You're the operations manager at a regional hospital and your management has set the goal that no one should wait more than | ||||||||||||||||
3 minutes to see a nurse in the Emergency Room. You've collected 40 samples of the average wait times at five different times during the day over 40 days as shown below. | ||||||||||||||||
a. construct an x-bar chart for this data. Is the process in control? | ||||||||||||||||
b. using 3 minutes as the upper tolerance limit and zero as the lower tolerance limit, calculate the process capability statistics ( Cp, Cpk, Cpu and Cpl). Is your process | ||||||||||||||||
capable of achieving management's goal, based on the data? | ||||||||||||||||
c. What percentage of ER visits would you expect to exceed the Upper Tolerance Limit? | ||||||||||||||||
Day | ||||||||||||||||
1 | 1.22 | 1.54 | 1.53 | 1.86 | 1.49 | |||||||||||
2 | 1.48 | 1.18 | 1.41 | 1.29 | 1.61 | |||||||||||
3 | 2.12 | 1.76 | 1.29 | 1.78 | 1.74 | |||||||||||
4 | 1.34 | 1.22 | 1.29 | 1.69 | 1.42 | |||||||||||
5 | 1.11 | 1.21 | 3.60 | 1.95 | 1.22 | |||||||||||
6 | 0.73 | 1.97 | 1.51 | 1.67 | 1.77 | |||||||||||
7 | 1.20 | 1.46 | 1.05 | 3.55 | 1.80 | |||||||||||
8 | 1.72 | 1.58 | 1.79 | 1.95 | 0.83 | |||||||||||
9 | 1.23 | 3.10 | 1.57 | 1.49 | 1.58 | |||||||||||
10 | 0.70 | 0.94 | 1.14 | 1.54 | 1.81 | |||||||||||
11 | 1.50 | 1.83 | 1.60 | 1.15 | 1.79 | |||||||||||
12 | 1.72 | 1.61 | 1.63 | 1.84 | 1.95 | |||||||||||
13 | 1.64 | 1.13 | 1.60 | 1.87 | 1.36 | |||||||||||
14 | 0.73 | 1.39 | 1.39 | 1.85 | 1.86 | |||||||||||
15 | 1.72 | 1.42 | 1.59 | 0.70 | 1.55 | |||||||||||
16 | 1.91 | 2.08 | 1.64 | 2.05 | 1.60 | |||||||||||
17 | 1.63 | 1.57 | 0.95 | 2.02 | 1.69 | |||||||||||
18 | 1.53 | 1.47 | 2.05 | 1.19 | 1.52 | |||||||||||
19 | 1.18 | 1.78 | 3.20 | 1.53 | 1.30 | |||||||||||
20 | 1.74 | 2.14 | 1.24 | 0.92 | 1.34 | |||||||||||
21 | 1.47 | 1.89 | 1.53 | 2.28 | 1.84 | |||||||||||
22 | 1.68 | 1.35 | 1.26 | 3.20 | 1.63 | |||||||||||
23 | 0.99 | 1.57 | 1.45 | 1.50 | 1.98 | |||||||||||
24 | 1.92 | 1.01 | 0.93 | 1.68 | 1.96 | |||||||||||
25 | 3.13 | 1.57 | 1.75 | 1.72 | 1.63 | |||||||||||
26 | 1.13 | 0.99 | 1.27 | 1.35 | 1.37 | |||||||||||
27 | 1.87 | 1.74 | 0.89 | 1.61 | 1.77 | |||||||||||
28 | 0.99 | 1.36 | 0.89 | 1.54 | 3.30 | |||||||||||
29 | 1.75 | 1.96 | 1.57 | 1.67 | 2.31 | |||||||||||
30 | 1.59 | 2.15 | 3.10 | 1.42 | 1.50 | |||||||||||
31 | 0.93 | 1.65 | 1.29 | 1.02 | 1.48 | |||||||||||
32 | 1.40 | 1.98 | 1.54 | 0.97 | 1.62 | |||||||||||
33 | 1.69 | 1.62 | 1.47 | 1.81 | 0.97 | |||||||||||
34 | 1.98 | 1.26 | 1.32 | 1.17 | 1.39 | |||||||||||
35 | 3.25 | 1.42 | 2.06 | 1.27 | 1.34 | |||||||||||
36 | 1.45 | 1.57 | 1.70 | 1.32 | 1.26 | |||||||||||
37 | 1.98 | 1.61 | 1.45 | 1.46 | 2.19 | |||||||||||
38 | 1.46 | 1.46 | 1.70 | 1.56 | 1.93 | |||||||||||
39 | 1.80 | 1.34 | 1.46 | 3.40 | 1.10 | |||||||||||
40 | 1.04 | 1.29 | 1.30 | 1.77 | 1.13 |
(a) Xbar and R chart are constructed as under:
Calculate sample Average (Xbar) and Range (R = difference of MAX and MIN value) for each day.
Calculate the Average of Xbars and R vaues of 40 days to get Xbarbar and Rbar
(b) Process standard deviation, = Rbar/d2 , where d2 is control chart constant = 2.326
= 1.12/2.326 = 0.4817
Cp = (USL-LSL)/6 = (3-0)/(6*0.4817) = 1.038
Cpu = (USL - )/3 = (3 - 1.6)/(3*0.4817) = 0.9688
Cpl = ( - LSL)/3 = (1.6 - 0)/(3*0.4817) = 1.1072
Cpk = MIN(Cpu, Cpl) = MIN(0.9608, 1.1072) = 0.9688
Cp and Cpk are less than 1.33. Therefore, the process is not capable of meeting specifications on 3 sigma quality.
c) z = (3-1.6)/0.4817 = 2.9064
P(z) = NORMSDIST(2.9064) = 0.9982
Percentage of ER visits expected to exceed the tolerance limit = 1 - 0.9982 = 0.0018 or 0.18 %
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