In this Post, we are going to discuss the Paper of Business Statistics and Mathematics Solved Paper 2013, Punjab University, BCOM, ADCI in which Measures of Central Tendency, Measures of Dispersion, Correlation & Regression, Index Numbers, Matrix, Arithmetic Progression, Geometric Progression, Simultaneous Linear Equations, Annuity, Quadratic Equation is discussed and solved.
Other solved papers of Business Statistics & Mathematics
 Solved Paper 2007 Punjab University
 Solved Paper 2008
 Solved Paper 2009
 Solved Paper 2010
 Solved Paper 2011
 Solved Paper 2012
Solved by Iftikhar Ali, M.Sc Economics, MCOM Finance Lecturer Statistics, Finance and Accounting
Business Statistics and Mathematics Solved Paper 2013, Punjab University, BCOM, ADC I
Table of Contents
Section I Business Statistics
Q.1 Weight of 175 male students at a university are given in the following frequency table…..Calculate Karl Pearson’s and Bowley’s Coefficient of Skewness.
Q.1 Weight of 175 male students at a university are given in the following frequency table:
Weight  Frequency 
118126  20 
127135  35 
136144  49 
145153  32 
154162  25 
163171  14 
Calculate Karl Pearson’s and Bowley’s Coefficient of Skewness
Solution:
Classes  Class Boundaries  X  Frequency (f)  fx  fx²  C.F 
118126  117.5126.5  122  20  2440  297680  20 
127135  126.5135.5  131  35  4585  600635  55 
136144  135.5144.5  140  49  6860  960400  104 
145153  144.5153.5  149  32  4768  710432  136 
154162  153.5162.5  158  25  3950  624100  161 
163171  162.5171.5  167  14  2338  390446  175 
∑f = n= 175  ∑f = 24941  ∑ fx² = 3583693 
Selection of Model Class for Median
Model Class for Mode
Maximum Frequency is 49 so L = 135.5, fm =49, f1 =35, f2 =32 & h=9
Selection of Model Class for Q1
Selection of Model Class for Q3
Q.2 Calculate weighted index number of prices for the year 2012 from the following data taking 2008 as base and using formulae recommended by: Laspeyre, Fisher, Paasche’s and Marshall….
Q.2 Calculate weighted index number of prices for the year 2012 from the following data taking 2008 as base and using formulae recommended by: Laspeyre, Fisher, Paasche’s and Marshall
Year  A  B  C  
Price  Quantity  Price  Quantity  Price  Quantity  
2008  5.0  80  3.6  90  3.1  20 
2012  8.7  100  5.7  95  4.6  30 
Solution:
Commodity  2008  2012  
Price Po  Quantity q0  Price P1  Quantity q1  p0q0  p1q1  P1q0  P0q1  
A  5  80  8.7  100  400  870  696  500 
B  3.6  90  5.7  95  324  541.5  513  342 
C  3.1  20  4.6  30  62  138  92  93 
Sum  786  1549.5  1301  935  
∑p0q0=  ∑p1q1=  ∑P1q0=  ∑P0q1= 
Q.3. A survey of 1600 families was conducted to observe that high and low income people send children to private and government school. The following results were obtained….Test whether income and type of school are independent at 5% level of significance (table Value is 3.841)
Q.3. A survey of 1600 families was conducted to observe that high and low income people send children to private and government school. The following results were obtained:
Income  School  Total  
Private  Government  
High  494  506  1000 
Low  162  438  600 
Total  656  944  1600 
Test whether income and type of school are independent at 5% level of significance (table Value is 3.841)
Solution
(i) Testing the Hypothesis
Ho: There is no Association between Income & School Choice.
H1: There is Association between Income & School Choice.
(ii) Level of Significance = α=0.05
(iii) Test Statistics is:
(iv) To find χ² first calculate expected frequencies fe:
Calculation of expected frequencies (fe)
Income  School  Total  
Private  Government  
High 

 1000 
Low 

 600 
Total  656  944  1600 
(v) Calculation of χ²:
Computation of χ²
fo  fe  fofe  (fofe)² 

494  410  84  7056  17.210 
506  590  84  7056  11.959 
162  246  84  7056  28.683 
438  354  84  7056  19.932 
77.784  

(vi) Critical Region: Degree of Freedom d.f= (R1)(C1)
So d.f= (21)(21)=1
The Value of Tabulated χ²(0.05,1)=3.841
The Critical Region χ²cal>3.841
(vii) Conclusion: The calculated value of χ² is 77.784 is greater than the tabulated value of χ² 3.841 or 77.784 falls in the critical region. We reject the Null Hypothesis and accept alternative hypothesis. We can say that there is relationship between Income level and choice of school.
Q.4. Given the six elements population 0, 3, 6, 12, 15 and 18. How many samples of size n = 3 can be drawn without replacement from this population. Form sampling distribution of sample means. Hence state and verify the relation between…..
Q.4. Given the six elements population 0, 3, 6, 12, 15 and 18. How many samples of size n = 3 can be drawn without replacement from this population. Form sampling distribution of sample means. Hence state and verify the relation between:
(i) Mean of the sampling distribution of the means and the population mean.
(ii) Variance of the sampling distribution of the mean and population variance.
Solution:
Population = 0,3, 6, 12, 15, 18
Population Size N = 6
Sample size n = 3
Sampling Distribution of Sample Means
S/No  Samples  Sum of Samples  Mean of Samples  S/No  Samples  Sum of Samples  Mean of Samples 
1  0,3,6  9  3  11  3,6,12  21  7 
2  0,3, 12  15  5  12  3,6,15  24  8 
3  0,3, 15  18  6  13  3,6, 18  27  9 
4  0,3, 18  21  7  14  3,12,15  30  10 
5  0,6, 12  18  6  15  3, 12, 18  33  11 
6  0,6, 15  21  7  16  3, 15, 18  36  12 
7  0,6, 18  24  8  17  6, 12, 15  33  11 
8  0, 12, 15  27  9  18  6, 12, 18  36  12 
9  0, 12, 18  30  10  19  6, 15, 18  39  13 
10  0, 15, 18  33  11  20  12, 15, 18  45  15 
Sampling Distribution of Sample Means
X̅  f  fX̅  fX̅² 
3  1  3  9 
5  1  5  25 
6  2  12  72 
7  3  21  147 
8  2  16  128 
9  2  18  162 
10  2  20  200 
11  3  33  363 
12  2  24  288 
13  1  13  169 
15  1  15  225 
20  180  1788  
∑f =  ∑fX̅ =  ∑fX̅²= 
Mean & Variance of Sampling Distribution
Mean & Variance of Population
X  X² 
0  0 
3  9 
6  36 
12  144 
15  225 
18  324 
54  738 
∑X =  ∑X² = 
Verification Formulas:
Section II Business Mathematics
Solution:
Calculation of Determinant
Calculation of Adj. A
Step 1 Calculation of Minors
Step 2: Calculation of Cofactors
Step 3 Transpose of Cofactors = Adjoint A
Q.6 (a) Solve for x and y: 4x – 3y = 10, 5x – 7y = 6
Solution:
4x – 3y = 10 (i)
5x – 7y = 6 (ii)
Multiply equation (i) by 5 and (ii) by 4, subtract & get:
Put y =2 in equation (i) to get the value of x
Prove:
4x – 3y = 10 (i)
4(4) – 3(2) = 10
10 = 10
(b) The area of a rectangular plot of land fenced all round is 2000 sq. yards and the total length of fencing is 180 sq. yards. Find the length and width of plot.
Solution:
Let the length of the plot be x
Let the width of the plot be y
Area = Length x Width
Area = 2000 sq. yards
Length of fencing = 180 sq. yards = Perimeter
Equation will be:
Put y = 90 – x into equation (i)
Hence
Length of the plot be x = 50
Width of the plot be y = 40
Q.7 (a) Show that the sum of geometric series of ten terms…..
Q.7 (a) Show that the sum of geometric series of ten terms:
Solution:
First we have to find 10^{th} term as:
Applicable Formula
(b) A company offers two alternatives for the payment of salary for the post of a high executive. Either one may receive Rs. 240,000 per year or Rs. 100 in the first month, Rs. 200 in the second month, Rs. 400 in the third month and so on. Which of the two alternatives should be prefer…..
Solution:
First Alternative = 240,000 per year
Second alternative = 100, 200, 400…..12 terms
Applicable Formula
First we have to find 12^{th} term:
Applicable Formula
Since the second option is greater than 1
409500>240,000 So Executive should prefer 2^{nd} option
Q.8 (a) Find the compound interest on Rs. 4500 in 3 years. If the rate of interest is 4% for the first year, 5% for the 2^{nd} year and 6% for the 3^{rd} year.
Solution:
Calculation
Compound Interest = 5208.84 – 4500 = 708.84
(b) Find the accumulated value of Rs. 5000 invested at the end of each quarter for 5 years at 8% compounded quarterly.
Solution:
Case = Ordinary Annuity, Quarterly Case
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Introduction to Statistics Basic Important Concepts
Measures of Central Tendency, Arithmetic Mean, Median, Mode, Harmonic, Geometric Mean
Correlation Coefficient, Properties, Types, Important Formulas for Correlation Coefficient