Lesson 14 Statistical Process Control Solutions Solved Problem #2: see textbook Solved Problem #4: see textbook Solved Problem #5: see textbook Solved Problem #6: see textbook (manual problem)

#1: Checkout time at a supermarket is monitored using a range and mean chart. Six samples which contain 20 observations per sample have been collected and the sample means and sample ranges have been computed as shown below. Sample 1 2 3 4 5 6 Mean 3.06 3.15 3.11 3.13 3.06 3.09 Range 0.42 0.50 0.41 0.46 0.46 0.45

Perform the manual calculations necessary to answer the following questions. a. What is the sample size? 20 b. Is the variability of the checkout time known or unknown? Unknown c. Which chart is used to analyze the checkout time variability? Range chart d. Which chart is used to analyze the checkout time? Mean chart e. Should the range or mean chart be analyzed first? Explain your answer. Range chart should be analyzed before the mean chart because the variability of the process must be in control before the mean chart can be analyzed. Analyze the range chart by answering the following questions. f. What is the grand range? .45 g. Calculate the centerline, upper and lower control limits for the range chart?

1

Centerline = grand range = R = .45 LCL = R * D3 = .45 * .415 = .18675 UCL = R * D4 = .45 * 1.585 = .71325

h. Plot the range chart showing the centerline, upper and lower control limits.

0.8 0.7 0.6 0.5 Sample Range 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 6 7 LCL Centerline UCL

i.

Are the sample ranges within the control limits? Yes

j.

Count the number of A/B and U/D runs for the range chart. A/B U/D BABAAB UDUDD 5 runs 4 runs

k.

What is the expected number of A/B runs?

E (r ) A / B =

l.

# Observations 6 +1 = +1 = 4 2 2

What is the variability of the expected number of A/B runs?

σ (r ) A / B =

#Observations − 1 = 4

6 −1 =1.118 4

m. What is the test statistic for the expected number of runs?

Z (r ) A / B =

n.

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