Homework 3-Z1863163 - Assignments PDF

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ISYE 630 Advanced Quality Control Homework #3 Autocorrelated Process Data Deadline: 11/19, 6pm The data shown in a table are concentration readings from a chemical process, made every 30 minutes. HW#3Data (Excel) in Blackboard. Please show the procedures (Excel, Minitab).

a) Calculate the sample autocorrelation function with k=1, 2, and provide an interpretation (Use both Excel and Minitab). For given data: 38.8 4 Variance= 29.2 8 ρ (k=1) 0.75 Covariance, k=1 25.1 9 ρ (k=2) 0.65 Covariance, k=2 MINITAB RESULTS:

EXCEL RESULTS:

Scatter Chart (k=1) 220 215 210 205 200 195 190 185 180 175 185

f(x) = 0.75 x + 50.17

190

195

200

205

210

215

220

Scatter Chart (k=2) 220 215 210 205 200 195 190 185 180 175 185

f(x) = 0.64 x + 72.32

190

195

200

205

210

215

220

Interpretation: The autocorrelation function for k=1 is 0.75 and for k=2 it is 0.65. We can interpret this to mean that there exists a significant relationship between the intervals. From the minitab plot of the autocorrelation function, we can conclude that k=1,2,3,4 are out of control and also subsequent k values for time series intervals are in control. b) Construct an individuals control chart (I chart) with the standard deviation estimated using the moving range method (Use Minitab). Provide an interpretation of this control chart.

Standard deviation=

´ 3.64 MR = =3.226 d 2 1.128

As we can observe from the Minitab results, the process is out of control as it has point outside the control limits. In order to correct this, less frequent sampling can be performed which results in no outof-control points. As we observed in autocorrelation function plot that there are out-of-control instances in the time series.

c) Fit a first-order autoregressive model to the data (time lag k =1). Set up an individuals control chart on the residuals from this model (use both Excel and Minitab). Interpret this control chart. EXCEL RESULTS:

I chart for residuals 20.00 15.00 10.00 5.00 0.00 1

-5.00

5

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

-10.00 -15.00 -20.00 Residuals

UCL-I chart

LCL-I chart

MR chart for residuals 18 16 14 12 10 8 6 4 2 0

1

5

9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

MR

UCL-MR chart

LCL- MR chart

MINITAB RESULTS:

Interpretation: We can observe from the control charts that the residuals obtained from regression model for the actual observations show that the process is in control as there are no out-of-control points. d) Are the residuals from the model in part (c) uncorrelated? Calculate the sample autocorrelation function (k=1,2) and provide an interpretation (use Minitab). For residuals: 17.23 Variance= Covariance, k=1 Covariance, k=2

-2.29 1.78 MINITAB RESULTS:

ρ (k=1) ρ (k=2)

-0.13 0.10

EXCEL RESULTS:

Scatter chart(Residuals,k=1) 10.00 5.00 -15.00

-10.00

-5.00

0.00 0.00 -5.00

5.00

10.00

-10.00 -15.00 Scatter chart(residuals,k=1)

Scatter chart(Residuals,k=2) 10.00 5.00

-15.00

-10.00

-5.00

0.00 0.00

5.00

10.00

-5.00 -10.00 -15.00 Scatter chart(Residuals,k=2)

Interpretation: As no points are out-of-control, there is no significant autocorrelation between the time series intervals....