Statistics 495, Applied Statistics for Industry I
Name:__________________________
Exam 2, Fall 2008
Site:____________________________
INSTRUCTIONS: You will have 1 hour and 30 minutes to complete the exam. There are 7
questions worth a total of 100 points. Not all questions have the same point value so gauge your
time appropriately. Read the questions carefully and completely. Answer each question and show
work in the space provided on the exam. Turn in the entire exam when you are done or when time
is up. For essay questions, think before you write.
1. [10 pts] The two basic principles of statistical thinking are:
•
•
Variability always is and always will be present.
The iterative nature of learning.
Explain briefly how each of these principles is incorporated into the construction and use of
control charts. Be sure to discuss both principles.
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2. [15 pts] Two assembly lines making the same item are monitored, one with an X chart and
s chart combination and the other with an X chart and R chart combination. The first
assembly line has X =108.4 and s =2.21 based on subgroups of size 4. The second
assembly line has X =109.2 and R =4.19 based on subgroups of size 5.
a. [5] Which assembly line exhibits more consistency (less spread)? Support your
answer statistically.
b. [5] Calculate the control limits for the X chart for assembly line 1.
c. [5] Calculate the control limits for the X chart for assembly line 2.
2
3. [25 pts] Ceramic boards for classrooms are produced in 4 foot by 8 foot sheets. The
production manager is concerned about surface defects on the sheets. Below are the data.
Day
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number
of Boards
1
5
2
3
2
2
3
1
1
3
4
3
3
4
Number
of Defects
2
5
3
8
1
4
8
0
2
8
5
7
4
6
Day
Number
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Totals
Number
of Boards
6
3
5
4
7
3
4
2
5
2
2
1
2
3
86
Number
of Defects
18
6
8
17
4
6
4
5
2
5
0
15
3
5
161
a. [3] Compute the average number of defects per day.
b. [5] Calculate the upper control limit (UCL) and lower control limit (LCL) for a cChart for the number of defects. Put these limits, along with the center line, on the
run chart on the next page.
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C Control Chart
Count for Number of Defects
20
15
10
5
0
0
5
10
15
20
25
30
Sample
c. [3] Are there any days that plot outside the control limits? If so, what days are they and
how many defects are there on those days?
d. [3] Another way to chart this data is with a U-Chart for the rate of defects per board.
Compute the average rate of defects per board.
e. [4] Because of the unequal number of boards inspected each day, the U-Chart will have
different limits for each day. Compute the upper and lower control limit for day 15.
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f. [3] Looking at the U-Chart of surface defects on ceramic boards, are there any days that plot
outside the control limits? If so, what days are they and what is the rate of defects on each
of those days?
U Control Chart
Count Per Unit for
Number of Defects
20
15
10
5
UCL
0
LCL
0
5
10
15
20
25
30
Sample
g. [4] Which chart, the c-Chart or the U-Chart gives a more accurate picture of the process that
is producing the ceramic boards? Explain briefly.
5
4. [10 pts] For the following, assume that we have an idealized process that produces items
with a measured quality characteristic that is normally distributed with process center μ
and process standard deviation σ .
a. [4] For an X chart with alarm rule of a single point outside 3 “sigma” control limits,
the ARL for the on target, in control process is 385. Suppose we wanted an on
target, in control process to have an ARL of 250, where should the control limits be
set?
b. [6]} For an X chart with alarm rule of a single point outside 2.6 “sigma” control
limits, the ARL for the on target, in control process is about 100. What subgroup
size should we take so that a 0.4 σ shift from center will be detected within 15
subgroups, on average?
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5. [15 pts] A rental car agency wants to improve the quality of the services they provide at an
airport. For each of the following situations indicate what type of control chart should be
constructed to monitor the quality characteristic of interest. Briefly explain your choice.
a. [5] The number of people riding the rental agency’s shuttle buses each day.
b. [5] The time it takes each customer from the moment she/he steps off the shuttle bus
till she/he exits the agency in a rental car.
c. [5] The number of vehicle rental contracts, each day, where the collision damage
coverage is declined.
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6. [20 pts] An important step of an aerosol manufacturing process is the filling of the aerosol
cans. The fill weights (in grams) of aerosol cans are recorded for forty subgroups of five
aerosol cans. Variables control charts are produced for the forty subgroups.
a. [3] Below is the R chart for the forty subgroups. Using the one point out of control
limits alarm rule, are there any subgroups that plot out of control? If so, what
subgroups are they?
R Chart
Range of Aerosol Fill Weight
6
5
4
UCL=3.47
3
2
Avg=1.64
1
0
LCL=0.00
-1
0
5
10
15
20
25
30
35
40
Sample
b. [3] What does this tell you about the aerosol can filling process?
c. [4] If the subgroups you identified in a) were removed, how would the center line
and control limits on the R chart change. You do not have to calculate the new
center line or control limits simply tell me how they would change and explain your
reasoning.
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d. [10] Our operational definition of “in control” for the X chart is passing Tests 1, 2,
5 and 6. Refer to the X chart below. Indicate if the X chart passes each test. If it
does not pass a test indicate every subgroup where the test fails.
X Chart
Mean of Aerosol Fill Weight
114.0
113.5
113.0
112.5
UCL=112.268
A
B
C
C
B
A
112.0
111.5
111.0
110.5
Avg=111.323
LCL=110.377
110.0
0
5
10
15
20
25
30
35
40
Sample
•
Test 1: One point beyond Zone A
•
Test 2: Nine points in a row in Zone C or beyond, on one side of center.
•
Test 5: Two out of three points in a row in Zone A or beyond, on one side of center.
•
Test 6: Four out of five points in a row in Zone B or beyond, on one side of center.
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7. [5 pts] Consider a hypothetical process that is producing defects at a rate of 4 per hour. A
theoretical c chart for the number of defects per hour has control limits given by:
LCL = 0.0 UCL = 4 + 3 4
If the process changed so that it started producing defects at a rate of c=9 per hour, what is
the average run length, ARL, before the c chart would detect the change?
Hint: Compute the approximate probability that, after the change, there would be a number
of defects in a given hour that is beyond the UCL given above. Recall that the number of
defects per hour is approximately normally distributed with mean = c and standard
deviation = c .
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