FINAL EXAM
COURSE: Quantitative Techniques and Simulation (BUAD 591)
EXAM DURATION: 3 HRS
INSTRUCTORS: Dr. Santiago Casamayor & Rosa Padilla DATE: 28/06/2015
INSTRUCTIONS: Read each question carefully and write the answers for questions number 1 and 4 , 5 on this paper, and write the answers for the question numbers 2 and 3 inside the answer booklets.
You may use a SPSS software, or calculator whenever necessary.
Section I: Matching
1. (8 marks). Please write the letter of the item from column “b” that matches the concept in column “a”:
Column a Answer Column b
CYCLICITY(d) 4 Trend a. Component very long term representing growth or decline of the data in an extended period TREND (a) 1 Random b. Very short-term component. Irregular or sporadic movements or short term
RANDM FACTR(b) 2 Season c.same time and almost with the same intensity. Periodic fluctuations in periods whose frequency is less than a year, about the
SEASON© 3
Cyclicity d. Cyclic movements are considered only if they occur in a time interval longer than one year
Section II: Case of study and decision making
2. (8 marks). Linear programming. A company produces two types of can opener: manual and electric. Each
requires in its manufacture the use of three machines: A, B, and C. Each manual can opener requires the use
of machine A for 2 hours, machine B for 1 hour, and machine C for 1 hour. An electric can opener requires
1 hour on A, 2 hours on B, and 1 hour on C. Furthermore, suppose the maximum number of hours available
per month for the use of machine A, B, and C are 180, 160, and 100, respectively. The profit on a manual
can opener is $4, and on an electric can opener it is $6. If the company can sell all the can openers it can
produce.
Manual Electric Hours available
A 2 hr 1 hr 180
B 1 hr 2 hr 160
C 1 hr 1 hr 100
Profit/Unit $4 $6
Apply Simplex Method to answer the question: How many of each type should be make in order to maximize the monthly profit?
3. (8 marks). Monte Carlo Simulation. The bakery UMURERWA distributes a random number of cakes each day to important restaurants in Kigali. Umurerwa manager would like determine a policy for managing his inventory of cakes.
Demand 2 3 4 5 6 7 8
Frequency 5 25 40 60 30 25 15
How many cakes should he prepare in 10 days? (Use the following 10 random numbers)
Random numbers 52 37 82 69 98 96 33 50 88 90
and customers over 60 years of age were selected. The number of ATM transactions last month was determined for each selected individual, and the results are shown below.
Under 25 10 10 11 15 7 11 10 9
Over 60 4 8 7 7 4 5 1 7 4 10 5
For answer the question use the following output of SPSS software statistics:
Group StatisticsAge N Mean
Std. Deviation
Std. Error Mean Number of ATM
transactions
Under 25 8 10.3750 2.26385 .80039
Over 60 11 5.6364 2.46060 .74190
a. Interpret the table above: __
Young adults (under 25 years) use the machines more than senior citizens, the variability are the same
b.
At the .01 significance level, can bank management conclude that younger customer use the ATMs more?
Independent Samples TestLevene's Test for Equality of
Variances t-test for Equality of Means
95% Confidence Interval of the
Difference
F Sig. t df
Sig. (2-tailed) Mean Difference Std. Error
Difference Lower Upper Number of
ATM transactions
Equal variances
assumed 0.534 0.475 4.282 17 0.001 4.73864 1.10661 2.40389 7.07338 Equal
variances not
assumed 4.342 15.953 0.001 4.73864 1.09135 2.42453 7.05275
Test statistic:__t=4.282 Sig:_0,001
Making a Decision and Interpreting the Result of the Test:
Reject null hypothesis.
You can see that the group means are statistically significantly different because the value in the "Sig. (2-tailed)" row is less than 0.05,therefore the younger customer use ATMs more than senior
citizens
c. Basic assumptions for Homogeneity
See the results of the Levene test for homogeneity of variance in the table above. Ho: the variances are equal
Ha: not assume equal variances
Test statistic:_F= 0.534, Sig: 0.475
Making a Decision and Interpreting the Result: Don’t reject null hypothesis, therefore the variance are equal in both group
Ha: the variables are not follow normal distribution
Test statistic:_Shapiro Wilk=.889, Sig: .230 for Under 25 years and for Over 60 is Shapiro Wilk=.957, Sig: .739 Making a Decision and Interpreting the Result:_ Don’t reject null hypothesis, therefore the variables follow Normal Distribution
Tests of Normality
Age
Kolmogorov-Smirnova Shapiro-Wilk Statistic df Sig. Statistic df Sig. Number of ATM
transactions
Under 25 .266 8 .100 .889 8 .230
Over 60 .165 11 .200* .957 11 .739
e. Interpret the following graphs:
Interpretation: In both group of age we can observe homogeneity and they follow normal distribution
5. (8 marks) The following data give the selling price, square footage, number of bedrooms, and age of houses
that have sold in a neighborhood in the last 6 months. Develop multiple correlation and regression analysis
for to predict selling price.
Selling Price ($)
Square
Footage Bedrooms Age
64000 1670 2 30
59000 1339 2 25
61500 1712 3 30
79000 1840 3 40
87500 2300 3 18
92500 2234 3 30
95000 2311 3 19
113000 2377 3 7
115000 2736 4 10
138000 2500 3 1
142500 2500 4 3
144000 2479 3 3
145000 2400 3 1
147500 3124 4 0
144000 2500 3 2
155500 4062 4 10
165000 2840 3 3
a. Interpret a Mean and Standard Deviation for at least 1 variable
The average of Selling Price of the houses that have sold in a neighborhood in the last 6 months is 114588.24 $, and the Standard deviation is the variability around mean is 35901.443 that indicate it is heterogeneous data (Coefficient of variation is 31%).
b. Interpret at least 1 bivariate correlation
Correlation between Selling price and Age
Ho: (there is no association between Selling price and Square Footage)
Ha: (there is an association between them)
Making a decision and interpret result: We reject the Null Hypothesis given a significance of .000, we conclude that there is a strong and direct association between Selling price and Square Footage (r = -.802** Sig. = .000 < .05)
c. Interpret Multiple Correlation, and how much is it?
Multiple Correlation = 0.941, there is high association between Selling price and Square footage, bedroom and age (the model improved by interacting independent variables).
d. Interpret Coefficient of Determination, and how much is it?
R square=.885, therefore 88.5% of the variation in the selling price, can be explained by variation in Square footage, bedroom and age of the houses that have sold in a neighborhood in the last 6 months.
Model Summaryb
Model R R Square
Adjusted R Square
Std. Error of the
Estimate Durbin-Watson
1 .941a .885 .859 13482.331 2.280
a. Predictors: (Constant), Age, Bedrooms, Square Footage b. Dependent Variable: Selling Price ($)
e. Interpret Durbin Watson, and how much is it? Descriptive Statistics
Mean
Std.
Deviation N Selling Price ($) 114588.24 35901.443 17 Square Footage 2407.29 618.457 17
Bedrooms 3.12 .600 17
Durbin Watson = 2.280 It is a value slightly greater than 2, indicating no evidence of autocorrelation. Interpret ANOVA REGRESSION:
Ho: β
1=β
2=β
3=0
Ha: At least one βj ≠0 F=33.484
Sig: .000
Making decision and interpret result:
As Sig <0.000 then Reject null hypothesis, indicating that at least one of the explanatory variables (Square Footage, number of Bedrooms or houses’ age) is related to the price that houses sold.
ANOVAa
Model Sum of Squares df Mean Square F Sig.
1 Regression 18259565356.891 3 6086521785.630 33.484 .000b Residual 2363052290.168 13 181773253.090
Total 20622617647.059 16
Dependent Variable: Selling Price ($) and Predictors: (Constant), Age, Bedrooms, Square Footage
f. Using multiple regression, and find a model that will help explain current sales, and which is the best predictor?
Y= 82373.7208 +25.859 Square Footage
– 2127.798 number of bedrooms – 1714.821 Age of house
Coefficientsa
Model
Unstandardized Coefficients
Standardized Coefficients
t Sig.
B Std. Error Beta
1 (Constant) 82373.720 23072.294 3.570 .003
Square
Footage 25.859 9.638 .445 2.683 .019
Bedrooms -2127.798 8872.079 -.036 -.240 .814
Age_houses -1714.821 328.054 -.623 -5.227 .000
a. Dependent Variable: Selling Price ($)
g. Use the Model you fount before and predict the selling price of a 10 years of the houses , and 2000 square foot house with 3 bedrooms
Y= 82373.7208 +25.859 x 2000 – 2127.798 x 3 – 1714.821 x 10
Y = 110560.1 dollars
Interpret: the Residuals follow a normal distribution Interpret: the residuals assumed homocedasticity
