1. X-bar and R charts are maintained on a certain quality characteristic based on a subgroup size of 4. The specifications require that the quality characteristic must be between 345 and 355 lb. On a certain day, data for 25 subgroups of size 4 were collected. EX-bar is found to be 8,745.2 and ER is found to be 62.8. Assume that the process was in statistical control that day.
a) Calculate the 3-sigma control limits for the X-bar and R control charts
b) If the mean of the process shifts to 347 lb the next day, compute the probability that the shift will not be detected on the X-bar chart on the first subgroup plotted after the shift takes place. Assume no change in a.
c) Assuming normal distribution of process output, what proportion of nonconforming units would be produced at this new value of the mean.
2. A product development group determines that it must have a fiber, which among other properties, has a minimum tensile strength of 2.500 gm in 99 percent of the fiber used. Manufacturer-ABC offers to supply such fiber and a contract is arranged. Manufacturer-ABC knows that the standard deviation (a) for the process is 0.020 gm. What minimum aimed at value of the mean is required to assure compliance with the contract? Assume the process is in statistical control and the distribution is normal.
3. Why is it important to examine the R chart first before looking at the X-bar chart?
4. A p-chart is used to monitor the quality of output of a mechanical part. The process has been operating with a mean p-bar of 0.02. An average of 350 units are produced each day and subjected to 100% inspection.
a) Calculate the control limits based on average daily output
b) On one particular day, 22 of the inspected 550 units were rejected. Was the process operating in control that day?
5. a) A Lot of 40 items contains exactly 3 defective items. If a sample of 10 items is to be drawn at random from the lot, find the probability of finding zero defective. items.
b) A process generating 7.5% nonconforming items is sampled at random intervals in subgroups of 10 items. What is the probability that a given subgroup will contain zero defective items? (assume a continuous process, hence a large lot size)
Five factors (X1, X2, X3, X4, X5) and all two-factor interactions are to be studied at two levels.
a. Construct an appropriate half-fractional-factorial design (show all work).
b. List confounding patterns for main and two-factor interactions (show all work).
7. X1, X2, X3, X4, X5, X1X2, and X1X3 effects are studied at two levels to maximize Y.
a. Analyze data using response tables. Plot main and interaction effects.
b. Construct the appropriate math model using the response table & Excel.
c. Comment on the goodness of the math model. (Is this a good math model? Can you tell? Why or why not?)
d. What can be done (if anything) to come up with a better answer for part c?
|
|
X1
|
X2
|
X3
|
X4
|
X5
|
Y
|
|
1
|
-1
|
-1
|
-1
|
-1
|
-1
|
136
|
|
2
|
-1
|
-1
|
1
|
1
|
1
|
145
|
|
3
|
-1
|
1
|
-1
|
1
|
1
|
155
|
|
4
|
-1
|
1
|
1
|
-1
|
-1
|
149
|
|
5
|
1
|
-1
|
-1
|
-1
|
1
|
143
|
|
6
|
1
|
-1
|
1
|
1
|
-1
|
126
|
|
7
|
1
|
1
|
-1
|
1
|
-1
|
125
|
|
8
|
1
|
1
|
1
|
-1
|
1
|
138
|
8. One is faced with a decision to purchase parts from three suppliers. The part you need has a specification of 15.00 + 0.09 mm. The SPC studies done by suppliers indicate that their processes were in statistical control with the following process parameters.
Supplier I : X-double bar = 15.00 mm, and Sigma-x = 0.03 mm
Supplier 2: X-double bar = 15.05 mm, and Sigma-x = 0.022 mm
Supplier 3: X-double bar - 14.95 mm, and Sigma-x = 0.015 mm
Which supplier would you select? Why?