1. Vacuum pumps are being audited to determine if the process is in control and also capable of meeting the customer specifications. Samples of size n=5 are taken each hour of the day and tested with the data on vacuum recorded. The results of 10 consecutive hours of testing are provided.

Sample #	Obs. 1	Obs. 2	Obs. 3	Obs. 4	Obs. 5	x	R
1	15	17	16	17	14
2	16	16	9	14	15
3	17	17	15	15	14
4	18	15	16	16	16
5	13	15	17	16	19
6	10	15	11	17	18
7	15	15	14	14	16
8	14	13	12	15	14
9	15	18	18	12	10
10	12	12	15	15	16

a. Find the and R of each sample and record above.

b. Find and record the grand mean ( ) and in the appropriate boxes.

c. Find the UCL and LCL for the chart.

d. Find the UCL and LCL for the R chart.

e. Sketch out a control chart for the x-bar chart. Is the process in control in terms of the "average" vacuum pressure? ______ Yes or No

f. Sketch out a control chart for the R chart. Is the process in control in terms of the "variation" in the process? _______ Yes or No

g. Find the process standard deviation ().

h. Find the appropriate process capability index for the process if the lower and upper specification limits are 8 to 20.

i. What is your interpretation of the capability index?

2. A company has been doing final inspection on hose assemblies for vacuum pumps. A loose hose on an orifice is considered a defect and must be repaired. The daily production records from a sample of 100 each day for 10 days show the following results.

Day	Defectives per sample	Daily pi = fraction defective
1	10
2	12
3	15
4	8
5	7
6	12
7	13
8	35
9	12
10	15
Total

a. Find the average fraction defective (p).

b. Find the UCL and LCL for the p chart.

c. Sketch out a plot of the fraction defective versus the day along with the upper and lower control limits and the individual daily fraction defective.

d. Is the process in control? Yes or No ______ If no, what should be done?

3. How could and R charts be used to determine if the grades assigned in different classes by different instructors were "in control"? What should be done to classes that are "out of control"?

4. If a process was sampled hourly for the past 30 hours with a sample size of 4 and the mean of the 30 samples was 52.43 and the mean of the 30 ranges was 3.45,

a. What would be the UCL and LCL for the x-bar chart?

b. What would be the UCL and LCL for the range chart?

c. What would be the value of the standard deviation of the process? d. If the specifications for the process were 50 7, draw a sketch of the process distribution with the specifications limits labeled.

e. Find the Cpk for the process.

f. What would be the capability index if the process was centered (Cp)?

g. Would the process be capable of 3 quality if the process was centered? h. Would the process be 6 capable with specifications if the process was centered?

5. The orders at a fast food restaurant have been filled over the past month with only 3% defectives (and the daily fluctuations have been in control). A total of 100 orders per day were sampled each day for 30 days. Find UCL and LCL for the fraction defective for a future baseline and for improvement.

6. Telephone inquiries of IRS "customers" are monitored daily at random. Incidents of incorrect information or other problems (impoliteness) are recorded. The data for the past week are below.

Day	1	2	3	4	5	6
No. Of problems	5	10	23	20	15	18

a. Calculate , , and . b. Is the process in control?

7. If a process has a mean of 45 and a standard deviation of 0.5, sketch out the distribution of output.

a. If the specification limits are 46 2, sketch those limits on the graph above and then find the process capability index (Cpk).

b. What measures can be taken to improve the Cpk of theprocess?