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NATIONAL UNIVERSITY OF SINGAPORE

DSC2008 BUSINESS ANALYTICS – DATA AND DECISIONS

(Semester 2 – AY2013/2014)

Time Allowed : 2 Hours

INSTRUCTIONS TO CANDIDATES 1. Please write only your student number below. Do not write your name.2. This booklet contains two (2) Sections and comprises fifteen (15) printed pages.3. Answer ALL questions. This is an OPEN Book examination. All materials are allowed.4. Write legibly. A dark pencil may be used.5. Graphic calculators or other calculators may be used, but not computers, tablets or phones.6. Write your answers in the spaces provided after each part of a question, except that answers

to Section A must only be entered into the Bubble Form provided. The first column for STUDENT NUMBER on the Bubble Form should be left blank.

7. Plan your answers to ensure they fit within the spaces provided. Other than this cover page and the spaces designated for providing your answers, you may do your “rough work” anywhere. Whatever you write outside of the answer spaces may be ignored.

Write your SEAT NUMBER and MATRICULATION NUMBER below.Seat No:

Matriculation No :

Question Max Marks

Section A 40

Section B

Question 1 15

Question 2 15

Question 3 18

Question 4 12

Total 100

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Section A (40 marks). Each question carries 2 marks. Choose the most appropriate answer.

1. Which of the following time series forecasting methods could NOT be used to forecast a time series that exhibits a linear trend with no seasonal or cyclical patterns? A. Dummy variable regression. B. Linear trend regression. C. Holt-Winters’ double exponential smoothing. D. Ratio-to-Moving-Average method with linear trend model.

2. To check for positive or negative first-order autocorrelation, we can use

A. the Durbin-Watson statistic. B. a one-sample t test. C. multiplicative Winters’ method. D. dummy variable regression.

3. If the errors produced by a forecasting method for 3 observations are 2, -1, and -6, then what is the closest value to their root mean squared error? A. -5/3B. 41/3C. 3D. 11/3

[Answer] RMSE = √(22+(−1 )2+(−6 )2)/3=3.70≈11 /3

4. Consider the following time series. Time period 1 2 3 4 5 6 7Sales 53 60 47 55 56 53 ?Compute sales forecast for time period 7 using

i. the 3-period weighted moving average giving a weight of 1/2 to the most recent period, a weight of 1/3 to the second most recent period, and a weight of 1/6 to the third most recent period and

ii. the exponential smoothing with smoothing constant of 1/2.What is the closest value to the difference between the two forecasts of i and ii?

A. 53

B. 5413

C. 113

D. 1

[Answer] SES(0.5) = 12×53+ 1

2

2

×56+12

3

×55+ 12

4

×47+ 12

5

×60+ 12

6

×53=53.02

WMA(3) ¿12×53+ 1

3×56+ 1

6×55=54.33

The difference = 1.31≈113

.

5. Which of the following statement is true concerning the autocorrelation function (ACF) and partial autocorrelation function (PACF)?A. The ACF will always be smaller than PACF. B. The PACF for an AR(q) model will be zero beyond lag q.C. The ACF for an AR(p) model will be zero beyond lag p. D. The ACF and PACF will always be identical for an ARMA(1,1) model.

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Answer questions 6 to 10 according to the following data analysis.

Lind fabric retailer sells a variety of fabrics and sewing supplies, craft materials, frames, home decorations, artificial floral items, and seasonal goods. It operates more than 300 stores, with most located in mall shopping centers. The quarterly revenues of Lind increased from about USD 5 million in Year 2009 to more than USD50 million in Year 2013.

2013Q4

2013Q2

2012Q4

2012Q2

2011Q4

2011Q2

2010Q4

2010Q2

2009Q4

2009Q20

10,000,000

20,000,000

30,000,000

40,000,000

50,000,000

60,000,000

Revenue

Lind invited you to predict the future revenues in Year 2014. Given the existence of seasonal pattern, you adopted a seasonal dummy model. Answer the following questions according to the Excel and SAS output below.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.978R SquareAdjusted R SquareStandard Error 3443218.65Observations 20

ANOVAdf SS MS F Significance F

Regression 4 3.88088E+15 9.70221E+14 81.835 5.30312E-10Residual 15 1.77836E+14 1.18558E+13Total 19 4.05872E+15

Coefficients Standard Error t Stat P-valueIntercept 17447690.73 2147703.318 8.123883116 7.12257E-07Time 1132370.43 8.319819526 5.29612E-07Q1 -17903183.80 2194629.884 -8.157723509 6.76499E-07Q2 -9331747.045 2181931.822 0.000661966Q4 11997271.06 2181931.822 5.498462848 6.12299E-05

The residuals of the fitted model are analyzed in SAS. Please refer to the SAS output.

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Augmented Dickey-Fuller Unit Root Tests for ResidualsType Lag Rho Pr < Rho Tau Pr < Tau F Pr > FSingle Mean

0 -13.2380 0.0261 -2.83 0.0722 4.05 0.11681 -22.8343 0.0002 -3.07 0.0477 4.70 0.07262 -353.7224 0.0001 -3.39 0.0264 5.76 0.0370

Autocorrelation Check of ResidualsTo Lag Chi-Square DF Pr > ChiSq Autocorrelations6 8.21 6 0.2234 0.271 -0.144 -0.385 -0.209 -0.118 -0.13412 13.82 12 0.3123 -0.037 -0.058 0.130 0.246 0.194 -0.07018 19.44 18 0.3655 -0.057 -0.063 -0.043 -0.163 0.023 0.094

6. What is the t-stat of Q2? Is Q2 significant at 5% level?A. -8.15. Insignificant.B. -8.15. Significant.C. -4.28. Insignificant.D. -4.28. Significant.

[Answer] d. The t-stat = Coef./SE = -9331747.045/2181931.822 = - 4.277.Given P-value less than 5%, the variable Q2 is significant.

7. Which quarter is considered as base season in your model and what is the fitted model for the base season?A. Q1. Revenue ¿17447690.73+1132370.43×Time−17903183.80×Q1.B. Q3. Revenue ¿17447690.73+1132370.43×Time.C. Q3. Revenue ¿17447690.73+1132370.43×Time−¿14105289×Q 3.D. Q4. Revenue ¿17447690.73+1132370.43×Time.

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8. What is the 95% interval forecast (using Normal approximation) for the first quarter of Year 2014 (2014Q1) with Time = 21?A. [16.6 million, 30.1 million].B. 23.3 million.C. [6.6 million, 30.1 million].D. 16.6 million.

[Answer] a. The point forecast = 17447690.73+1132370.43×21−17903183.80=¿23 324 286.The 95% interval forecast = 23324286±1.96×3443218.65=[16 575 577 ,30 072 995 ] .

9. Are the residuals stationary?A. Yes. The ADF test rejects the null hypothesis.B. No. The ADF test rejects the null hypothesis.C. Yes. The Ljung-Box test indicates that the residuals are independent.D. No. The Ljung-Box test indicates that the residuals are independent.

10. Is the model adequate?A. Yes, given the adjusted R square being 0.978. B. Yes, since the ADF test rejects the null hypothesis.C. Yes, given that the Ljung-Box tests do not reject the null hypothesis.D. No. The residuals are dependent.

11. One-way Analysis of Variance (ANOVA) is not which of the following?A. A generalization of the 2-sided pooled t-test for means, when variances are unknown

but equal.B. A method for comparing multiple entire distributions of variables.C. A component of the usual multiple regression output.D. A simple model for the design of experiments.

12. P-value in hypothesis testing is not which of the following?A. Given the truth of the Null Hypothesis, the probability of a fresh sample favoring the

Alternative Hypothesis at least as much as the current sample.B. When compared with the preset probability of Type I Error, leads to whether the Null

Hypothesis is rejected.C. In a stepwise regression, can be used for selecting the variable to be deleted from a

regression model.D. The probability that the Null Hypothesis is true.

13. How many percent of the data are between the second and third quartiles?A. 25%B. 68%C. 50%D. Depends on the Standard Deviation.

14. Which of the followings is not a measure of spread for a variable distribution?A. Interquartile rangeB. Median absolute deviationC. RangeD. The average of the first and third quartiles

15. Which of the following is a description of the correlation coefficient?A. It measures the correlation between two regression coefficients.B. It is the square root of the Adjusted R2 in regression.

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C. For X and Y variables standardized, it measures the extent of clustering around either the positively or negatively sloping 45-degree line through the origin.

D. It is not symmetrical in the two variables involved.

16. Logistic regression isA. the regression of log(Y) on log(X).B. frequently used when deciding between only two possible outcomes.C. where the regression error term is assumed to follow the Logistic rather than Normal

distribution.D. the regression of log(Y) on X.

17. The Central Limit Theorem says that:A. in the limit, everything tends toward the centre.B. the limiting distribution of the sample average is Normal as the sample size is

increased.C. in a different round, a corresponding observation tends to be relatively closer to the

centre.D. nowhere is the central tendency of a distribution more limiting than at the average.

18. Indicator or dummy variables may not be used forA. identifying important outliers.B. imputing missing values.C. indicating dummy collinearity.D. representing categorical variables.

19. Residual plots may sometimes provide useful regression diagnostics, except for identifyingA. inconstant error variance.B. correlated errors.C. regression effect.D. departure from Normality assumption for errors.

20. Simpson’s Paradox warns againstA. rampant use of transformed X variables.B. failure to identify any regression effect.C. confusion between the power and the strength of a test.D. carelessly adding numbers from disparate groups.

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Section B (60 marks). There are 4 questions.

Question 1 (15 marks)Two scientists Flury and Riedwyl collected and analyzed 200 old Swiss banknotes. Each banknote was measured (unit: centimeter), as indicated in the figure,

X1 = length of the banknoteX2 = height of the banknote (left)X3 = height of the banknote (right)X4 = distance of the inner frame to the lower borderX5 = distance of the inner frame to the upper borderX6 = length of the diagonal of the central picture

The aim was to study how these measurements may be used to distinguish genuine and counterfeit banknotes.

The K-means method was used to cluster these banknotes. Analyst YC repeated the analysis with K = 2 and K = 5. The SAS output is displayed as follows.

Cluster Means K = 2Cluster X1 X2 X3 X4 X5 X61 214.82 130.30 130.19 10.55 11.13 139.442 214.97 129.95 129.72 8.31 10.18 141.50

Cluster Means K = 5Cluster X1 X2 X3 X4 X5 X61 215.27 130.28 130.01 8.81 10.18 141.222 214.91 130.30 130.21 9.52 11.57 139.293 214.75 130.30 130.18 11.36 10.75 139.594 214.85 129.84 129.61 7.83 10.49 141.545 214.95 129.84 129.67 8.95 9.32 141.85

X1

X6X2

X3

X5

X4

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a) Choose an appropriate K-means method and justify your selection. [4 marks]

Since the aim was to determine whether a banknote is genuine or counterfeit, it is natural to set K = 2. One represents the genuine notes and the other is for the counterfeit notes.

b) Discuss the differences among the clusters of your selected method, in terms of the 6 measurements. [4 marks]

According to the K-means result, the length, height (left and right) and distance of the inner frame to the upper border, i.e. X1, X2, X3 and X5, of genuine and counterfeit banknotes are quite similar, while there are relatively big differences between the two centers on distance of the inner frame to the lower border at 2.24cm and the diagonal length of the central picture at 2.06cm, i.e. X4 and X6. It implies that one can focus on X4 and X6 to distinguish genuine or counterfeit banknotes.

c) Analyst YC also tried Hierarchical method. The result is consistent with your selection. However YC lost the distance measure between two banknotes. The measures of the two banknotes are given:

Banknot X1 X2 X3 X4 X5 X61 214.8 131.0 131.1 9.0 9.7 141.02 214.6 129.7 129.7 8.1 9.5 141.7

What is the city-block distance between the two banknotes? [4 marks]

City-block distance ¿|214.8−214.6|+|131−129.7|+|131.1−129.7|+|9.0−8.1|+|9.7−9.5|+|141.0−141.7|=4.7 .

d) List 3 popular distance measures in cluster analysis and discuss their pros and cons.[3 marks]

I. Euclidean Distance: scale dependent. Not affected when including new objects.II. Manhattan Distance: scale dependent. Not affected when including new objects.

Manhattan Distance, compared to Euclidean Distance, dampens effect of outliers as differences are not squared.

III. Mahalanobis Distance: doesn’t dependent on scales of measurement used. However it will be influenced when new objects are included as the standard deviation needs to recalculate given the increased sample.

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Question 2 (15 marks)Poverty is a state of a lack of goods and services that are commonly taken for granted by members of mainstream society. The most common measure of poverty in the U.S. is the poverty level set by the U.S. government according to total income received. For example, the poverty level in Year 2012 was set at $23,050 (total yearly income) for a family of 4 in the U.S.

Consider the time series of poverty rates, percent of all the individuals under the age of 18 living below the poverty level, in the U.S. from Year 1959 to 2003:

Analyst YC conducted Holt-Winters’ method with season period of M = 4 years to forecast the poverty rates (i.e. some component pattern repeats every 4 years). The initial forecasts of level and trend are 21.18 and -1.26. The initial forecasts of seasonality are 1.09, 1.09, 0.99 and 0.83. The optimal smoothing constants are α=1 , β=0.16 , γ=0.a) Explain whether or not Holt-Winters’ method is appropriate to analyze the series of

poverty rates. [4 marks]

The time series plot displays a time varying trend with possible seasonal/cyclical variations. However the underlying dependence is not very clear. The HW method is flexible to forecast time series data when the underlying dynamics is not apparent and may have trend and seasonal variations.

b) Interpret the meaning of the optimal smoothing constants, α ,β , γ . [4 marks]

The magnitude of smoothing constants tells the influence of the recent values on forecast. Given α=1 , the level of time series continuously updates when new observation arrives. On the contrary, γ=0 indicates the initial values of seasonality are quite accurate and there is no need to update season forecast. The trend forecast slowly updates given β=0.16 .

c) Are there seasonal/cyclical variations in the time series? [3 marks]

Yes, there are seasonal/cyclical variations with stable season forecasts of 1.09, 1.09, 0.99 and 0.83. In each period, the poverty rate in the first two years are 9% higher than the average, while 1% and 17% lower in the rest two years.

d) List 2 alternative models that are appropriate to analyze the time series of poverty rates.

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[4 marks]

We can try multiplicative seasonal model and additive seasonal model.

Question 3 (18 marks)Salaries for employees at a bank are analyzed using regression, with the following results:

Regression StatisticsMultiple R 0.816645142R Square 0.666909288Adjusted R Square 0.653518706Standard Error 6625.6718Observations 208

ANOVAdf SS MS F Significance F

Regression 8 17491101398 2186387675 49.80435632 1.64008E-43Residual 199 8736005833 43899526.8Total 207 26227107231

Y: Salary Coefficients Standard Error t Stat P-valueIntercept 31829.30731 3405.456736 9.346560469 1.89825E-17YrsThisBank 1495.083406 159.6484805 9.364845828 1.68338E-17YrsThisBank*Female -1109.41783 204.3454289 -5.429129666 1.63858E-07YrsPriorBank*Female 988.3191454 356.2003226 2.774616087 0.006053119ComputerRelated 4018.024602 1674.774669 2.399143405 0.017355991YrsPriorBank -585.1391184 305.4275282 -1.91580347 0.056823454Female 1896.763788 4320.417514 0.439023262 0.661120987Age*Female -46.21756033 142.7899567 -0.323675148 0.746523894Age 2.591559621 119.9336832 0.021608272 0.982782086

Please note that Female and ComputerRelated are indicator/dummy variables, with 1 for their respective categories and 0 otherwise. All employees with ComputerRelated jobs are female. YrsThisBank and YrsPriorBank records the time an employee has worked for this bank and for any prior bank. “Variable”*Female is the product of the corresponding Variable with the Female indicator variable.

(a) Using the regression results above, what are the separate prediction equations for the Salaries of Male and Female employees at the bank? Please round the coefficients to whole numbers. [3 marks]

Male: Salary = 31829 + 1495*YrsThisBank – 585*YrsPriorBank + 3*Age

Female: Salary = (31829+1897) + (1495-1109)*YrsThisBank + (-585+988)*YrsPriorBank + (3-46)*Age + 4018*Computer Related= 33726 + 386*YrsThisBank + 403*YrsPriorBank - 44*Age + 4018*Computer Related

(b) Based on your answers to (a), comment on expected yearly salary increments for male and female employees of this bank. How would these numbers be commonly interpreted?

[3 marks]Male: 1495+3 = 1498 (3 is the coefficient of Age)Female: 386-44 = 342

Female employees’ average yearly salary increment is only 23% of that of male employees. This probably reflects the relatively junior positions of female employees. Note: The fact that female employees’ intercept is 1897 larger wasn’t really an issue.

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(c) Explain how you might improve the model by discarding a variable. Please be specific.[3 marks]

Age has the largest P-value; in fact it is almost 1. This variable should be deleted and a new model fitted.

A variables-selection technique is employed, resulting in the new model below:

Regression StatisticsMultiple R 0.816241456R Square 0.666250115Adjusted R Square 0.657988979Standard Error 6582.791103Observations 208

ANOVAdf SS MS F Significance F

Regression 5 17473813213 3494762643 80.64873091 3.12637E-46Residual 202 8753294018 43333138.7Total 207 26227107231

Y: Salary Coefficients Standard Error t Stat P-valueIntercept 32171.74593 924.3653924 34.80414368 2.80414E-87YrsThisBank 1485.262977 75.76029523 19.60476754 1.24796E-48YrsThisBank*Female -

1127.60382987.24034017 -12.92525713 3.03196E-28

YrsPriorBank*Female 1005.09522 286.5924527 3.507054043 0.000558289ComputerRelated 4092.763696 1639.307593 2.4966417 0.013337044YrsPriorBank -608.86039 248.3842939 -2.451283777 0.015084658

(d) Please explain whether the above is indeed a better model. [3 marks]

The Significance F, P-values and number of predictors are all smaller, and Adjusted R 2 is a bit higher, hence a better model.

(e) How come Age is now not a factor in explaining Salary? [3 marks]

Age was probably collinear with both YrsPriorBank & YrsThisBank. In the presence of these 2 variables, Age is no longer needed for explaining Salary.

(f) Based on this new model, compare the expected yearly salary increments for male and female employees of this bank. Please comment on the negative coefficient of YrsPriorBank. [3 marks]

Male: Salary = 32172 + 1485*YrsThisBank – 608*YrsPriorBank

Female: Salary = 32172 + (1485-1128)*YrsThisBank + (-608+1005)*YrsPriorBank + 4093*Computer Related

= 32172 + 358*YrsThisBank + 396*YrsPriorBank + 4093*Computer Related

The ratio of female and male employee salary increments is now 358 to 1485, which is 24%, almost the same as for the first model. For female employees, with Age out of the way, it seems that each year of work is worth more in any prior bank than in the current bank!

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The -608 negative coefficient of YrsPriorBank might reflect that, among male employees, the older recruits occupy more junior positions in the bank, since YrsPriorBank is probably a surrogate for Age.

Question 4 (15 marks)U.S. law school graduates’ pays have been studied by Forbes magazine in March 2014. Law schools whose graduates volunteered their pay information are ranked. Schools having the highest 25 average annual starting pays at graduation also have their graduates’ mid-career pays reported. The following plot scatters the latter pays versus the initial pays.

70000 80000 90000 100000 110000 120000 130000 140000 150000110000

120000

130000

140000

150000

160000

170000

180000

190000

200000

210000

220000Mid-Career vs Starting Pay

The boxplot for the two data series is as follows:

0 50000 100000 150000 200000 250000

Chart Title

US$ Anually

Axi

s Ti

tle

The very top (hence rich) U.S. law schools encourage graduates to take on low-paying judicial clerkships in the public sector, partly through awarding study-loan-forgiving post-graduate fellowships. Some of these top graduates continue into a career of public service, while others become academics (with pays nowhere near what private law firms offer). However, sought-

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after clerkships (especially at the U.S. Supreme Court) are also stepping stones to very high-paying high-street law firms, so many top students who are not primarily interested in public-sector positions also elected for those low-paying jobs, for a stint, upon graduation.

(a) How do you explain the shape of the boxplot for law graduates average starting pays?[3 marks]

If there are many law schools in the US, the top 25 average salaries would represent only the right tail of average salary distribution. As we start moving left from the right tail (perhaps representing law schools with few graduates starting in public-sector), the clustering of salaries will initially become increasingly dense. Hence, the lower boxplot represents a left-truncated distribution with very short left tail compared to the long right tail where there are outliers.

(b) Please comment on the data summary depicted by the two boxplots. [3 marks]

The Starting-pay boxplot skews to the right, whereas the more-symmetrical Mid-career-pay boxplot skews to the left, probably because some schools continue to have large number of mid-career graduates in the public sector. The top-half of the first plot has a range bigger than the top-half of second plot, although overall spread is larger for the Mid-career group. The difference in the medians of starting and mid-career pays is almost US$100,000. Perhaps graduates in top law firms, who are not in the public sector at their mid-career, cause their schools to be in the top half of the second boxplot, in which case the schools end up having graduates with rather similarly high average salaries of nearly US$200,000.

(c) Why is the scatterplot without the usual oval or elliptical shape? Please comment on how this might affect the prediction of Mid-career pay from Starting pay. [3 marks]

The fact of the left truncation of the Starting-pay distribution largely contributed to the unusual scatterplot, i.e. the high degree of skewness of the X variable led to the absence of the oval. Given the small sample size, the one big outlier on the right also makes it hard for the ellipse to appear. All these make for low forecast accuracy from average graduate Starting pay of a school from among those with the highest 25 average Starting pay, to the corresponding average Mid-career pay of the school’s graduates. [In fact, the R2 is only 0.2.]

(d) Please write an executive summary for the readers of Forbes magazine. [3 marks]

Top U.S. law graduates do not all go for the highest-paying jobs in the first instance, since a stint as a judicial clerk can pay very high dividends later when switching to a private law firm. This voluntary hardship posting is also encouraged by generous study-loan-payback subsidies offered by the very best law schools that encourage their graduates to go into public service. Hence, the median of the top 25 average-starting-salary law schools is only about US$80,000 yearly. These salaries are self-reported by graduates, so may represent a bias sample.

By mid-career, however, only lawyers who are committed to low-paying public service remain in the judicial system, so the median of the average salaries of lawyers from those same 25 schools more than doubled to about US$175,000. Since the proportions of public-sector-moved-to-private-sector graduates are not indicated for the schools, it is difficult to predict those top schools’ graduates’ average mid-career salaries from their respective

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average starting salaries; presumably, schools with the higher proportions of those “career switches” saw the larger jumps in average salaries.

========= END OF PAPER =========