IE 361 EXAM #1 - Department of Industrial and Manufacturing … · 2014-12-19 · = 9.9 and 𝑠 2 = 0.56. ... Without doing any calculations or using any mathematical symbols, ...

IE 361 EXAM #1 FALL 2013

Show your work: Partial credit can only be given for incorrect answers if there is enough information to clearly see what you were trying to do. There are two additional blank pages on the back of the exam you can use, but please label your work so it is easy to follow.

I have neither given nor received unauthorized assistance on this exam.

________________________________________________________________________ __________________________ Name Signed Date ________________________________________________________________________ Name Printed

True/False Questions _______ / 20

Multiple Choice Questions _______ / 20

Multi-part Questions _______ / 60

TOTAL

_______ / 100

Below are 10 true/false questions. Each is worth 2 point each. Clearly write either “T” or “F” in front of each.

_________ 1. If a process is stable or consistent, it is producing high quality goods.

_________ 2. The first step in a good quality assurance project is to formulate appropriate measures of process performance and develop corresponding measurement systems.

_________ 3. A speedometer that always reads 31 mph is not a precise instrument.

_________ 4. You cannot calculate 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 with a data set obtained from one operator measuring a single part many times.

_________ 5. A standard (an item corresponding to a known measurand) is needed to determine σdevice.

_________ 6. Calibration of a device is designed to improve the accuracy of your measurement system.

_________ 7. A measurand device displays a constancy of bias if the average measurement equals two times the measurand.

_________ 8. You will get a different value of 𝜎𝑅&𝑅 in a go/no-go Gauge R&R study depending on if you code a non-conforming part as “0” or “1”.

_________ 9. Reducing a company’s number of vendors (i.e. suppliers of raw materials) should help reduce the variation in the associated products the company makes.

_________ 10. A positive correlation between the numbers associated with two types of problems suggests that a single root cause may be responsible for both types of problem.

Below are 4 multiple choice questions. Each is worth 5 point each. Circle the letter corresponding to the single best response for each of these questions. If extra space is needed, use the extra pages provided (clearly label the work with the problem number).

11. Which of the following is considered a hallmark of good QA practices?

a) Data collected from a single period is sufficient to give a snapshot of process performance. b) Utilize historical data to identify the problem and guide process change suggestions when starting a

new QA project c) Have an unbiased third-party collect data rather than the workers doing the work or running the

equipment. d) A large amount of data is necessarily more informative than a moderate amount. e) none of the above

12. You measured the thickness of some washers produced by a stable production process. The thickness of 12 different washers was measured and the resulting values had a sample mean of 0.171 inches with a standard deviation of .024 inches. The thickness of the 13th washer was measured 5 times and the resulting values had a mean of 0.138 inches with and standard deviation of .0096 inches.

Given this data, what can you calculate?

a) 95% two-sided confidence intervals for the sum of the washer thickness and your measurement bias. b) 95% two-sided confidence limits for a true mean washer thickness for this process c) 95% two-sided confidence intervals of the repeatability standard deviation. d) Two of the above e) All of the above

13. A Gauge R&R Study found that the 95% two-sided confidence limits of 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 were 0.4 to 0.7 and the 95% two-sided confidence limits of 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 were 3.1 to 3.9. Based on these results, which of the following statements is true?

a) The first place to look for substantial variation reduction is procedure standardization b) Train each operator to use the system more consistently, even if each operator does it differently. c) Differences between operator biases are a relatively minor part of the measurement variation. d) Exactly 2 of the responses (a) through (c) are correct e) None of the above responses are correct.

14. Below is a JMP report from a “one-way random effects” analysis of a study where an operator used the same gauge to measure the weights of 5 sacks of coffee beans from a sack-filling process. The operator measured each sack of beans 4 times apiece. Units of the data were in pounds.

Given the data above, the 95% confidence limits for the standard deviation of the sack-filling process are

a) 0 to 2.74 pounds b) 0 to 7.49 pounds c) 1.78 to 3.74 pounds d) 3.18 to 13.97 pounds e) not determinable from this data

Below are 4 multi-part questions. Each part is worth 5 points. If extra space is needed, use the extra pages provided (clearly label the work with the problem number).

15. One student collects data as show below. First the student collects n=5 measurement on a single measurand. Then the student collects a single measurement on each of m=7 measurands.

a) In terms of 𝜎𝑅&𝑅2 , 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 , 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 , and 𝜎𝑥2, write the variance of the distribution resulting

from measuring item x1 multiple (n) times.

b) In terms of 𝜎𝑅&𝑅2 , 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 , 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 , and 𝜎𝑥2, write the variance of the distribution resulting

from measuring items x2 through xm+1 once time each.

c) You are given the following data: 𝑦1�� = 10.4 and 𝑠1� = 0.16. Furthermore, 𝑦2�� = 9.9 and 𝑠2� = 0.56.

Estimate the item-to-item standard deviation.

d) Given the data from the above problem, estimate the 95% two-sided confidence interval of the standard deviation. Compute: • Degrees of freedom

• Confidence interval

16. A production process is stable, properly aimed, and capable with respect to the engineering specifications. You have developed a measurement system and run a Gauge R&R study. The measurement system is both precise enough and accurate (after calibration) enough to check for conformance to engineering specifications.

a) On the scale below, draw the distribution of 99% of the samples of the measurement of the mean.

b) Explain both the location and the width of the distribution. You should not need to do any calculations, but clearly describe your logic in two sentences or less.

17. A different Gauge R&R study was conducted on the weight (in grams) of a granola bar. The study produced the ANOVA table below. The same measurement device was used throughout, where each of I=4 items was measured m=3 times by each of J=3 different operators.

a) Calculate the single number estimate of the standard deviation that would be experienced by a single operator repeatedly re-measuring the same granola bar.

b) Calculate the 95% two-sided confidence limits for the standard deviation that would be experienced by a single operator repeatedly re-measuring the same granola bar.

c) If 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 = 4.179, calculate the single number estimate of the standard deviation that would be experience by many operators measuring the same granola bar once each.

18. A go/no-go Gauge R&R study has 3 inspectors evaluating 2 parts, 10 times each. The fraction of items each inspector declared “non-conforming” is given in the table below.

Part 1 Part 2 Operator 1 .2 0 Operator 2 .3 .2 Operator 3 .15 .2

a) Based on the data calculate terms of 𝜎𝑅&𝑅2 .

b) Based on the data calculate 𝜎𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟2 .

c) Without doing any calculations or using any mathematical symbols, what does the go/no-go data look like for situations where there is no repeatability variation?


Show your work: Partial credit can only be given for incorrect answers if there is enough information to clearly see what you were trying to do. There are two additional blank pages on the back of the exam you can use, but please label your work so it is easy to follow.



Section I. True/False Questions _______ / 20

Section II. Multiple Choice Questions _______ / 20

Section III. Multi-part Questions _______ / 60

TOTAL

_______ / 100

SECTION I. TRUE/FALSE QUESTIONS Below are 10 true/false questions. Each is worth 2 point each. Clearly write either “T” or “F” in front of each.

_________ 1. Based on the same data, tolerance confidence limits are bigger than the prediction confidence limits

_________ 2. A process that is unstable and violates control chart limits on charts will not produce parts within engineering specifications.

_________ 3. When an operator tweaks an otherwise stable process, the process variation will likely increase.

_________ 4. If Cp = Cpk, then the process will be within the upper and lower specifications.

_________ 5. The same variance component is used in the calculation of Gauge Capability Ratio and the Process Capability Ratio.

_________ 6. Common cause variation is stable and predictable variation.

_________ 7. For a standards-given X-chart for the number of nonconforming items in a sample of n items, the centerline varies with the sample size.

_________ 8. The control limits for a retrospective R chart does not depend on the samples size (i.e. number of observation taken during one data collection period).

_________ 9. Engineering feedback control is one method for keeping a process stable.

_________ 10. Confirmation of a normal distribution shape is required before making any form of statistical prediction limits.

SECTION II. MULTIPLE CHOICE Answer four (4) of the five (5) questions in this section.

Circle the letter corresponding to the single best response for each of these questions. If extra space is needed, use the extra pages provided (clearly label the work with the problem number).

11. Which of the following statements regarding control charting is most universally true:

a) Control charting tells one when intervention in the process is not justified b) Control charting tells one when the process output is within engineering specifications c) Control charting tells one when something in the process has changed d) Exactly two of the responses (a) through (c) are true e) All of the responses are true.

12. Given a �̅� charts for a normally distributed process with standards are μ = 75 and σ = 10, which of the following statements is most universally true:

a) The control limits for charts of n = 5 or n = 10 of Shewhart �̅� chart would be the same. b) The ARL for charts of n = 5 or n = 10 would be the same if process parameter remain at standard

values. c) The ARL for charts of n = 5 or n = 10 would change if the process mean changes from the standard

value. d) Exactly two of the responses (a) through (c) are true e) All of the responses are true.

13. Given that you can only take one sample per day from a low-yield process, which of the following statements is most universally true:

a) It is sensible to plot individual sample on an x-chart. b) A moving range will be affected only by common cause variation, c) The moving range is typically an under-estimate of the process standard deviation. d) Exactly two of the responses (a) through (c) are true e) All of the responses are true.

14. The figure below is a normal quantile plot (where the y-axis is normal probability instead of normal

quantiles). Which statement is most universally true?

a) The plot provides clear evidence that this data comes from a normal distribution b) The plot indicates that the process capability ratio will be a meaningful measure of process

capability. c) The average value of the data is approximately 2.3. d) Exactly two of the responses (a) through (c) are true e) All of the responses are true.

15. The lengths of tacks are measured by a gauge. A particular type of tack has specifications of 3.2 ± 0.2 cm. You can assume that the tacks are produced by a physically stable process, but are not normally distributed. If you have data on n=10 tack lengths, how confident are you that the 11th tack produced by this process would measure at least 3.0 cm?

a) Approximately 81% b) Approximately 91% c) Approximately 95% d) Approximately 99% e) There is not enough information to answer this question.

SECTION III. MULTI-PART QUESTIONS Below are 3 multi-part questions. Each part is worth 5 points. If extra space is needed, use the extra pages provided (clearly label the work with the problem number).

16. Jars of peanut butter are produced with a fill specification of 16 ± 0.8 ounces.

(a) If the 95% two-sided confidence interval for the process capability “6σ” is (0.264, 0.814), calculate the 95% two-sided confidence limits for the process capability ratio Cp.

(b) Is the process capable? Explain.

(c) The standards for this fill process are given as �̅� = 15.8 and s = 0.067. For a data set based on n = 8 samples, calculate the two sided 95% confidence limits for the process capability index that measures current process performance, Cpk .

d) What does your confidence limits for Cpk (calculated in part (c) above) tell you about whether you have a “six sigma” process?

17. Below are the means and ranges of samples of size n = 4 flight times (in units of seconds) of paper

airplanes launched from the third floor balcony in the Atrium of Howe Hall. The engineering specifications for this paper airplane design are 9.9 ± 4.1.

Sample 1 2 3 4 5 6 7 8 9 10 Sum �̅� 11.4 12.5 10.3 12.2 11.3 6.2 7.7 8.4 9.6 10.4 100.0 R 4 2 3 5 4 4 4 4 5 6 41

(a) Based on a computation of the appropriate control limits for �̅�, is there evidence of process instability? Show your calculations, and explain which samples (if any) are evidence of process instability and why (using all the Western Electric Alarm Rules, which can be found on the last page of the exam).

(b) If we assume that the short-term process variation is constant across time, is the process capable of meeting engineering requirements (if properly aimed and monitored)?

(c) Based on an estimate of process standard deviation, give the 90% predication limits for the next flight time sample.

(d) Make two-sided normal distribution 95% tolerance limits for 90% of all flight times produced by the paper airplanes.

18. This question provides two ways to judge if the process is stable: number of non-conformities and

number nonconforming.

• Over 10 shifts of production, a total of 167 non-conformities were identified in the 3270 wine glasses that were made and inspected.

• Furthermore, of these 3270 wine glasses, 81 of the glasses were judged to be nonconforming. • The process was analyzed retrospectively and found to be stable, and thus these historical rates can

be considered standard values.

In the subsequent shift 11, 300 glasses were made and inspected. A total of 19 non-conformities were identified, and 8 of the glasses were judged to be nonconforming

(a) What is the expected number of non-conformities in shift 11, given the standards established during the first 10 shifts?

(b) Does the count of non-conformities in Shift 11 provide clear evidence of a process change? Using on your answer in (a), calculate the appropriate control limits to justify your answer.

(c) What is the expected number of nonconforming glasses in shift 11, given the standards established

during the first 10 shifts?

(d) Does the count of nonconforming in Shift 11 provide clear evidence of a process change? Using on your answer in (c), calculate the appropriate control limits to justify your answer.

INFORMATION SHEET

WESTERN ELECTRIC RULES • A single data point outside the 3 sigma control limit • 2 out of 3 consecutive points outside the 2 sigma control limit on one side of center • 4 out of 5 consecutive points outside the 1 sigma control limit on one side of center • 8 consecutive points on one side of center

THE Z-DISTRIBUTION TABLE

#


Show your work: Partial credit can only be given for incorrect answers if there is enough information to

clearly see what you were trying to do. There are two additional blank pages on the back of the exam you can

use, but please label your work so it is easy to follow.



Section I. True/False Questions _______ / 20

Section II. Multiple Choice Questions _______ / 30

Section III. Multi-part Questions _______ / 50

TOTAL

_______ / 100

2

SECTION I. TRUE/FALSE QUESTIONS

Below are 10 true/false questions. Each is worth 2 point each. Clearly write either “T” or “F”

in front of each.

_________ 1. If a calibration study produced a regression analysis with an estimate for the 95%

confidence limits for β1 as (0.12 ± 0.15), the bias for this measurement device is constant.

_________ 2. A prediction interval is likely to contain a single additional observation drawn from a stable process.

_________ 3. In a go/no-go experiment, if all the calls for a single operator on a single part are the same, then there is no repeatability variation.

_________ 4. A stable process will never produce a plotted value outside the Shewhart control limits.

_________ 5. If a fitted interaction effect ab22 = 20, then ab11 = -20.

_________ 6. The one-way normal model assumes that while means may vary, the standard deviation does not change between different process conditions.

_________ 7. A 27-4 fractional factorial experiment will only run 1/8 of the combinations of a full factorial experiment.

_________ 8. In a 22-1 fractional factorial experiment, if one tested combination a, then the only sensible second combination to test is combination b.

_________ 9. One can always assume that higher order interaction effects are less important than the main effects of the factors.

_________ 10. Gauge R&R studies are used to assess the accuracy and precision of the gauge.

3

SECTION II. MULTIPLE CHOICE

Answer six (6) of the eight (8) questions in this section (if you answer more than 6 questions,

your top-scoring 6 questions will count towards your score).

Circle the letter corresponding to the single best response for each of these questions. If

extra space is needed, use the extra pages provided (clearly label the work with the problem

number).

11. In the data collection scheme below, which of the following variances can you estimate?

a) 𝜎𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑏𝑖𝑙𝑖𝑡𝑦2

b) 𝜎𝑟𝑒𝑝𝑒𝑎𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦2 and 𝜎𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑏𝑖𝑙𝑖𝑡𝑦

2

c) 𝜎𝑥2 and 𝜎𝑟𝑒𝑝𝑒𝑎𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦

2

d) 𝜎𝑥2 and 𝜎𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑏𝑖𝑙𝑖𝑡𝑦

2

e) 𝜎𝑥2, 𝜎𝑟𝑒𝑝𝑒𝑎𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦

2 , and 𝜎𝑟𝑒𝑝𝑟𝑜𝑑𝑢𝑐𝑖𝑏𝑖𝑙𝑖𝑡𝑦2

12. The potential of an assistive system to boost performance is tested experimentally. In the experiment the

system was either on or off, and the user was in either low or high workload situations. Given the plot

below, which of the following conclusions are supported by the data?

a) When the system is on, performance is detectably improved in low workload over high workload

b) There is no detectable difference (system on vs. off) in performance at low workload

c) Use of the system only improves performance at high workload

d) Exactly two of the responses (a) through (c) are true

e) All of the responses are true.

dorneich

Text Box

see next page for figure

4

13. A 24 full factorial experiment was conducted (with factors A, B, C, D). It was determined from the data

that the set of b2 = -7, cd22 = 4, d2 = 2 represent all detectable fitted effects (where the other effects have

been determined to be not detectable). Which statement is most true?

a) Factor A has no effect on the response y, either alone or in combination with other effects

b) Factor B has no effect on the response y, either alone or in combination with other effects

c) Factor C has an effect on the response y that is independent of the effect of other factors

d) Factor D has an effect on the response y that is independent of the effect of other factors

e) All of the above

14. For the data presented in Question 13, if the goal is to find a combination of levels of the four factors that

makes the mean response y as small as possible, what levels of the factors should be used?

a) A low, B low, C low, D low

b) A low, B low, C high, D low

c) A high, B high, C low, D low

d) A high, B high, C high, D low

e) None of the above

15. For the data presented in Question 13, if �̅�…. = 20 , what is �̅�(1)?

a) 7

b) 19

c) 21

d) 29


16. A 26-2 fractional factorial study was conducted. Two generators were used to choose levels of Factors D

and E are shown below. What is the defining relation for this study?

E ↔ ABD and F ↔ ABC

a) I ↔ ABCDE

b) I ↔ ABD ↔ ABC

c) I ↔ ABDE ↔ ABCF

d) I ↔ ABDE ↔ ABCF ↔ CDEF


5

17. Given the generators in Question 16, when the Factors A and B are at their low levels, and Factors C and

D are at their high levels, what levels are the other two factors?

a) E low and F low

b) E low and F high

c) E high and F low

d) E high and F high

e) Cannot be determined without more information

18. Given the generators in Question 16, which statement about the experimental results is most true?

a) The main effect E is confounded only with the ABC three-factor interaction

b) The main effect E is not confounded with any 2-factor interaction.

c) The main effect E is confounded with only 2 other factorial effects.

d) Exactly two of the responses (a) through (c) are true

e) None of the above.

6

SECTION III. MULTI-PART QUESTIONS

Below are 2 multi-part questions. Each part is worth the points indicated. If extra space is

needed, use the extra pages provided (clearly label the work with the problem number).

19. A ball of radius R is placed into a rectangular box of dimensions a x b x c. The quantity

𝑉 = 𝑎𝑏𝑐 −4

3𝜋𝑅3

represents the volume of air left inside the box. Suppose that the measured variables a, b, c, and R are in fact

random with means and standard deviations given in the table below.

Input a b c R 𝜇𝐼𝑛𝑝𝑢𝑡 10 cm 25 cm 50 cm 3 cm

𝜎𝐼𝑛𝑝𝑢𝑡 .1 cm .1 cm .1 cm .2 cm

a) [10 pts] Calculate the mean and standard deviation with the experimentally determined volume V.

b) [5 pts] Which measured variable above is the largest contributor of uncertainty to the Volume V?

Explain.

7

20. A 23 factorial experiment is conducted. The table below shows the data collected.

setup Factor A Factor B Factor C y �̅� s

1 - - - 14, 14, 15 14.333 0.577

2 + - - 16 16

3 - + - 7 7

4 + + - 12 12

5 - - + 13, 16, 12 13.667 2.082

6 + - + 15, 15, 13 14.333 1.155

7 - + + 8, 9, 8 8.333 0.577

8 + + + 15 15

a) [5 pts] Compute a single estimate of the standard deviation of the response y for any of the

experimental setups. What are the appropriate degrees of freedom for your estimate?

Spooled: ______________________________ ν: ______________________

b) [5 pts] Give two-sided 95% confidence limits for the standard deviation of the response y for any fixed

experimental setup.

c) [5 pts] Calculate the margin of error based on the two-sided 95% confidence limits for the mean response

time of the "all low" combination.

8

d) [10 pts] The quantity 𝐿 =1

4(𝜇8 − 𝜇4) +

1

4(𝜇7 − 𝜇3) +

1

4(𝜇6 − 𝜇2) +

1

4(𝜇5 − 𝜇1) is a measure of the

effect on the response y due to change in Factor C from low to high. Make a 95% two-sided

confidence interval for this quantity. Does Factor C impact the response in a statistically detectable

way? Explain.

e) [5 pts] Fill in missing values in the Yates algorithm table below to compute the fitted factorial effects

corresponding to the “all high” treatment combinations.

Factor A Factor B Factor C Combination �̅� cycle #1 cycle #2 cycle #3 cycle#38

- - - (1) 14.333 30.333 49.333

+ - - a 16.000 19.000 51.333

- + - b 7.000 28.000 6.667

+ + - ab 12.000 23.333 7.333

- - + c 13.667 1.667 -11.333

+ - + ac 14.333 5.000 -4.667

- + + bc 8.333 0.667 3.333

+ + + abc 15.000 6.667 6.000

f) [5 pts] Calculate the “margin of error” for the factorial effects calculated in the Yates algorithm above.

Which factors are "clearly larger than the background variation"?

IE 361 EXAM #1 - Department of Industrial and Manufacturing … · 2014-12-19 · = 9.9 and 𝑠 2 = 0.56. ... Without doing any calculations or using any mathematical symbols, ...

Documents