The table below gives information on coded length measurements in a machining process. The data shown are ¯x and R values (coded) for 24 samples of size n = 4. Only the last three decimals of a measurement are recorded (i.e. 24.5 should be 0.30245).
|
Sample no.
|
X
|
R
|
|
1
|
24.5
|
4
|
|
2
|
24.2
|
3
|
|
3
|
21.6
|
3
|
|
4
|
25.5
|
4
|
|
5
|
25.0
|
5
|
|
6
|
24.1
|
6
|
|
7
|
22.6
|
4
|
|
8
|
23.8
|
3
|
|
9
|
24.8
|
7
|
|
10
|
23.6
|
8
|
|
11
|
21.9
|
3
|
|
12
|
28.6
|
9
|
|
Sample no.
|
X
|
R
|
|
13
|
25.4
|
8
|
|
14
|
24.4
|
6
|
|
15
|
24.0
|
5
|
|
16
|
26.8
|
7
|
|
17
|
24.9
|
4
|
|
18
|
23.5
|
3
|
|
19
|
21.7
|
8
|
|
20
|
24.0
|
4
|
|
21
|
25.1
|
2
|
|
22
|
23.7
|
4
|
|
23
|
22.8
|
3
|
|
24
|
23.5
|
2
|
|
sum
|
580
|
115
|
(a) Plot ¯X and R charts based on the data above. What center line and control limits would you recommend for controlling future production? You can assume that out control points are due to assignable causes.
(b) Suppose specifications are 0:30250 _ 0:00100. Calculate cp.
(c) Discuss briefly the capability of the process and how process performance can be improved.
2. Parts manufactured by an injection molding process are subjected to a compressive strength test. Twenty samples of five parts each are collected and the compressive strengths (inpsi) are shown in the table that follows.
|
x1
|
x2
|
x3
|
x4
|
x5
|
2
|
S
|
|
83.0
|
81.2
|
78.7
|
75.7
|
77.0
|
79.1
|
2.99
|
|
88.6
|
78.3
|
78.8
|
71.0
|
84.2
|
80.2
|
6.65
|
|
85.7
|
75.8
|
84.3
|
75.2
|
81.0
|
80.4
|
4.79
|
|
80.8
|
74.4
|
82.5
|
74.1
|
75.7
|
77.5
|
3.88
|
|
83.4
|
78.4
|
82.6
|
78.2
|
78.9
|
80.3
|
2.49
|
|
75.3
|
79.9
|
87.3
|
89.7
|
81.8
|
82.8
|
5.78
|
|
74.5
|
78.0
|
80.8
|
73.4
|
79.7
|
77.3
|
3.22
|
|
79.2
|
84.4
|
81.5
|
86.0
|
74.5
|
81.1
|
4.53
|
|
80.5
|
86.2
|
76.2
|
84.1
|
80.2
|
81.4
|
3.86
|
|
75.7
|
75.2
|
71.1
|
82.1
|
74.3
|
75.7
|
4.01
|
|
80.0
|
81.5
|
78.4
|
73.8
|
78.1
|
78.4
|
2.89
|
|
80.6
|
81.8
|
79.3
|
73.8
|
81.7
|
79.4
|
3.31
|
|
82.7
|
81.3
|
79.1
|
82.0
|
79.5
|
80.9
|
1.57
|
|
79.2
|
74.9
|
78.6
|
77.7
|
75.3
|
77.1
|
1.94
|
|
85.5
|
82.1
|
82.8
|
73.4
|
71.7
|
79.1
|
6.14
|
|
78.8
|
79.6
|
80.2
|
79.1
|
80.8
|
79.7
|
0.81
|
|
82.1
|
78.2
|
75.5
|
78.2
|
82.1
|
79.2
|
2.85
|
|
84.5
|
76.9
|
83.5
|
81.2
|
79.2
|
81.1
|
3.11
|
|
79.0
|
77.8
|
81.2
|
84.4
|
81.6
|
80.8
|
2.55
|
|
84.5
|
73.1
|
78.6
|
78.7
|
80.6
|
79.1
|
4.12
|
(a) Is there evidence to support the claim that compressive strength is normally distributed?
[If your statistical package does not have a subprogram that allows you to answer this, you can briefly explain how normality can be verified or tested.]
(b) Construct X and s charts to show that the process in statistical control. (If you are constructing the charts manually, you only need to use the last two columns)
(c) After establishing that the process is in control, the charts were used for future monitoring. The readings for the next 13 samples are shown on the next page. Plot the X and s values against the control limits from part (b). What do the charts indicate re the process mean and variability?
|
|
X2
|
X3
|
X4
|
X5
|
2
|
S
|
|
79.2
|
84.4
|
81.5
|
86.0
|
74.5
|
81.1
|
4.53
|
|
80.2
|
86.2
|
76.5
|
84.1
|
80.2
|
81.4
|
3.78
|
|
75.7
|
75.2
|
71.1
|
82.1
|
73.4
|
75.5
|
4.11
|
|
80.0
|
84.0
|
78.4
|
73.8
|
78.1
|
78.9
|
3.68
|
|
80.6
|
81.8
|
79.3
|
73.5
|
79.0
|
78.8
|
3.19
|
|
87.3
|
76.9
|
87.0
|
80.2
|
77.4
|
81.8
|
5.08
|
|
79.2
|
88.2
|
96.1
|
92.1
|
81.5
|
87.4
|
7.08
|
|
79.1
|
86.1
|
86.0
|
78.2
|
81.4
|
82.2
|
3.74
|
|
80.1
|
75.5
|
80.1
|
81.6
|
85.5
|
80.6
|
3.59
|
|
90.4
|
82.2
|
97.5
|
81.5
|
80.3
|
86.4
|
7.38
|
|
70.0
|
73.0
|
76.1
|
82.3
|
86.5
|
77.6
|
6.75
|
|
84.2
|
84.3
|
90.4
|
80.2
|
80.8
|
84.0
|
4.05
|
|
93.0
|
84.5
|
88.3
|
86.6
|
93.4
|
89.2
|
3.93
|
3. The manager of an accounting office at a large hospital was interested in studying the problem of errors in the entry of account numbers into the computer system. A group of 200 account numbers was selected from each day's output and each was inspected to determine whether it was incorrect (i.e. a nonconforming item). A total of 155 nonconforming items were found over 25 days.
(a) Calculate the center line and 3-sigma control limits for the p (or np) chart.
(b) Out-of-control signals are also indicated by "at least two out of three consecutive points beyond 2-sigma". Construct also 2-sigma limits.
(c) Suppose there were 10, 10, 13 nonconforming items on the next three consecutive days. Would these observations be unusual?
(d) A new operator had been employed but unknown to the manager her error rate was 0.08. What is the probability that this higher rate will be detected by her third day?
4. The owner of a dry cleaning business, in an effort to measure the quality of the services provided, would like to study the number of dry-cleaned items that are returned for rework per day. Records were kept for four weeks (Monday to Saturday) with the following results (read across from left to right):
4 6 3 7 6 8 6 4 8 6 5 12
5 8 3 4 10 9 6 5 8 6 7 9
(a) Define the random variable of interest and state the distribution you would expect this random variable to have.
(b) Set up the appropriate control chart(s) to monitor the random variable of interest.
(c) Suppose that the previous long run average for the number of items returned was 5 per day. Recalculate the control limits and evaluate the results in light of this information.
5. A u chart is to be used to control a corrugated paper product line. End product is produced in rolls of varying length 1 m wide. Nonconformities include surface imperfections, improper gluing, improper tension setting on the corrugated inner core, etc. The control statistic is nonconformities per 100 m with one roll constituting a sample. To determine the initial control limits, 20 rolls were inspected and the total count of nonconformities was 326 in a total of 9000 m.
(a) Find the value of ¯u in nonconformities per 100 m.
(b) The next roll inspected is of length 250 m and has 15 nonconformities. Is there reason to believe that this roll is significantly worse than the other rolls?