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# Quantitative Management

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TABLE OF CONTENTS Question 1: ……………………………………………..…………………..………......1 - 3 1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 Question 2: ……………………………………………………………..…………...….4 - 5 2.1 2.2 2.3 2.4 Question 3: ………………………………………………………………..………………..6 Question 4: …………………………………………………………..................…..…7 - 8 Question 5: ……………………………..……………………………………………..9 - 11

Bibliography: ………………………………………………………...…….……………..12

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QUESTION 1

1.1 Class Interval 20 up to 50 50 up to 80 80 up to 110 110 up to 140 140 up to 170 Frequency 06 12 14 04 04 40 Less than type Cumulative Frequency 00+06 = 06 06+12 = 18 18+14 = 32 32+04 = 36 36+04 = 40

1.2

Draw to scale 1.2.1 Histogram 1.2.2 Frequency Polygon 1.2.3 “Less-than” Ogive

REFER TO THE GRAPH ON THE NEXT PAGE FOR ANSWERS

2

3

1.3

Use the Ogive to determine 1.3.1. the 65th percentile To determine the 65th percentile we calculate :65% of 40 = 26. The 65% percentile occurs at position 26. From the graph it gives the value of 82 1.3.2 The inter-quartile range is a measure of dispersion and is equal to the difference between the third and first quartiles. Half of the inter-quartile range is called semi inter-quartile range or Quartile deviation. Symbolically it is defined as; Q.D = (Q3 - Q1)/ 2 where Q1 and Q3 are the first and third quartiles of the data.
!""#\$%&'(%#)*\$ !"#\$% ! !""!!"# ! ! !"#\$%&'( !!!"#\$%&'( ! !

������������������������������������������������ ������������������������������������������������������ = Q1 =
!"!!" !

=

= 61 and Q3 =

= 101.5

������������������������������������������������ ������������������������������������������������������ =

!"!.!!!"

= 81.25

4

QUESTION 2 2.1 Compute the mean and standard deviation for the daily trading volumes to use as estimates of the population mean and standard deviation. Shares in millions ������ 723 766 783 813 836 917 944 973 983 992 1046 1057 10833 MEAN VARIANCE SD Calculations
!! ! !"#\$\$ !"

������ − ������ -179.75 -136.75 -119.75 -89.75 -66.75 14.25 41.25 70.25 80.25 89.25 143.25 154.25

(������ − ������ )2 32310.06 18700.56 14340.06 8055.06 4455.56 203.06 1701.56 4935.06 6440.06 7965.56 20520.56 23793.06 143420.25 902.75 13038.20 114.18

Mean

= ������ =

=

= 902.75

Standard Deviation

= ������ =

������!������ ������ !!!

=

!"#"\$%.!" !"!!

= 114.18

5

Shares 723 766 783 813 836 917 944 973 983 992 1046 1057 10833 2.2
!"# !"#\$\$ !"" !"#\$\$ !"# !"#\$\$ !"# !"#\$\$ !"# !"#\$\$ !"# !"#\$\$ !"" !"#\$\$ !"# !"#\$\$ !"# !"#\$\$ !"" !"#\$\$ !"#\$ !"#\$\$ !"#\$ !"#\$\$

Probabilities = = = = = = = = = = = = 0.066741 0.070710 0.072279 0.075048 0.077172 0.084649 0.087141 0.089818 0.090741 0.091572 0.096557 0.097572 1.00

The probability that on a particular day the trading volume will be less than 800 million shares: 0.066741 + 0.070710 + 0.072279 = 0.21

2.3

The probability that on a particular day the trading volume will exceed 1 billion shares: 0.096557 + 0.097572 = 0.19

2.4

6

Question 3

2007 2008 2009 TOTAL

Hertz 30 29 32 91

Avis 25 23 30 78

National Budget 20 14 18 20 15 21 53 55

Other 6 9 7 22

TOTAL 95 99 105 299

H0: There is no evidence of change in market shares of the car rental firms HA: There is evidence of change in market shares of the car rental firms α = 0.05 Reject H0 if the test calculated statistic exceeds the critical chi-square value. i.e. Reject if the calculated x2 > 15.50731 (from table) Contingency Table f0 30 29 32 25 23 30 20 18 15 14 20 21 6 9 7 299 fe 28.31 30.13 31.96 24.78 25.83 27.39 16.84 17.55 18.61 17.47 18.21 19.31 6.99 7.28 7.73 x2 0.041 0.042 0.00 0.02 0.309 0.248 0.593 0.012 0.701 0.691 0.176 0.147 0.140 0.404 0.068 3.592

Chi-Square = 3.592 Degree of Freedom: df = (k-1)(k-2) = (5 – 1)(3 – 1) = 4 x 2 = 8 We accept Ho , (that there is no evidence of change in market shares of the car rental firms) because the test static does not exceed the critical value.

7

QUESTION 4 4.2 1 QUARTER YEAR 2 DAYS LOST 3 4-Q -M-A 5 CENTERED 6 7 8 DESEASONA LIZED (Using median) 823.4 897.7 864.5 3 4 1 1999 2 3 4 1 2000 2 3 4 1 2001 2 3 4 TOTAL 911 845.5 801 813.3 819 781.3 722 710.0 783 712.0 516 746.5 827 742.5 860 795.0 767 785.8 726 750.8 790 745.3 720 741.0 745 709 12442 715.0 767.3 12445 712.8 812.7 12449 743.1 96.9 759.5 707.2 748.0 105.6 726.8 742.3 768.3 94.5 785.7 832.2 790.4 97.0 736.1 733.8 768.8 111.9 907.2 844.7 744.5 111.1 760.8 777.1 729.3 70.8 558.4 591.5 711.0 110.1 751.4 749.1 745.6 96.8 761.6 709.1 797.3 102.7 753.4 769.6 829.4 96.6 866.9 918.2 855.0 106.5 874.3 871.6 DESEASONA LIZED (Using mean) 841.0 835.8
8

1 1998 2

895 851

%

Calculation of Seasonal Indexes Summary Table (Using the median) Year 1998 1999 2000 2001 Unadjusted (Median) Adjusted (Typical Index) Summary Table (Using the mean) Year 1998 1999 2000 2001 Unadjusted (Mean) Adjusted (Typical Index) 4.3 1 102.7 111.1 105.6 106.5 x 0.9995002 106.4 2 96.8 111.9 96.9 101.9 x 0.9995002 101.9 3 106.5 110.1 97 4 96.6 70.8 94.5 4 1 102.7 111.1 105.6 111.1 x 0.977995 108.7 2 96.8 111.9 96.9 96.9 x 0.977995 94.8 3 106.5 110.1 97 4 96.6 70.8 94.5

106.5 x 0.977995 104.2

94.5 x 0.977995 92.4

409

400.0

104.6 x 0.9995002 104.5

87.3 x 0.9995002 87.2

400.2

400.0

To do seasonal forecast per quarter for 2010, we code the time period and use the method of least squares to obtain the trend equation value. We then multiply each trend value by the seasonal index for that period.

9

QUESTION 5 A Decision Tree represents a series of decisions that are linked through time. That is, after one decision has been made, there may be any number of subsequent decisions. Moreover, the choices that are available for future decisions depend on past choices. A diagram is very helpful in keeping track of all of these details, and explaining the situation to others.
Probabilities

1.0 0.3 0.4 0.3 0.5 0.3 0.2 0.6 0.3 0.1

5.1

Decision Tree
Profits
R200 000 R200 000

tio

n 1

R234 000

R300 000 R240 000 R160 000 R200 000 R300 000 R200 000

R234 000

Op t
Op

A t 2 Op

Op

R230 000 ion 2B

tio n 3

R230 000

R200 000 R300 000 R200 000

5.2

Expected monetary value computation for the town’s parking facilities

We compute the expected monetary value for the town’s parking facilities problem displayed with the associated probabilities. The calculations for determining the expected monetary value for the decision alternative option 1: Expected value for (1.0 probability) = (1.0) x (R200 000) = R 200 000

Expected monetary value of investing in option 1 is: R200 000

10

The calculations for determining the expected monetary value for the decision alternative option 2A: Expected value for (0.30 probability) Expected value for (0.40 probability) Expected value for (0.30 probability) = (0.30) x (R300 000) = R90 000 = (0.40) x (R240 000) = R96 000 = (0.30) x (R160 000) = R48 000

Expected monetary value of investing in option 2A is: R90 000 + R96 000 + R48 000 = R234 000 The calculations for determining the expected monetary value for the decision alternative option 2B: Expected value for (0.50 probability) Expected value for (0.30 probability) Expected value for (0.20 probability) = (0.50) x (R200 000) = R100 000 = (0.30) x (R300 000) = R90 000 = (0.20) x (R200 000) = R40 000

Expected monetary value of investing in option 2B is: R100 000 + R90 000 + R40 000 = R230 000 The calculations for determining the expected monetary value for the decision alternative option 3: Expected value for (0.60 probability) Expected value for (0.30 probability) Expected value for (0.10 probability) = (0.60) x (R200 000) = R120 000 = (0.30) x (R300 000) = R90 000 = (0.10) x (R200 000) = R20 000

Expected monetary value of investing in option 3 is: R120 000 + R90 000 + R20 000 = R230 000

11

The owners of the town’s parking facilities, as decision makers using expected monetary value as a strategy will choose the maximum of the expected monetary values computed for each decision alternative. Maximum of {R200 000, R240 000, R230 000, R230 000} = R240 000 The owners will use option 2A

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BIBLIOGRAPHY 1. Bancroft, G and O’Sullivan G (1981) Mathematics and statistics for Accounting and Business Studies. McGraw Hill Book Coy. (Uk) Ltd. 2. Black,K. (2000) Business Statistics: Contemporary Decision Making (3rd ed) South-Western College Publishing. 3. Buglear, J. (2005) Quantitaive Methods For Business. Elsevler Butterworth Heinemann. 4. Daniel, W. and Terrel, J. (1995) Business Statistics for Management and Economics (7th edition) Houghton Mifflin Company, Boston. 5. Lind, Marshall and Wathen (2005). Statistical Techniques in Business and Economics (12th edition), New York, McGraw Hill. 6. Schonberger, R.J. & Knod, E.M., Jr. (1985). Operations Management: Serving the Customer. (3rd ed). Homewood, Illinois: BPI Irwin. 7. Stevenson, W.J. (1999). Production/Operations Management. 6th Ed. Irwin: McGrawHill. 8. Wegner, T. (2005). Applied Business Statistics. Methods and Applications. Kenwyn:Juta & Co, Ltd. 9. Wisniewski, M. and Stead R. (1996). Foundation Quantitative Methods for Business.London : Prentice Hall

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