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STATISTICS AND BUSINESS MATHEMATICS
B.com-1 Private Annual Examination 2015
Compiled & Solved By: JAHANGEER KHAN
B.com-1
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 1
(SECTION – A)
Q.1 (a): Find the distance between the points (1 , 2), (4 , 5).
SOLUTION (1-a): ( ) ( ) ( ) ( ) As we know that
√( ) ( )
Substituting values:
√( ) ( )
√( ) ( )
√
√
√
√
√
√
Q.1 (b): Find the roots of the function .
SOLUTION (1-b): As we know that
√
By substituting values
( ) √( ) ( )( )
( )
√
√
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 2
Hence Solution Set = { , }
Q.2 (a): Find the equation of straight line passing through the points (3 , -2) , (5 , 1).
SOLUTION (2-a): ( ) ( ) ( ) ( ) As we know that
( )
Substituting values:
( ) ( )
( )
( )
( )
( ) ( )
Q.2 (b): Find the derivative of the function (i); √ (ii)
SOLUTION (2-b-i): √
( )
( )
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 3
( )
( )
( )
( )
( )
( )
√
SOLUTION (2-b-ii):
( )
( )( ) ( )( )
( )
( )
Q.3 (a): Solve the equation by Cramer’s rule: ,
SOLUTION (3-a):
|
|
( )( ) ( )( )
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 4
|
|
( )( ) ( )( )
|
|
( )( ) ( )( )
As we know that
Substituting values:
Hence Solution Set = { , }
Q.3 (b): Given two matrices
*
+ , *
+ Show that ( )
SOLUTION (3-b): *
+ *
+
[( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )
]
*
+
*
+
( ) *
+
( ) *
+
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 5
*
+
*
+
*
+ *
+
[( )( ) ( )( ) ( )( ) ( )( )( )( ) ( )( ) ( )( ) ( )( )
]
*
+
*
+
( )
*
+ *
+
(SECTION – B) Q.4 (a): Fill in the blanks: (i) Statistics mainly concerned with __________ data. (ii) Ogive is used to find __________. (iii) The standard deviation is positive square root of ____________. (iv) The limit of coefficient of correlation is _____________.
(v) The base year index is always ______________.
SOLUTION (4-a): (i) Statistics mainly concerned with sample data. (ii) Ogive is used to find Median. (iii) The standard deviation is positive square root of Variance. (iv) The limit of coefficient of correlation is . (v) The base year index is always 100%.
Q.4(b): Find Mean, mode and comments on the shape of distribution 12 18 27 20 17 6
SOLUTION (4-b): 12 5 60 12 18 15 270 30 27 25 675 57 20 35 700 77 17 45 765 94 6 55 330 100
Total 100 ----- 2800 -----
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 6
(
)
(
( ) )
(
)
(
)
( )
Hence Solution Set = { , }
Comment: Since Mean > Mode so, the distribution is skewed to the right
Q.5(a): Calculate Mean absolute deviation about mean: 32 , 28 , 47 , 63 , 75 , 39 , 10 , 60 , 96
SOLUTION (5-a): | |
32 -18 18 28 -22 22 47 -3 3 63 13 13 75 25 25 39 -11 11 10 -40 40 60 10 10 96 46 46
450 0 188
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 7
( ) | |
( )
( )
Q.5(b): Find the variance of the numbers; 1 , 2 , 3 , 4 , 5.
SOLUTION (5-b): 1 1 2 4 3 9 4 16 5 25
15 55
( )
( )
( ) ( )
( )
( )
Q.5(c): Find the coefficient of variation: 3 5 7 9 11 2 5 10 2 1
SOLUTION (5-c): 3 2 9 6 18 5 5 25 25 125 7 10 49 70 490 9 2 81 18 162
11 1 121 11 121 35 20 ----- 130 916
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 8
√
(
)
√ (
)
√ ( )
√
√
Q.6(a): Find Regression line y on x. X 6 2 10 4 8 y 9 11 5 8 7
SOLUTION (6-a): 6 9 36 54 2 11 4 22
10 5 100 50 4 8 16 32 8 7 64 56
30 40 220 214
( )
( ) ( )( )
( ) ( )
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 9
( )
( )( ) ( )
Q.6(b): Calculate Karl Pearson coefficient of correlation: x 1 2 3 4 5 6
y 6 4 3 5 4 2
SOLUTION (6-b): 1 6 1 36 6 2 4 4 16 8 3 3 9 9 9 4 5 16 25 20 5 4 25 16 20 6 2 36 4 12
21 24 91 106 75
√* ( ) +* ( ) +
( ) ( )( )
√* ( ) ( ) +* ( ) ( ) +
√* +* +
√* +* +
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 10
√
Q.6(c): Shift the base year from 1991 to 1995: Year 1991 1992 1993 1994 1995
Price Index % 100 103 105 107 110
SOLUTION (6-c):
1991 100
1992 103
1993 105
1994 107
1995 110
(SECTION – C)
Q.7(a): Find the number of permutations of the words (i) KARIM (ii) ACCOUNTANT
SOLUTION (7-a-i): n
( )
5
( )
5
5
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 11
SOLUTION (7-a-ii):
n
10
10
10
10
Q.7(b): From a group of 5 men and 3 women. How many combinations are possible with 2 men and 1 woman?
SOLUTION (7-b): n
( )
2 men and 1 woman from 5 men and 3 women can be selected into following ways: 5 x 3
( ) ( )
Q.7(c): Two unbiased coins are tossed once: Find the probability of (i) No head (ii) Two tails
SOLUTION (7-c-i):
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 12
* + ( ) * + ( )
( ) ( ) ( )
( )
( )
SOLUTION (7-c-ii): * + ( ) * + ( )
( ) ( ) ( )
( )
( )
Q.8(a): A random sample of 36 students selected from different colleges showed an average number of marks 62 with S.D of 5. Construct 95% confidence interval for the average marks of all students.
SOLUTION (8-a):
(95% Confidence Interval)
5
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 13
√ ( )
√
( √
) ( ) ( √
)
( ) ( ) (
)
( ) ( ) ( ) ( ) ( )
Q.8(b): A plant manufactures 8% of defective items. A sample of 6 is selected. What is the
probability that: (i) At most two are defective (ii) all are defective.
SOLUTION (8-b-i): ( ) ( ) n
( ) ( ) ( ) ( ) ( ) 6 ( )
( ) 6 ( ) ( ) 6 ( )
( )
( ) ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( )
SOLUTION (8-b-i):
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 14
( ) ( ) n
( ) 6 ( )
( ) ( ) ( )( ) ( ) ( )( ) ( )
Q.9(a): Draw all possible samples each of size 2 from population 9, 11, 15, 21. Show that E( )=µ.
SOLUTION (9-a): If Repetition is not Allowed.
S.no Samples Sample Mean ( ) 1. 9 , 11 10 2. 9 , 15 12 3. 9 , 21 15 4. 11 , 15 13 5. 11 , 21 16 6. 15 , 21 18
Total ----- 84
E ( ) = µ
Verified
If Repetition is Allowed.
S.no Samples Sample Mean ( ) 1. 9 , 9 9 2. 9 , 11 10 3. 9 , 15 12 4. 9 , 21 15 5. 11 , 9 10 6. 11 , 11 11 7. 11 , 15 13 8. 11 , 21 16 9. 15 , 9 12
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 15
10. 15 , 11 13 11. 15 , 15 15 12. 15 , 21 18 13. 21 , 9 15 14. 21 , 11 16 15. 21 , 15 18 16. 21 , 21 21
Total ----- 224
E ( ) = µ
Verified
Q.9(b): A random variable follows Poisson Dist. With mean is 1.5 Find: (i) ( ) (ii) ( )
SOLUTION (9-b-i): ( )
( )
( ) ( ) ( ) ( )
( ) ( )
( )
( )
( )
( )
( )
( ) ( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
SOLUTION (9-b-ii): ( )
( )
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 16
( ) ( )
( )
( ) ( ) ( )
( )
( )
Q.10(a): A random sample of size 16 was found to have and . Test the hypotheses against at
SOLUTION (10-a):
1. H0 : 2. HA : 3. α 1% OR 0.01 4. Use –test:
√ ⁄
√ ⁄
5. Critical Value:
6. Decision/Conclusion: Since | | > | | so reject H0 and accept HA. it means that population
mean is not equal to 32. Q.10(b): Given two random samples of size with and
Test the Hypotheses against at .
SOLUTION (10-b):
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 17
1. H0 : 2. HA : 3. α 0.05 4. Use –test:
√( )
( )
√( )( )
( )( )
√( )( ) ( )( )
√
√
√
( )
√
( )
√
√
( )
Compiled & Solved By: JAHANGEER KHAN
STATISTICS AND BUSINESS MATHEMATICS - 2015 B.com1 Private Annual Examination 2015
pg. 18
5. Critical Value:
6. Decision/Conclusion:
Since | | > | | so reject H0 and accept HA. it means that .
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