Download Exercise 3 - Ryerson University

Ryerson Polytechnic University
QMS400: Introduction to Management Statistics
Assignment # 2
Instructor: Moez Hababou
Exercise 1:Finance and Accounting (35points)
Problem II: Finance and Accounting (35 points)
(a) the joint frequencies table of grades for both students is: (5pts)
grade in Finance
grades in Accounting
0-50
50-70
70-100
Totals
(b) (3pts)
Grades in Accounting
frequency
0-30
30-60
60-80
80-100
Totals
0
3
1
4
3
0
2
5
2
7
6
15
3
10
3
16
8
20
12
40
0-50
8
Grades in Finance
frequency
(c) (5pts)
Grades in Accounting
probability
0-30
4
Grades in Finance
frequency
0-30
10%
0-50
20%
50-70
20
30-60
5
70-100
12
60-80
15
50-70
50%
30-60
12.5%
80-100
16
70-100
30%
60-80
37.5%
80-100
40%
total
40
Total
40
total
100%
Total
100%
In absolute terms, it is easier to have higher grades in Finance.
Yes, as there are more students with high grades in Finance (in the interval 80 to 100) whereas students
obtain grades that are lower (mostly in the interval 50-70) in Accounting.
(d) (5pts)
Let GF be the grade received in Finance and GA refer to the grade received in Accounting.
P(GF>85)=40%*(100-85)/(100-80)=30%
P(GA>75)=30%*(100-75)/(100-70)=25%
There is thus a higher probability of getting an A in Finance. Most students would thus enroll in Finance if
both courses were elective.
(e) (7.5pts)
grade in Finance
grades in Accounting
0-50
50-70
70-100
Totals
0-30
30-60
60-80
0.00%
7.50%
2.50%
10.00%
7.50%
0.00%
5.00%
12.50%
5.00%
17.50%
15.00%
37.50%
80-100
7.50%
25.00%
7.50%
40.00%
Totals
20.00%
50.00%
30.00%
100.00%
From the joint probability table, we see that there is a probability of 7.5% for someone to have a grade in
accounting higher than 70 and a grade in Finance higher than 80. If we assume that these students are
evenly distributed in the interval 70-100, with respect to their grade in Accounting, we can deduce that the
proportion if people receiving both grades higher than 80 is equal to:
P(GF>80 and GA>80)=7.5%*(100-80)/(100-70)=5%.
Now, build the joint probability distribution table. What is the probability of someone
P(GF>80 given that GA>80)= P(GF>80 and GA>80)/P(GA>80)
By linear extrapolation, P(GA>80)=30%*(100-80)/(100-70)=20%
We obtain thus
P(GF>80 given that GA>80)= 5%/20%=25%
We know from (c) that P(GF>80)=40%.
P(GF>80 given that GA>80)= 25/40=5/8
Therefore, P(GF>80 given that GA>80)P(GF>80). These two events are not independent.
(f) (5pts)
There is no real way to compute a coefficient of correlation. However, we observe that most observations
are
located
in
entries
with
high
grades
in
both
courses
(call it southeastern cells of the joint probability table). For instance, 60% of the population has grades
higher than 50 in accounting and higher than 60 in Finance. We can conclude then that grades are more
plausibly positively correlated.
(g) (5pts)
It is somehow unrealistic, as grades in absolute terms are irrelevant. For instance, a final grade of 80 is
equivalent to an A in accounting. However, this grade is worth only a B in Finance!!! Consequently,
correlation can not really be tested and can be tested only based upon final letter grades.
Exercise 2: (35 pts)
Returns 1990
1991
1992
1993
1994
1995
1996
1997
1.13%
1998
1999
A
28.57% -13.33% 2.56%
26.00%
5.16%
B
C
-35.83% 14.29% 27.27%
8.93%
11.48% -7.35% 23.81% 12.82% 14.20%
11.94% 4.17% 42.40%
-75.00% -50.00% 233.33% -48.00% 150.00% -60.00% 200.00% 8.97%
Statistics average
A
8.86%
B
6.08%
C
50.05%
geo mean
N/A
N/A
N/A
variance
2.94%
3.62%
145.34%
return period
10.67%
8.64%
11.76%
Covariance A
-0.00846
-0.03502
5.88%
Sharp ratio
0.353828
0.091228
0.28681
Based on the Sharpe coefficient, which stock would you invest in? Stock A
Explain your approach in computing the Sharpe ratio=> compute first standard-deviation and average
return for the whole period 1990-1999
Exercise 3 (30 pts)
The probability distribution of the number of Computer systems sales for a salesman at Future Shop is
given by the following:
X=number of daily sales
Probability
1
.20
2
.35
3
.30
4
.15
(a) Plot the above probability distribution and briefly describe it. Almost symmetrical
Probability
0.4
0.35
0.3
0.25
0.2
Probability
0.15
0.1
0.05
0
1
2
3
4
(b)-(c)
X=number of daily sales Probability
1
2
3
4
0.2
0.35
0.3
0.15
EV(X)
x.P(x)
0.2
0.7
0.9
0.6
2.4
x-EV(x)
-2.2
-1.7
-1.5
-1.8
Variance
st-dev
(x-EV(x))^2
4.84
2.89
2.25
3.24
13.22
3.64
CV=1.52 => very high variation
(b) Let p refer to the probability that at least 3 daily sales are achieved. Out of 5 salesmen, what is the
probability that:
 At least 4 of them succeed in getting 3 daily sales?
 Exactly 2 employees achieve 3 daily sales?
 Strictly less then one employee achieves 3 daily sales
p=.45
 Binomial process with p=.45 and n=5
 At least 4 of them succeed in getting 3 daily sales? => Prob(X4)=1-P(X3)=1-.869=.131
 Exactly 2 employees achieve 3 daily sales? => Prob(X=2)= P(X3)- P(X2)=.593-.256=.337
Strictly less then one employee achieves 3 daily sales => Prob(X<1)=Prob(X=0)=.050
(c) Assuming that the previous distribution is asymptotically normal (that is if the number of salesmen
increase, it becomes more and more symmetrical), if you have 100 salesmen, using the normal table,
what is the probability that:
X follows normal distribution with mean (nEV(Xi)=100*2.4=240) and standard-deviation (nStdev(Xi)=100*3.64=364)
Prob(between 250 and 350 computers are sold) = P(250X350)=P(.022Z.30)=.6179-.5080=.1099


Prob(less than 100 computers are sold)=P(X100)= P(Z-.3846)=0
Prob(more than 400 computers are sold)=P(X400)=P(Z.4395)=1-.6664=.3336