May 4 2010
IE330 Spring 2010
Final Exam
Part 1: Mandatory (Chapters 15 and 16)
No calculators, closed book, closed notes.
Do not tear off any pages.
1
1. CHAPTER 15 - True/False questions (3 points each, 15 points total)
a. (TRUE or FALSE) In a sign test, the effect size is computed by dividing
the difference in means by the standard error of the estimator.
b. (TRUE or FALSE) For a Wilcoxon rank-sum test, tied values are
discarded.
c.
(TRUE or FALSE) Comparisons between levels of the factor cannot be
made when using a Kruskal-Wallis test.
d. (TRUE or FALSE) If the data is symmetric but not normally distributed,
the Wilcoxon signed-rank test is preferred over the sign test.
e.
(TRUE or FALSE) Non-parametric tests should not be used when there
are many outliers in the data that cannot be removed.
2. CHAPTER 15 - Multiple choice questions (3 points each, 15 points total)
a. Which of the following is useful for testing the stability of an alloy
deformed by four different methods if the residuals are not normally
distributed?
i. Z-test
ii. Sign test
iii. Kruskal-Wallis test
iv. Wilcoxon signed-rank test
v. None of the above
b. You have collected data on customer transaction times using one of two
ATM display layouts. For each customer, you have a time when using
display 1 and display 2, and you are interested in the effect of layout on
the time it takes the customer to complete their transaction. The
populations are not normally distributed. Which of the following tests
could you use?
i. Paired test
ii. Sign test
iii. Wilcoxon rank-sum test
iv. Wilcoxon signed-rank test
v. None of the above
c. You are testing whether the mean of a population is equal to zero or not,
but the data is not normally distributed. (The population is continuous.)
Which of the following tests could you use for this purpose?
i. 22 factorial test
ii. Simple linear regression
iii. Kruskal-Wallis test
iv. Wilcoxon rank-sum test
v. None of the above
2
d. In running a simple linear regression, you find that the residuals have nonconstant variance, which cannot be stabilized using transformations.
Which of the following non-parametric methods can be used instead?
i. Wilcoxon signed-rank test
ii. Wilcoxon rank-sum test
iii. Sign test
iv. Kruskal-Wallis test
v. None of the above
e. Which of the following are true concerning the Wilcoxon signed-rank test
(choose all that are true)?
i. The test compares the data against a hypothesized median (if its
mean then its correct)
ii. The data should be free of outliers
iii. The counts of values above and below the median are compared.
iv. The data should come from a uniform distribution
v. None of the above
3. CHAPTER 16 - True/False questions (3 points each, 15 points total)
a. (TRUE or FALSE) During the baseline process, it is recommended that
25-50 subgroups are used.
b. (TRUE or FALSE) While monitoring a process using statistical process
control, the process must be free of special causes of variation for the
analysis to be valid.
c.
(TRUE or FALSE) It is recommended that subgroup sizes be increased
during the monitoring process.
d. (TRUE or FALSE) A centered process is one where the process mean is
centered between the upper and lower control limits.
e. (TRUE or FALSE) If the process is free of special causes of variation, one
should never see an out-of-control point.
4. CHAPTER 16 - Multiple choice questions (3 points each, 15 points total)
a. We are tracking the specific gravity of the root beer we are producing, as
the correct mixture has a particular specific gravity. Which of the
following is the best control chart to use? (Pick one only.)
i. x-bar and r chart
ii. P-chart
iii. NP-chart
iv. U-chart
v. C-chart
vi. None of the above can be used
3
b. Which of the following are true regarding specifications (choose all that
are true)?
i. Specifications are used in computing the control limits. (Wrong!)
ii. Specifications are used in computing the process capability ratio.
iii. Specifications are engineering tolerances established by the
customer.
iv. Specifications are established by computing tolerance intervals.
v. None of the above is true.
c. We are tracking the number of defective welds on a part produced by an
automated welding process. Which of the following attribute control
charts should be used for statistical process control? (Choose all that can
be used effectively.)
i. s chart
ii. P-chart (Wrong!) P-chart is tracking proportion!
iii. NP-chart
iv. U-chart
v. C-chart
vi. None of the above can be used
d. During the monitoring phase of statistical process control, which of the
following are true? (Select all that are true.)
i. Whenever special causes of variation are found, the control limits
should be re-computed.
ii. 25-50 subgroups should be used.
iii. During the monitoring phase, we are trying to identify when
special causes of variation have been introduced into the process.
iv. Specification limits should be plotted along with the control limits.
v. The control limits should always be symmetric with respect to the
process mean (3).
vi. None of the above are true
e. During the monitoring phase, we find a point outside our control limit on
the s-chart. Which of the following might be an explanation? (Select all
that could be true.)
i. The mean of the process has shifted.
ii. The standard deviation of the process has been reduced.
iii. A special cause of variation has been introduced into the process.
iv. It is due to normal random variation.
v. None of the above could be true.
4
5. CHAPTER 15 Problem (20 points)1
The nickel content, in parts per thousand by weight, is measured for six welds.
Due to two outliers, a t-test is not appropriate. It is desired to test
H 0 : = 12 vs. H 0 : 12 . The data are as follows:
Nickel content (ppt): 11.5
13.9
9.3
9.0
21.7
0.9
SummaryforNickelContent( ppt)
AndersonDarlingNormality Test
A Squared
PV alue
Mean
StDev
V ariance
Skew ness
Kurtosis
N
0
5
10
15
0.26
0.574
11.050
6.812
46.407
0.16278
1.41153
6
Minimum
1stQ uartile
Median
3rdQ uartile
Maximum
20
0.900
6.975
10.400
15.850
21.700
95% ConfidenceI nterv alforMean
3.901
18.199
95% ConfidenceI nterv alforMedian
3.793
18.914
95% ConfidenceI nterv alforStDev
9 5 % Conf ide nce I nt e r v a ls
4.252
16.708
Mean
Median
5.0
7.5
10.0
12.5
15.0
17.5
20.0
A Wilcoxon Signed-Rank test is performed, with the following results:
Wilcoxon Signed Rank Test: Nickel Content (ppt)
Test of median = 12.00 versus median not = 12.00
Nickel Content (ppt)
N
6
N for
Test
6
Wilcoxon
Statistic
7.0
1
P
0.529
Estimated
Median
11.30
Taken from Navidi, W. (2008). Statistics for Engineers and Scientists, 2nd Edition, New York: McGrawHill, pp. 444-445.
5
a. State the assumptions for using the Wilcoxon Signed-Rank test and the
evidence you have for those assumptions being met.
b. Why was the sign test not used?
c. Based on the results, draw a conclusion about the hypothesis? (Explain
your answer)
d. In this case N for test is equal to N. Can you think of a case where
these will not be equal?
6
6. CHAPTER 16 Problem (20 points)
You are tracking the wait time for patients who are getting a flu shot at your
clinic. During a baseline phase, you found the mean wait time to be 7 minutes
and the standard deviation to be 4 minutes. Assume C4 = 1.
XbarRChartofWaitTime( min)
+ 3SL= 13
Sa m p l e M e a n
12
+ 2SL= 11
9
+ 1SL= 9
_
_
X= 7
6
1SL= 5
3
0
2SL= 3
3SL= 1
1
2
3
4
5
6
7
8
Sa m ple
9
10
11
12
13
14
15
Sa m p l e R a n g e
20
+ 3SL= 18.79
15
+ 2SL= 15.27
10
+ 1SL= 11.75
_
R= 8.24
5
1SL= 4.72
0
2SL= 1.20
3SL= 0
1
2
3
4
5
6
7
8
Sa m ple
9
10
11
12
13
14
15
a. Is the process in control? Why or why not? (Use the Western Electric
rules shown on the last page, and circle any out of control points on the
chart.)
b. What is the subgroup size?
7
c. If our goal is to have people wait no more than 10 minutes, what is the
process capability ratio of this process? (Approximate, there is no need to
be precise.)
d. How many people out of 100 would have to wait more than 10 minutes?
8
4 May 2010
IE330 Spring 2010
Final Exam
Part 2: Optional (cumulative Chapters 8 - 14)
No calculators, closed book, closed notes.
Do not tear off any pages.
9
1. Chapter 8 (2 points each, 6 points total)
a. Which of the following is not a measure of location?
i. Quartile
ii. Mode
iii. Inter-quartile range
iv. Median
b.
(TRUE or FALSE) In choosing n observations from N ( , ) , a larger
would result in a wider confidence interval on population mean if all else
is equal.
c. Suppose I will choose a number at random from a standard normal
distribution. (Call that number x.) Which of the following is the
probability that the number chosen is greater than a number z, where z =
x2
2
0? (The standard normal distribution is one whose PDF is F ( x ) = e
.)
2
x2
z
P ( x > z ) = 0.5
i.
e2
ii.
this is correct
P( x > z) =
2
if the limit is from z to infinity
x2
iii.
2
ze
z2
iv.
P( x > z) =
e2
z 2
P( x > z) =
2
2. Chapter 9 (2 points each, 6 points total)
a. Suppose you are evaluating which of two processes results in the greatest
productivity. The populations of the productivity from the two processes
are N ( 1 , 1 ) and N ( 2 , 2 ) where 1 2 , and you know the value of
both standard deviations. Which of the following could be the appropriate
test statistic? (Choose all that are possibly correct given the information
above.)
i.
iv.
X X 2 0
X X 2 0
T0 = 1
Z0 = 1
2
2
11
1 2
Sp
+
+
n1 n2
n1 n2
v.
ii.
P P2
X X 2 0
1
Z0 =
T0 = 1
2
2
s1 s2
1 1
+
P 1 P +
n1 n2
n1 n2
D 0
vi.
None of the
T0* =
iii.
above.
SD / n
(
10
)
b.
(TRUE or FALSE) Power is the probability of rejecting the null
hypothesis given that it is false.
c. Given that the two-sided 95% confidence interval on a mean is [9.25,
12.75], which of the following can we say are true (select all that are
true)?
i. The population mean is between and 9.25 12.75.
ii. If we used the same sample to create a 95% upper one-sided
confidence bound, the bound would be lower than 12.75
iii. The sample mean is 11. (?) isnt this right???
iv. If we used the same sample to conduct a hypothesis test of H0: =
8 vs. H1: 8, the null hypothesis would be rejected for = 0.05.
3. Chapter 10 (2 points each, 6 points total)
a.
(TRUE or FALSE) In a two-sample test of means, I should pool the
variances of the samples if I believe that the population variances are
different.
b. A scientist reports that a null hypothesis of
H 0 : 1 2 = 0 vs. H1 : 1 2 0 is rejected at =0.01 and with =
0.20. What is the chance that this conclusion is incorrect?
i. < 5%
iv. < 10%
ii. < 2.5%
v. < 1%
iii. < 20%
i. None of the above
11
c. Which statements are true regarding paired t-tests (select all that are true)?
i. For the paired t-test, we compute the mean of each sample, then
compare whether these means are the same or not.
ii. Use paired t-tests when specific observations in each sample can
be matched.
iii. Use paired t-tests whenever you have the same number of
observations in each sample.
iv. Paired t-tests may not be more powerful than a two-sample t-test if
the variances of the two populations are small.
v. None of the above is true.
4. Chapter 11 (2 points each, 6 points total)
a. (TRUE or FALSE) In a simple linear regression, the residuals must be
normally distributed, which can be checked by examining the normal
probability plot of the residuals.
b. Suppose a 95% confidence interval on the coefficient of the predictor
variable is [-1.7, 9.8]. Which of the following are true? (Choose all that
are true.)
i. The regression is not significant.
ii. If we took a large number of samples and did regressions on each,
95% of the coefficients for the predictor would be in the interval.
(Wrong!)
iii. The actual coefficient of the predictor is close to 4.05.
iv. The R2 value is 5%.
v. None of the above is true.
c.
(TRUE or FALSE) In a simple linear regression model (one predictor),
the p-value of the ANOVA is always equal to the p-value of the
coefficient of the predictor.
5. Chapter 12 (2 points each, 6 points total)
a. To check the normality assumption in a regression analysis, which of the
following plots can be used (choose all that are true)?
i. Residuals versus fits plot
ii. Boxplot
iii. Normal probability plot
iv. Residuals versus order plot
v. All of them can be used
b.
(TRUE or FALSE) In multiple linear regression, at least one of the
coefficients must be significantly different than zero for the regression to
be significant.
c. (TRUE or FALSE) In a multiple linear regression model, the value of R2
(adj) must be equal to or lower than the value of R2.
12
6. Chapter 13 (2 points each, 6 points total)
a. (TRUE or FALSE) The results of a one factor ANOVA where the factor
has exactly two levels will be the same as a two factor t-test on the same
data.
b. Which are valid interpretations of a low (< 0.05) p-value for a one-factor
ANOVA (choose all that are correct)?
i. The factor has an effect on the response.
ii. The treatment effects are all zero.
iii. The response is different at each of the different levels of the
factor.
iv. At least one treatment mean is different than zero.
v. None of the above is correct.
c.
(TRUE or FALSE) In a one factor ANOVA, the variance of the residuals
must be the same at all levels of the factor.
7. Chapter 14 (2 points each, 6 points total)
a. Which of the following ANOVA models can be used to analyze the results
of a 23-1 fractional factorial experiment? (Choose all that could be used.)
Yijkl = + i + j + k + ijkl
i.
ii.
Yijkl = + i + j + k + ( ) ij + ijkl
iii.
Yijkl = + i + j + k + ( ) ij + ( ) ik + ijkl
iv.
v.
Yijkl = + i + j + k + ( ) ijk + ijkl
None of the above are correct
b.
(TRUE or FALSE) In a two-way factorial experiment, one can always
test if the interaction is significant.
c.
(TRUE or FALSE) A balanced factorial experiment is one where the
variances are constant across the different factors.
13
8. Chapter 10 problem (19 points)
Conduct a hypothesis test (=0.05) for H0: 1- 2 = 0 vs. H1: 1- 2 0, given
that the variances of the two populations are unknown but equal. You have
samples from both populations, each containing 32 observations. For sample 1,
the mean is 26.7 with a variance of 0.8, and sample 2 has a mean of 25.7 with a
variance of 1.2.
14
9. Chapter 12 problem (19 points)
Your supervisor has asked you, the companys star industrial engineer, to produce
information on the effect of water temperature and corn syrup content on the
specific gravity of the root beer that you produce. You decide to conduct a
multiple linear regression on data collected over the past 6 months. The following
is the result of that regression:
ResidualPlotsforSpecificgravity
NormalProbabilit yPlot
VersusFit s
99.99
0.0050
90
Re sidua l
Pe r cent
99
50
10
1
0.0025
0.0000
0.0025
0.0050
0.01
0.008
0.004
0.000
Residual
0.004
0.008
1.010
Hist ogram
1.020 1.025
FittedValue
1.030
VersusOrder
150
0.0050
100
Re sidua l
Fr eque ncy
1.015
50
0.0025
0.0000
0.0025
0.0050
0
0.006 0.004 0.002
0.000
Residual
0.002
0.004
0.006
1 00 00 00 00 00 00 00 00 00 00 00 00
1 2 3 4 5 6 7 8 9 10 11 12
Obser vationOr der
Regression Analysis: Specific gravity versus Water Temp (C), Corn Syrup Content (grams per liter)
The regression equation is
Specific gravity = 0.870 + 0.000015 Water Temp (deg C)+ 0.000997 Corn Syrup Content (g/l)
Predictor
Constant
Water Temp (deg C)
Corn Syrup Content (g/l)
S = 0.00197442
Coef
0.870141
0.00001511
0.00099685
R-Sq = 79.7%
SE Coef
0.002542
0.00005837
0.00001454
T
342.32
0.26
68.58
P
0.000
0.796
0.000
R-Sq(adj) = 79.7%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
2
1197
1199
SS
0.0183386
0.0046663
0.0230049
MS
0.0091693
0.0000039
F
2352.11
P
0.000
a. Do you believe that the regression is valid? Why or why not? Be as
specific and complete as possible.
15
b. Do you believe there a relationship between either or both of the factors
on the specific gravity of the root beer? Why or why not?
c. What do the R-sq and R-sq (adj) values tell you about the regression?
d. The target specific gravity for the root beer is 1.02. Show that the
regression equation produces a value close to this for a water temperature
of 20C and a corn syrup content of 150 grams per liter.
e. How precisely (approximately) does the regression suggest we need to be
in terms of corn syrup content (i.e. what in grams per liter) to stay
between a specific gravity of 1.015 and 1.025, assuming a constant water
temperature of 20C?
16
10. Chapter 13 problem (20 points)
Three different types of polyethylene sockets can be used for hip replacement, but it is
believed that they have different wear rates. Thirty observations of each type are taken in
lab experiments, and an ANOVA is run, with the following results:
One-way ANOVA: Wear rate (mm per year) versus Plastic type
Source
Plastic type
Error
Total
DF
2
87
89
S = 0.04096
R-Sq = 23.89%
Level
A
B
C
N
30
30
30
Mean
1.0125
0.9937
0.9581
SS
0.04582
0.14599
0.19181
StDev
0.0481
0.0359
0.0378
MS
0.02291
0.00168
F
13.65
P
0.000
R-Sq(adj) = 22.14%
Individual 95% CIs For Mean Based on
Pooled StDev
---+---------+---------+---------+-----(-----*-----)
(----*-----)
(-----*-----)
---+---------+---------+---------+-----0.950
0.975
1.000
1.025
Pooled StDev = 0.0410
Tukey 95% Simultaneous Confidence Intervals
All Pairwise Comparisons among Levels of Plastic type
Individual confidence level = 98.06%
Plastic type = A subtracted from:
Plastic
type
B
C
Plastic
type
B
C
Lower
-0.04403
-0.07962
Center
-0.01882
-0.05442
Upper
0.00638
-0.02921
+---------+---------+---------+--------(-----*------)
(-----*------)
+---------+---------+---------+---------0.080
-0.040
0.000
0.040
Plastic type = B subtracted from:
Plastic
type
C
Plastic
type
C
Lower
-0.06080
Center
-0.03559
Upper
-0.01039
+---------+---------+---------+--------(-----*-----)
+---------+---------+---------+---------0.080
-0.040
0.000
0.040
17
a. Explain, as specifically and completely as possible, what the ANOVA results
above tell you about whether wear is different between plastic types A, B, and C.
Be sure to indicate which you think is best.
18
b. What is the expected difference in wear between type A and type C?
c. Identify the assumptions you need to check to confirm the ANOVA is valid and
your conclusions based on the plots below.
ResidualPlotsforWearrate( mmperyear)
NormalProbabilit yPlot
VersusFit s
99.9
0.10
90
0.05
Residua l
P e r ce nt
99
50
10
1
0.1
0.10
0.05
0.00
Residual
0.05
0.00
0.05
0.10
0.10
0.960
0.975
Hist ogram
1.005
1.020
VersusOrder
0.10
15
0.05
Residua l
20
Fr e que ncy
0.990
FittedValue
10
5
0
0.08
0.04
0.00
0.04
Residual
0.08
0.12
19
0.00
0.05
0.10
1
10
20
30 40 50 60 70
Obser vationOr der
80
90
Helpful formulas:
Confidence intervals
x z / 2 / n
x t / 2, n 1s/ n
( n 1) s 2 2 ( n 1) s 2
2
/ 2,n 1
p z / 2
2
1 / 2,n 1
p (1- p)
p p + z / 2
n
p(1- p)
n
Prediction intervals
x t / 2,n 1s 1 +
1
1
X n +1 x + t / 2,n 1s 1 +
n
n
Tolerance intervals
x ks
Single sample tests
X 0
Z0 =
/ n
X 0
T0 =
s/ n
0
2
( n 1) S 2
=
Z0 =
02
X np0
np0 (1 p0 )
Two sample tests
Z0 =
X1 X 2 0
or T0 =
12 22
+
n1 n2
X1 X 2 0
S12 S 22
+
n1 n2
X X 2 0
T0 = 1
11
Sp
+
n1 n2
or T0 =
D 0
SD / n
20
2
where S p
S12
F= 2
S2
( n1 1) S12 + ( n2 1) S22
=
n1 + n2 2
Z0 =
P P2
1
1 1
P 1 P +
n1 n2
(
)
Process control
PCR =
USL LSL
6
USL LSL
PCRk = min
,
3
3
Western Electric rules
Any point outside of the control limits
2 out of 3 consecutive points beyond a 2 sigma limit
4 out of 5 consecutive points beyond a 1 sigma limit
9 consecutive points on one side of centerline
21
22
23
24
25
26