable shift have begun? II. When is the unfavorable shift detected? 116 riation / Managing Chaos Understanding Va Formulas for Charts for Individual Values and Mov Use the individual values to compute the Avera This value will be the central line for the X Find the moving ranges and ge Moving Range, mR e will be the central line for the mR chart. compute the Avera T his value WI To find the Upper Natural Process Limit for the X-chan multiply the Average Moving Range by 2.66 and add the product to the Average UNPL = X + (2.66 x mR) To find the Lower Natural Process Limit for the X-chart: multiply the Average Moving Range by 2.66 and subtract the product from the Average LNPL = X-(2.66 x mR) To find the Upper Range Limit for the mR chart: multiply the Average Moving Range by 3.27. URL -3.27 x mR The multiplicative constants of 2.66 and 3.27 seen in the equations above are scaling factors needed to convert the Average Range into the values you need to obtain the appropri for each portion of the XmR chart Useful lmits may be constructed with as few as five or six values .The uncertainty in the computed limits decreas ate limits data used to compute the limits increases as the amount o

Answer 1

Solution

Final answers are given below. Back-up Theory and Calculation Details follow at the end.

Part (2)

Moving Range

i	xi	MRi = |xi - xi - 1|
1	10.5	-
2	11.2	0.7
3	9.2	2.0
4	10.1	0.9
5	10.4	0.3
6	10.5	0.1

Answer 1

Part (3)

Average Trade Deficit for the last half of 1988 (xbar) = 10.32 Answer 2

Part (4)

Average Moving Range = 0.8   Answer 3

Part (5)

Natural Process Limits for x-chart:

CL: 10.32

LCL: 8.19

UCL: 12.44

Answer 4

Part (6)

Upper Range Limit: 2.61   Answer 5

Part (6)

Control Status: All points are within limits. Answer 6

Back-up Theory

xi = measurement on the ith uint
k = number of samples
xbar = (1/k)Σ(i = 1, k)xi
MRi = |xi - xi -1|
MRbar = {(1/(k - 1)}Σ(i = 2, k)MRi
Control Limits
I-Chart
Center Line CL = xbar
Lower Control Limit LCL = xbar - 3MRbar/d2
Upper Control Limit UCL = xbar + 3MRbar/d2
MRChart
Center Line CL = MRbar
Lower Control Limit LCL = 0
Upper Control Limit UCL = D4MRbar

Calculation Details

k		6
Σxi		61.9
ΣMRi		4
xbar		10.31667
MRbar		0.8
d2		1.128
D4		3.267
σhat = MRbar/d2		0.70922
I Chart
CL = xbar		10.31667
LCL = CL - 3σhat		8.189007
UCL = CL + 3σhat		12.44433
MR Chart
CL = MRbar		0.8
LCL		0
UCL = D4MRbar		2.6136
Control Status
All points within limits

DONE