QUANTATIVE TECHNIQUES WORKSHEET SEMESTER -I

Transcription

DATTA MEGHE INSTITUTEOF MANAGEMENT STUDIES
SDMP CAMPUS, Atrey Layout, Nagpur -22
QUANTATIVE TECHNIQUES WORKSHEET
SEMESTER -I
D.M.I.M.S-----Quantative Techniques Worksheet---Semester-I
Page 1
UNIT I: MEASURES OF CENTRAL TENDENCY AND DISPERSION
CLASS WORK PROBLEMS
1. Calculate mean from the following data:
R No. 1 2 3 4 5 6 7 8 9 10
Marks 40 50 55 78 58 60 73 35 43 48
(soln - = 54 marks)
2. Calculate mean from the following data:
Value
1 2 3 4 5 6 7 8 9 10
Frequency 21 30 28 40 26 34 40 9 15 57
(soln - = 5.72)
3. From the following find out the mean profits:
Profits per shop Rs.
Number of Shops
100 – 200
10
200 – 300
18
300 – 400
20
400 – 500
26
500 – 600
30
600 – 700
28
700 – 800
18
(soln -
= 486 )
4. The following are the marks scored by 7 students, find out the median marks:
Roll Number
Marks
1
45
2
32
3
18
4
37
5
65
6
38
7
46
(soln - M = 45)
5. Find out the median from the following:
57
58
61
42
38
65
72
66
(soln - M = 59.5)
6. Locate median from the following:
Size of Shoes
Frequency
5
10
5.5
16
6
28
6.5
15
7
30
Page 2
7.5
8
40
34
(soln - M = 7)
7. Calculate the median from the following table:
Marks
Frequency
10 – 25
6
25 – 40
20
40 – 55
44
55 – 70
26
70 – 85
3
85 – 100
1
(soln - M = 48.18 marks)
8. The following table shows age distribution of persons in a particular region:
Age (years)
No. of Persons
Age (years)
No. of Persons
(‘000)
(‘000)
Below 10
2
Below 50
14
Below 20
5
Below 60
15
Below 30
9
Below 70
15.5
Below 40
12
70 and above
15.6
Find the median age.
(soln - M = 27 years)
9. Calculate the mode from the following:
Size
Frequency
10
10
11
12
12
15
13
19
14
20
15
8
16
4
17
3
18
2
(soln - Z = 13)
10. Calculate the mode from the following:
Size of Item
Frequency
0–5
20
5 – 10
24
10 – 15
32
15 – 20
28
20 – 25
20
25 – 30
16
30 – 35
34
35 – 35
10
40 – 45
8
(soln - Z = 13.33)
Page 3
11. Calculate the semi inter-quartile range and quartile coefficient from the following:
Age in Years
No. of members
20
3
30
61
40
132
50
153
60
140
70
51
80
3
(soln – Q.D =10 years, Co-eff. Q.D =0.2)
12. Calculate the range and semi-inter quartile range of wages:
Wages (Rs.)
Labours
30 – 32
12
32 – 34
18
34 – 36
16
36 – 38
14
38 – 40
12
40 – 42
8
42 – 44
6
Also calculate the quartile coefficient of dispersion.
(soln – Q.D =2.85, Co-eff. Q.D =0.08)
13. Calculate mean deviation from mean and median for the following data.
100
150 200 250 360 490 500 600 671
Also calculate coefficient of mean deviation.
(soln – M.D =174.44, Co-eff. M.D =0.48)
14. Calculate mean deviation from the following data:
x
2
4
6
8
10
f
1
4
6
4
1
(soln – M.D =24, Co-eff. M.D =1.5)
15. Calculate the mean deviation from mean for the following data:
Class interval
2-4
4-6
6-8
80-10
Frequency
3
4
2
1
(soln – M.D =5.2, Co-eff. M.D =1.48)
16. Calculate the standard deviation from the following data:
14
22
9
15
20
17
12
11
(soln – σ = 4.18)
Page 4
17. Find the missing frequency from the following data, given mean is 16.82.
Marks
Frequency
0–5
10
5 – 10
12
10 – 15
16
15 – 20
?
20 – 25
14
25 – 30
10
30 – 35
8
(soln – missing frequency is 18.235 or 18)
18. Calculate standard deviation from the following:
Marks
No. of Students
10
8
20
12
30
20
40
10
50
7
60
3
(soln – σ = 13.45)
19. Compute the standard deviation and mean deviation from the following:
Class (x)
0-10
10-20 20-30 30-40 40-50 50-60
Frequency (f)
8
12
17
14
9
7
(soln – σ = 16.67)
60-70
4
20. The daily temperature recorded in a city in Russia in a year is given below:
Temperature ‘C’
No. of Days
-40 to -30
10
-30 to -20
28
-20 to -10
30
-10 to 0
42
0 to 10
65
10 to 20
180
20 to 30
10
Calculate the mean and standard deviation.
(soln – σ = 14.73ºC)
21. The index numbers of prices of cotton and coal shares in 2007 were as under:
Month
Index number of prices of Index number of prices of
cotton shares
coal shares
January
188
131
February
178
130
March
173
130
Page 5
April
164
129
May
172
129
June
183
120
July
184
127
August
185
127
September
211
130
October
217
137
November
232
140
December
240
142
Which of the two shares do you consider more variable in price?
(soln- σ = 12.28%, 4.42%)
22. In two factories A and B, engaged in the same industrial area the average weekly wages (in
rupees) and the standard deviations are as follows:
Factory
Average
Standard Deviation
No. of Workers
A
34.5
3
476
B
28.5
4.5
524
a. Which factory A or B pays out a larger amount as weekly wages?
b. Which factory A or B has greater variability in individual wages?
(soln- 15.79%)
SELF PRACTICE PROBLEMS
Q.1
Calculate Mean from the following data.
Years
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
Q.2
Production
12
17
19
23
19
25
28
19
27
26
28
12
14
18
19
Height in inches
60
62
63
64
65
66
No. of Persons
5
13
18
20
21
30
Page 6
67
68
69
70
23
12
4
2
Q.3 Calculate Mean from the following data.
Income in Rs.
0-100
100-200
200-300
300-400
400-500
500-600
600-700
700-800
No. of Persons
5
10
12
16
27
10
15
5
Marks
10-20
20-30
30-40
40-50
50-60
60-70
70-80
80-90
90-100
100-110
No. of Students
1
3
6
11
9
11
7
6
3
1
Income
1-50
51-100
101-150
151-200
201-250
251-300
301-350
351-400
401-450
No. of Persons
10
22
29
40
55
32
21
17
12
Marks
0-9
10-19
20-29
30-39
40-49
50-59
60-69
70-79
No. of students
2
4
23
30
40
45
35
25
Page 7
Central Value
7
12
17
22
27
32
37
Q.8
Frequency
8
18
27
21
10
28
7
Income
100-200
200-300
300-400
400-500
500-600
600-700
700-800
800-900
900-1000
Frequency
10
5
15
10
20
20
9
6
5
Q.9
The no. of run score by Kapil Dev and Gavaskar during 10 one day international matches
th
of 4 world cup shown below –
Kapil Dev Gavaskar
5
40
20
35
90
60
76
62
102
58
90
76
6
42
108
30
20
30
16
20
a) Who is the better run getter?
b) Who is more consistent (stayable)?
Q.10
In 10 test matches Gavaskar and Vengasarkar scored the following runs –
Gavaskar
Vengasarkar
65
73
105
17
110
92
27
99
30
13
70
2
80
13
90
94
95
97
97
98
Page 8
Q.11
Q.12
Q.13
Calculate the standard deviation from the following data.
Income in Rs. 2542 2522 2534 2532 2545 2566 2550
Calculate the standard deviation from the following data:Marks
20
22
24
26
28
30
No. of students
9
13
19
22
24
21
34
14
36
10
No. of students
6
21
44
27
4
Find out range and it’s co-efficient
Size of Item
15
20
22
25
30
32
35
Q.15
32
18
Calculate the standard deviation
Marks
10-25
25-40
40-55
55-70
70-85
Q.14
2530
Frequency
2
5
7
8
3
5
6
Find out range and it’s co-efficient
Marks
10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90
Frequency
2
3
7
20
25
10
8
2
HOME WORK PROBLEMS
Q.1
Sr. No
1
2
3
4
5
6
7
8
9
10
Q.2
Income
12
14
16
14
14
18
16
10
16
14
Wages
2
3
4
5
6
7
No. of employment
7
13
17
20
21
19
Page 9
8
9
10
11
12
Q.3
Wages in Rs.
3
10
15
20
25
30
35
40
Q.4
16
11
6
4
2
No. of works
5
5
15
10
15
10
3
2
Wages
10
15
20
25
30
35
40
45
50
55
No. of workers
5
7
9
13
17
17
13
9
7
2
Income
300-325
325-350
350-375
375-400
400-425
425-450
450-475
475-500
Frequency
5
17
80
227
326
248
88
9
Class
1-3
3-5
5-7
7-9
9-11
11-13
13-15
15-17
Frequency
6
53
85
56
21
16
4
4
Q.7 Calculate Median from the following data.
No. of eggs
No. of Hens
Page 10
0-29
30-59
60-89
90-119
120-149
150-179
180-209
210-239
240-269
270-300
3
4
12
33
69
92
50
25
11
1
Marks
0-10
10-20
20-22
22-27.5
27.5-30
30-40
40-47
47-50
50-60
60-65
65-70
No. of students
10
15
5
17
3
25
8
2
10
3
2
Income
0-10
10-20
20-30
30-35
35-40
41-50
50-60
60-66
66-6+8
68-70
70-80
80-90
No. of Pensions
1
4
10
12
10
30
35
6
3
1
7
1
Size
10-15
16-17.5
17.5-20
22-30
30-35
35-40
45 and onwards
Frequency
10
15
17
25
28
30
40
Q.11 Calculate Mode from the following data.
Mid Value
15
25
35
45
55
Frequency
5
10
14
21
21
Page 11
55
75
85
Q.12
Calculate Mode from the following data.
Age
20
30
40
50
60
70
80
Q.13
No. of students
3
61
132
154
140
51
2
Calculate Mode from the following data.
Marks
More than 70%
More than 60%
More than 50%
More than 40%
More than 30%
More than 20%
Q.14
15
11
3
Students
7
18
40
40
63
65
Following are the share prices of two companies
Amar & Co
318
322
325
312
324
315
308
319
Vijay & Co
2542
2522
2534
2532
2545
2530
2566
2550
Q.15 Prices of a particular commodity in five years in two cities are given belowPrice in city ‘A’
20
22
19
23
16
Price in city ‘B’
10
20
18
22
25
Find from the above data the city which had more stable prices?
Q.16 Find out standard deviation by direct method.
Income in Rs. 3000 4000 4200 4400 4600 4800 5800
Q.17 Calculate the standard deviation from the following
dx
-4
-3
-2
-1
0
+1
+2
+3
Q.18
Frequency
3
6
9
19
24
25
9
5
Find out range and its coefficient
Age
30
Frequency
2
Page 12
40
50
60
70
80
90
100
110
Q.19
3
9
10
17
12
3
3
3
Find out range and its coefficient from the following data
Marks
Less than 10
Less than 15
Less than 20
Less than 25
Less than 30
Less than 35
Less than 40
Student
7
12
18
24
35
49
61
UNIT II: REGRESSION ANALYSIS
CLASS WORK PROBLEMS
Q.1) Calculate the coefficient of correlation and obtain the lines of regression for the
following:
Year
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
No. of
15
18
17
20
23
25
29
33
36
40
Students
Obtain an estimate of y which should correspond to the average x=6.2
(ans –y=013.14)
Q.2) Find the regression equations from the following data
Age of Husband 18 19
20
21
22
23
Age of wife
17 17
18
18
19
19
24
19
Q.3) Given data
X
1
Y
6
3
5
5
1
3
0
2
0
1
1
1
2
7
1
25
20
26
21
27
22
a) Fit a regression line of Y on X and hence predict “X” if Y=2
b) Fit a regression line of X and Y and hence predict Y if X=5
c) Karl Pearson’s coefficient of correlation
Q.4) Given
‘X’
Mean
18
Standard deviation 14
‘Y’
100
20
Page 13
‘r’ between X and d Y = 0.8
a) Find out most probable value of Y if X is 70 and most probable value of X if Y = 90
b) If regression coefficient are 0.8 and 0.6 what would be the value of coefficient of
correlation.
Q.5) From the following data, write down the equations to the regression lines
Average
S.D
Marks in Mathematics
48.4
8.4
Marks in English
35.6
10.5
Correlation co-efficient = 0.8
Estimate the marks in mathematics corresponding to 70 marks in English
HOME WORK PROBLEMS
Q.1) From the following data of the age of husband and age of wife form two regression
lines and calculate the husband’s age when the wife’s age is 16.
Age of Husband
36
23
27
28
28
29
30
31
33
35
Age of wife 29
18
20
22
27
21
29
27
29
28
Q.2) Find the regression equation of X on Y and Y on X from the following data
X
10
20
30
40
50
60
Y
15
5
10
25
30
40
Obtain the estimate of Y when X = 22
Q.3) The following scores were worked out from a test in Mathematics and English in an
annual examination.
Marks in Mathematics
Marks in English
Mean
39.5
47.5
Standard deviation
10.8
16.8
Coefficient of correlation = 0.42
a) Find both regression equations
b) Estimate the value of Y when X=50
c) Value of X when Y = 30
Q.4) The coefficient of correlation between production and sales of a certain product was
found to be 0.8 over a period of 50 weeks. Average production per week was 25 tonnes and
average sales per week were Rs. 20 lakhs. Coefficient of variation was 16 and 25
respectively. By using linear regression, estimate.
a) The production when sales would amount to Rs. 30 lakhs
b) The sales when the production would become 25 tonnes.
Q.5) Obtain the regression equation of Y on X and X on Y and the value of coefficient of
correlation from the following table giving the marks in Accountancy and statistics
Marks
in
Marks in Accountancy
statistics
5-15
15-25
25-35
35-45
Total
0-10
1
1
2
Page 14
10-20
20-30
30-40
40-50
Total
3
1
5
6
8
3
18
5
9
9
4
27
1
2
3
4
10
5
20
15
8
60
Q.6) Following is the distribution of students according to their height and weight
Height
in
Weight in Lbs
inches
90-100
100-110
110-120
120-130
50-55
55-60
60-65
65-70
4
6
6
3
7
10
12
8
5
7
10
6
2
4
7
3
Q.7) Calculate correlation coefficient and regression coefficient for the following data
X
2
4
6
8
10
12
14
Y
4
2
5
10
4
11
12
Find the estimate of y when x = 13
Q.8) Calculate the following:
a) The regression equation of X on Y and Y on X from the following data.
b) Estimate X when Y = 20
X
10
12
13
17
18
Y
5
6
7
9
13
Q.9) From the following find: the two lines of regression and correlation coefficient
Marks in 25
28
35
32
31
36
29
38
34
economics
Marks in 43
46
49
41
36
32
31
30
33
statistic
32
39
Page 15
Unit III: Correlation Analysis
CLASS WORK PROBLEMS
23. Calculate the coefficient of correlation and obtain the lines of regression for the
following:
x
1
2
3
4
5
6
7
8
9
y
9
8
10 12 11 13 14 16 15
(Ans –y=013.14) (problem 13.7, page 481)
24. Calculate coefficient of correlation from the following data:
x
12 9
8
10 11 13 7
y
14 8
6
9
11 12 3
(Ans – + 0.95) (problem 12.2, page 402)
25. Find if there is any significant correlation between the heights and weights given
below:
Height in inches
57
59
62
63
64
65
55
58
57
Weight in lbs
113 117 126 126 130 129 111 116 112
(Ans – + 0.98) (problem 12.3, page 402)
26. Find out the coefficient of correlation in the following case:
Height of Father (in inches)
65
66
67
67
68
69
71
73
Height of Son(in inches)
67
68
64
68
72
70
69
70
(Ans – 0.472) (problem 12.5, page 405)
27. Following are the rank obtained by 10 students in two subjects. Statistics and
Mathematics. To what extent the knowledge of the students in the two subjects is
related?
Statistics
1
2
3
4
5
6
7
8
9
10
Mathematics
2
4
1
5
3
9
7
10
6
8
(Ans – +0.76) (problem 12.13, page 417)
28. A random sample of 5 college students is selected and their grades in Mathematics and
Statistics are found to be:
1
2
3
4
5
Mathematics
85
60
73
40
90
Statistics
93
75
65
50
80
(Ans – +0.8) (problem 12.14, page 418)
29. From the following data calculate the rank correlation coefficient after making
adjustment for tied ranks.
x
48
33
40
9
16
16
65
24
16
57
y
13
13
24
6
15
4
20
9
6
19
Page 16
(ans – 0.733) (problem 12.15, page 419)
30. Calculate coefficient of correlation between the marks obtained by a batch of 100
students in Accountancy and Statistics as given below:
Marks in Marks in Accountancy
Statistics 20-30
30-40
40-50
50-60
60-70
Total
15-25
5
9
3
17
25-35
10
25
2
37
35-45
1
12
2
15
45-55
4
16
5
25
55-65
4
2
6
Total
5
20
44
24
7
100
(ans – 0.7953) (problem 12.10, page 411)
31. Calculate coefficient of correlation between the marks obtained by a batch of 100
students in Accountancy and Statistics as given below:
Income (Rs.)
Savings (Rs.)
50
100
150
200
Total
400
8
4
12
600
12
24
6
42
800
9
7
2
18
1000
10
5
15
1200
9
4
13
Total
8
25
50
17
100
Calculate the coefficient of correlation between income and savings
(Ans – 0.523) (problem 12.18, page 424)
32. Compute the coefficient of correlation between dividends and price of Securities as
given below:
Secutiry Price
Amount Dividends (in Rs.)
(Rs.)
6-8
8-10
10-12
12-14 14-16 16-18
130-140
1
3
4
2
120-130
1
3
3
3
1
110-120
1
2
3
2
100-110
2
3
2
90-100
2
2
1
1
80-90
3
1
1
70-80
2
1
(Ans – 0.755) (problem 12.36, page 438)
SELF PRACTICE PROBLEM
Q.1 – Find out correlation by the help of following information.
Import – 21, 29, 20,22,18,16,20,24,26
Price Index – 125,128,120,118,118,125,130,125,120
Page 17
Q.2 – Find out correlation
Age of Husbands – 20, 25, 30,35,40,45,50,55,60
Age of wives – 16, 20, 23,25,33,38,46,50,55
Q.3 – Calculate coefficient of correlation from the following table
x/y
30-40 4050-60 60-70
Total
50
30-40
3
1
1
5
40-50
2
6
1
2
11
50-60
1
2
2
1
6
60-70
1
1
1
3
Total
6
10
5
4
25
Q.4 – Find out coefficient of correlation and probable error. Calculate coefficient of
correlation from the following table
Age of wives
Age of
1525-35 35455565Total
Husband
25
45
55
65
75
15-25
1
1
2
25-35
2
12
1
15
35-45
4
10
1
15
45-55
3
6
1
10
55-65
2
4
2
8
65-75
1
2
3
Total
3
17
14
9
6
4
53
Q.5 –Calculate correlation and probable error
Salary 6070-80 80-90 90-100 100-110
70
Age
20-30 4
3
1
30-40 2
5
2
1
40-50 1
2
3
2
1
50-60 1
3
5
2
60-70 1
1
5
Total 7
11
10
9
8
Total
8
10
9
11
7
45
Q.6 Find out rank coefficient of correlation
X – 80 91 99 71 61 81 70 59
Y – 123 135 154 110 105 134 121 108
Q.7 Find out correlation
X – 78 90 97 69 59 79 68 67
Y – 125 37 156 112 107 136 123 108
X – 75 88 95 70 60 80 71 50
Page 18
Y – 120 134 150 115 110 140 142 100
HOME WORK PROBLEMS
Q.1 – Find out correlation and probable error
Year - 1925,1926,1927,1928,1929,1930,1931,1932,1933,1934
Av. Daily – 368,384,385,361,347,384,395,403,400,385
No. of workers (in hundred)
Consumer – 32,21,24,20,22,26,29,
(in lakh) 26,28,27
Q.2 - Find out correlation and probable error
City – 1,2,3,4,5,6,7,8,9,10,11,12
Employment – 22,31,90,82,43,65,59,16,61,48,35,50
Annual sales – 250,200,980,850,710,280,680,180,670,920,190,960
Q.3 Find out correlation and probable error
X – 8.0, 7.8, 7.5, 7.5, 6.8, 6.7, 6.0, 5.9
Y – 1.2, 1.3, 1.4, 1.4, 1.4, 1.6, 1.5, 1.7
Q.4 – Find out coefficient correlation
Marks in B.O
55-65
45-55
35-45
25-35
15-25
20-30
5
30-40
8
12
Marks in statistics
40-50
50-60
24
15
29
-
60-70
6
1
-
Q.5 – Find out coefficient of correlation
Test “B”
140-149
150-159
160-169
170-179
180-189
110-119
27
7
9
5
1
120-129
20
28
18
9
-
Q.6 – Find out coefficient correlation
Total cultivable area (in area)
Area of Wheat 0 500
1000
0
12 6
200
2 18
4
400
4
7
600
1
800-1000
Test “A”
130-139
140-149
10
3
20
8
28
10
10
10
1
7
1500
2
3
2
1
150-159
3
5
6
160-169
5
1
2000-2500
1
1
2
Page 19
X – 60 77 70 80 77 75 70 70 70 60
Y – 50 69 65 75 69 65 65 60 55 50
X – 81 79 80 78 79 75 77 73 74
Y – 79 78 81 77 77 73 78 75 73
X – 44 37 40 37 32 30 26 25 22 20
Y – 48 42 41 30 32 40 29 26 25 23
Q.10 Find Karl Pearson’s coefficient of correlation from the following dataWages 100 101 102 102 100 99 97 98 96 95
Cost
98 99 99 97 95 92 95 94 90 91
of
living
Q.10 Find coefficient of correlation in the following case
Height
65 66 67 68 69 71 73
of
father
(in
inches)
Cost of
67 68 64 72 70 69 70
living
Q.11 A random sample of 5 college students is selected and their grades in Mathematics
and Statistics are found to be –
Statistics
1 2 3 4 5 6 7 8 9 10
Mathematics 2 4 1 5 3 9 7 10 6 8
Q.12 From the following data calculate the rank correlation coefficient after making
adjustment for tied ranksX 48 33 40 9 16 16 65 24 16 57
Y 13 13 24 6 15 4 20 9 6 19
Q.13 Calculate the Karl Pearson’s coefficient of correlation from the following data-(Winter
2002)
X
1 2 3 4 5 6 7
Y
3 5 6 8 10 11 13
Q.14 Calculate the Karl Pearson’s coefficient of correlation between the advertising
expenses and the sales given below-(Summer 2003)
Advertising 39 65 62 90 82 75 25 98 36 78
exp (Rs)
Sales (Rs) 47 53 58 86 62 68 60 91 51 84
Page 20
Q.15 Calculate the coefficient of correlation for the following data of marks obtained by 10
students in QTM and MSD (Winter 2005)
Roll 1 2 3 4 5 6 7 8 9 10
No
QTM 80 38 9 5 30 74 84 91 60 66
MSD 36 06 17 14 25 10 32 00 03 20
Q.16 Find Karl Pearson’s coefficient of correlation between sales and expenses of following
10 firms. Interpret your result: (Winter 2012)
Firms
1 2 3 4 5 6 7 8 9 10
Sales
50 50 55 60 65 65 65 60 60 50
Expense 11 13 14 16 16 15 15 14 13 13
Q.17 Ten competitors in a beauty contest are ranked by three judges in following order.
Use spearman’s rank correlation coefficient to determine which pair of judges has nearest
approach to common test in beauty (Winter 2012)
Judge 1
1 6 5
10
3
2
4
9
7
8
Judge 2
3 5 8
4
7
10 2
1
6
9
Judge 3
6 4 9
8
1
2
3
10 5
7
Page 21
UNIT IV: TIME SERIES ANALYSIS AND FORECASTING
CLASS WORK PROBLEMS
1 Draw a trend line by the method of semi averages.
Year
2002 2003 2004 2005 2006
Sales (000)
60
75
81
110
106
(problem 15.2, page 597)
2007
120
2 Draw a trend line by the method of semi averages.
Year
2001 2002 2003 2004 2005
Sales (000)
110
105
115
112
120
2006
118
3
2007
130
Calculate the coefficient of correlation and obtain the lines of regression for the
following:
Year
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
No. of
15
18
17
20
23
25
29
33
36
40
Students
(Ans –y=013.14) (problem 15.4, page 599)
4
The following figures relate to the profits of a commercial concern for 8 years
Year
2000 2001 2002 2003 2004 2005 2006 2007
Profits (Rs)
15420 14470 15520 21020 26120 31950 35370 34670
Find the trend of profits by the method of moving averages.
5
Year
1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Production 464 515
518
467
502
540
557
571
586 612
(in million
lb)
6
Calculate trend values by the method of least square from the data given below and
estimate the
Year
2003 2004 2005 2006 2007
Sales of Co. A (Rs. 70
74
80
86
90
Lakhs)
Page 22
7
Calculate trend value from the following data using the method of least square
Year
2002 2003 2004 2005 2006 2007
Production 7
9
12
15
18
23
8
Year
2001- 2002- 2003- 2004- 2005- 20062002 2003 2004 2005 2006 2007
Asset
83
92
71
90
169
191
Also estimate the figures for 2011-12,(Ans Rs. 285.50 crores) (problem 15.9, page 605)
Q.1 – Draw a trend line by the method of semi – averages
Year
1997 1998 1999 2000 2001
2002
Sales
60
75
81
110
106
120
(‘000)
Q.2 – Draw a trend line by the method of semi –averages
Year
1996
1997
1998
1999
2000
2001
2002
Sales
110
105
115
112
120
118
130
(‘000)
Q.3 – Calculate three yearly moving averages of the following data:
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
Year
No. of
15
16
17
20
23
25
29
33
36
40
students
Q.4 – Assuming a four yearly cycle calculate the trend by the method of moving averages
from the following data relating to the production of tea in India
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
Year
Prod.
464 515 518 467 502 540 557 571 586 612
(Quintals
Q.5 – Calculate trend value by the method of least square from the data given below and
estimate the sales for 2006
Year
2000
2001
2002
2003
2004
Sales (Rs.
76
74
80
86
90
Page 23
Lakhs)
Q.7 - Calculate trend value from the following data using the method of least square.
Year
1994
1995
1996
1997
1998
1999
Prod
7
9
12
15
18
23
(Kg)
Q. 8 – Calculate the trend values by the method of least squares from the data given below
and estimate the sales for 1983 : (Winter 2003)
Year
1976
1977
1978
1979
1980
Sales (Rs.
Lakhs)
70
74
80
86
90
HOME WORK PROBLEMS
Q.1 – Calculate three yearly and five yearly moving averages of the following data
Year
Prod (Kg)
Year
Prod
(Kg)
1991
15
1998
56
1992
21
1999
63
1993
30
2000
70
1994
36
2001
74
1995
42
2002
82
1996
46
2003
90
1997
50
2004
95
Q.2 – Calculate five yearly moving averages of the following data;
Year
Students
Year
Students
1991
332
1998
427
1992
317
1999
428
1993
357
2000
407
1994
392
2001
438
1995
402
2002
450
1996
405
2003
470
1997
410
2004
480
Q.3 – Calculate three yearly and four yearly moving averages of the data given below to
obtain trend value: (Summer 2005)
Year
1
2
3
4
5
6
7
8
9
10
Figure 110 104 78 105 109 120 115 110 114 122
Page 24
Year
11 12 13 14 15 16 17 18 19 20
Figure 130 127 122 118 130 140 135 130 127 135
Year
21 22 23 24 25 26 27 28 29 30
Figure 146 142 138 135 145 155 150 148 143 156
Q.4 – Below are given the figures of production (in million tones) of a sugar factory:
Year
2000 2002 2003 2004 2005 2006 2009
Prod
77
88
94
85
91
98
90
(Million
tones)
Fit a straight line by the method of least square and tabulate the trend values.
Q.5 - (Winter 2012) – Fit a trend line equation using the least square method to estimate
production for the year 1973 & 1985:
Year 1975 1977 1979 1981 1982
Prod 18
21
23
27
16
Q.6 – (Winter 2012) – From the following series of annual data, find the trend by method of
semi-averages. Also estimate values for 2000.
Year 1990 1991 1992 1993 1994 1995 1996 1997 1998
Value 170 231 261 267 278 302 299 298 340
Page 25
UNIT V: PROBABILITY AND STATISTICAL DECISION THEORY
CLASS WORK PROBLEMS
1 Union of sets A and B is the set of all elements which either in A or in B or both. It is
denoted AυB
2
Intersection of two sets A and B denoted A∩B, is defined as a set whose elements
belong to both A and B symbolically
3
Two coins are tossed simultaneously. What is the probability of getting a head and a
tail?
(Ans – 4/52 =1/13) (problem 18.22, page 740)
4
An urn contains 8 white and 3 red balls. If two balls are drawn at random. Find the
probability that (a) both are white, (b) both are red (c) one is of each colour
(Ans – 24/55) (problem 18.26, page 742)
5
Two cards are drawn from a pack of cards at random. What is the probability that it
will be (a) a diamond and a heart, (b) a king and a queen (c) two kings
(Ans – 24/55) (problem 18.24 & 18.26, page 740 & 743)
6
The probability that X and Y will be alive ten years hence is 0.5 and 0.8 respectively.
What is the probability that both of them will be alive ten years hence?
7
A university has to select an examiner from a list of 50 persons, 20 of them women
and 30 men, 10 of them knowing Hindi and 40 not, 15 of them being teachers and
the remaining 35 not. What is the probability of the University selecting a Hand
knowing woman teacher?
8
A ball is drawn at random from a box containing 6 red balls, 4 white balls and 5
blue balls. Determine the probability that it is:
i) Red, ii) White, iii) Blue, iv) Not Red and v) Red or White
Page 26
9
You note that your officer is happy on 60% of your calls, so you assign a probability
of his being happy on your visit as 0.6 or 6/10. You have noticed also that if he is
happy, he accedes to your request with a probability of 0.4 or 4/10, whereas if he is
not happy he accedes to the request with a probability of 0.1 or 1/10. You call one
day and he accedes to your request. What is the probability of his being happy?
(Ans – 0.857) (Problem18.39, page 750)
10 Box I contains three defective and seven non-defective balls and Box. If contains
one defective and non-defective balls. We select a box as random and then draw one
ball at random from the box.
a. What is the probability of drawing a nen-defective ball?
b. What is the probability of drawing a defective ball?
c. What is the probability that box 1 was chosen, given a defective ballis drawn?
(Problem 18.40, page 750)
1. Find out the number of ways 6 books in a library can be arranged taking all books at
a time.
Soln:- [720]
2. A Company received 9 applications for 3 posts. Find out, in how many ways 3 people
can be selected for the nine applications.
Soln:- [504]
3. A coin is tossed 7 times. Find the probability of obtaining (i) 6 Heads (ii) 7 heads and
(iii) 5 or more Heads.
Soln:- [P (6) =0.055, P (7) =0.0078,
Probability of getting more than 5 heads=0.227]
4. Find out the probability of getting (a) Three Heads (b) at least Two Heads and (c) at
least 1 Head, when 5 coins are tossed simultaneously.
Soln:- [P (3) = 0.313, P (2) = 0.813, P(1)
= 0.969]
5. The average percentage of pass in a test paper is 70. Find out the probability that
out of a group of 8 students at least 5 passed in the examination.
Soln:- [P (5) = 0.846]
6. The probability of success in an examination of a village is 0.3. What is the
probability that out of 7 students (i) none (ii) one (iii) two and (iv) at least one will
be succeed in the examination?
Soln:- [Probability of none success =0.082, Probability of one success
=0.245,Probability of two success =.0315,Probability of at least on success =
0.918]
Page 27
7. 15 coins are tossed at a time. Find out the probability in a single tossing for the
following outcomes. i) Less than 5 heads
ii) 12 or more heads iii) at least 10 heads
iv) no heads
8. An Urn contains 5 white, 3 red and 7 black balls. Find out the probability of getting a
white or red ball in a single draw.
9. A card is drawn from a well shuffled pack of 52 cards. What is the probability of the
card being either red or a jack?
Soln:- [7/13]
10. A box contains 15 balls numbering from 1 to 15. Find the probability that a ball
selected at random is a ball with number that is a multiple of 3 or 5.
Soln:- [7/15]
11. The probability that a contractor will get a building contract is ¼ and the probability
of not getting a road contract is 2/3. If the probability of getting at least one contract
is 2/5, find out the probability that he will get either of the two contracts.
Soln:- [11/60]
12. Two students A and B are independently solving a problem. The probability of A
solving the problem is 45 and probability of B solving it is 1/3. What is the
probability that both of them will solve the problem?
Soln:- [4/15]
HOME WORK PROBLEMS
1 A box contains 5 red, 4 black and 3 white balls. What is the probability of getting a
white, a black and a red ball?
Soln:- [5/144]
2 A bag contains 6 white and 4 red balls. The balls are drawn simultaneously without
replacement of the first ball. What is the probability that both balls are of white
colour?
Soln:- [1/3]
3 From a well shuffled pack of 52 cards, 2 cards are drawn at random. What is the
probability that both of them are Queen Cards? Assume that there is no replacement
of the card after the first draw.
Soln:- [3/663]
4 Two persons are competing for a manager post in a company. The probabilities that
the first and second will win are 0.7 and 0.3 respectively. If the first person wins, the
probability of opening a new branch is 0.8 and the corresponding probabilities is 0.4
if the second person wins. What is the probability that the new branch is opened?
Soln:- [0.68]
Page 28
5 An urn contains 8 red and 3 blue marbles. If 2 marbles are drawn at random, find
the probability that (i) both are red and (ii) one of them is red and the other is blue?
Soln:- [Probability of getting 2 red marbles=28/55, Probability
of getting one red marble and one blue marble 24/55]
6 A bag contains 7 white and 2 black balls. Two balls are drawn in succession at
random. What is the probability that one of them is white and the other is black?
Soln:- [14/81]
7 Explain the theorem of total probability he probability that a job applicant for the
post of an accountant with a post graduate degree is 0.3, that he has both is 0.2. Out
of 3000 applicants what number would have either a post has some previous
experiences is 0.7 and the mobility that he has both graduate degree and previous
experience.
Soln:- [2,400]
8 A bag contains 7 white and 9 black balls. Two balls are drawn in succession at
random. What is the probability that one of them is white and the other is black ?
Soln:- [21/40]
9 Two cards are drawn randomly from the well shuffled pack of cards. Find the
probability that (i) Both are diamond cards (ii) One is an Ace and the other is a King
and (iii) both are not club cards.
Soln:- [1/17,4/663,16/17]
10 A bag contains 4 red, 3 white and 2 black balls. Two balls are drawn at a time. What
is the probability that both are either red or white balls?
Soln:- [1/4]
11 The following probabilities are to be computed from a well-shuffled pack cards. (i)
The probability of drawing an Ace and Club cards. (ii) The probability of drawing an
Ace and Club cards. (ii) The probability of drawing a Heart card and a Picture card.
(iii) The probability of drawing a TEN given that a Spade card is drawn. (iv) The
probability of drawing either a King or a Queen or a Diamond card. (v) The
probability of drawing a Picture and an Ace.
Soln:- [1/51, 13/204, 1/52, 0.42, 1/51]
12 In a bag there are 4 white and 10 yellow balls. Two balls are drawn at random. What
is the probability that of these two balls one is white and the other is yellow?
Soln:- [40/91]
13 From a set of 20 discs numbered 1 to 20, one disc is chosen randomly. Find the
probability that the number on it is divisible by (i) 3 and (ii) 5 or 7.
Page 29
UNIT VI: LINEAR PROGRAMMING AND PROBLEM FORMULATION
CLASS WORK PROBLEMS
1 A manufacturer of furniture makes two products chairs & tables. Processing of
these products is done on two machines A & B. A chair requires 2 hrs on Machine A
& 6 hrs on Machine B. A table requires 5 hrs on machine A and no time on machine
B.There are 16 hrs of time per day available on machine A & 23 hrs on machine B.
Profit gained by the manufacturer from a chair & table is Rs 2 & Rs 10 respectively.
What should be daily production of each of the two products.
2 An animal feed company must produce 200 kg of mixture of consisting of
ingredients X1 & X2 . X1 cost Rs 3 per kg & X2 cost Rs 8 per kg. No more than 80 kg
of X1 can be used and atleast 60 kg of X2 must be used. Formulate LPP.
3 An electric Co. produces products P1 & P2 . Products are produced & sold on a
weekly basis. The weekly production cannot exceed 25 for products P1 & 35 for
prodct P2 because of limited available facilities. The company employs total of 60
workers. Product P1 requires 2 man weeks of labour, while P2 requires one man
week of labour. Profit margin on P1 is Rs 60 & on P2 is Rs 40. Form LPP.
4 A company is making two products A & B. The cost of producing one unit of product
A and B is Rs 60 & 80 respectively. As per the agreement, the company has to
supply at least 200 units of product B to its regular customers. One unit of product
A requires one machine hours whereas product B has machine hrs available
abundantly within the company. Total machine hours available for product A are
400 hrs. One unit of each product A & B requires one labour hr each & total of 500
labour hrs are available. The company wants to minimize the cost of production by
satisfying the given requirements. Formulate the problem as LPP.
GRAPHICAL SOLUTIONS
1
Solve LPP
Maximize Z= 5x1 + 3x2
Subject to
3x1 + 5x2 ≤ 15
5x1 + 2x2 ≤ 10
x1,x2 ≥ 0
2
Maximise Z = 6x1 + 7x2
Subject to
2x1 + 3x2 ≤ 12
2x1 + x2 ≤ 8
x1, x2 ≥ 0
3
Maximise Z = 80 x1 + 100 x2
Subject to
X1 + 2x2 ≤ 720
5x1 + 4x2 ≤ 1800
Page 30
x1, x2 ≥ 0
4
5
3x1 + x2 ≤ 900
Min Z = 40x1 + 24 x2
Subject to,
20x1 + 50x2 ≥ 4800
80x1 + 50x2 ≥ 7200
x1,x2 ≥ 0
Maximise Z = 40x1 + 80 x2
Subject to
2x1 + 32 x2 ≤ 48
x1 ≤ 15
x2 ≤ 10
x1,x2 ≥ 0
5 Min Z = -x1 + 2x2
Subject to,
X1 + x2 ≤ 6
X1- x2 ≤ 2
X1, x2 ≥ 0
6 Max Z = 3x1 + 2x2
Subject to,
where, x1, x2 ≥ 0
-2x1 + x2 ≤ 1
x1 ≤ 2
x1 + x2 ≤ 2
7 The owner of Metro sports wishes to determine how many advertisements to place
in the selected three monthly magazinesA, B & C. His objective is to advertise in
such a way that total exposure to principal buyers of extensive sports goods is
maximized. Percentage of readers for each magazines are known. Exposure in any
particular magazine is the number of advertisement placed multipled by number of
principal buyers. The following data is given:
Requirements
Magazines
A
B
C
Readers
1 lakh
0.60 lakh
0.40 lakhs
Principal buyers
15%
15%
7%
Cost per advt. 5000
4500
4250
The budgeted amount is at the most Rs 1 lakh for advertisements. The owner has
already decided that magazine A should have no more than 6 advertisements and B
& C each have at least two advertisements. Formulate LPP.
8
A caterers knows that he will need 80 napkins on a given day and 140 napkins the
day after. He can purchase napkins @ Rs 8 each and after they are purchased, he
Page 31
can have dirty napkins laundered @ Rs 2 for the use the next day. In order to
minimize his cost how many dirty napkins should he have laundered.
9
A company produces two types of products say type A & B. Product B is of superior
quality & product A is a lower quality. Profits on the two types of products are Rs 30
& Rs 40 respectively. The data about resources required & availability of resources
are given below.
Requirement
ProductA
Product B
capacity available
Per month
Raw material (kgs)
60
120
12000
Machine Hrs(per piece)
8
5
630
Assembly
3
4
500
10
11
How should the company manufacture the two types of products in order to get at
maximum overall profits.
A firm manufactures two products A and B on which the profits earned per unit are
Rs. 3 and Rs. 4 respectively. Each product is processed on two machines M1 and M2.
Product A require one minute of processing time on M1 and two minutes on M2
while B require one minute on M1 and one minute on M2. Machine M1 is available
for not more than 7 hours 30 minutes while machine M2 is available for 10 hours
during any working day. Find the number of units of product A and B to be
manufactured to get maximum profit (use graphical method)
Solve graphically the following linear programming problem.
Objective function
Minimise cost Z = 6x + 8y
Subject to the constraints
3x + 6y ≥ 48
y≥5
x≥0
y≥0
12 Maximize Z = 40x1 + 80x2
Subject to
2x1 + 32x2 ≤ 48
x1 ≤ 15
x2 ≤ 10
x1,x2 ≥ 0
13 Min Z = 2x1 + x2
Subject to
5x1 + 10x2 ≤ 50
x1 + x2 ≥ 1
x1 ≤ 4
x1, x2 ≥ 0
Page 32
14 Min Z = 4x1- 2x2
Subject to
X1 + x2 ≤14
3x1 + 2x2 ≥ 36
2x1 + x2 ≤ 24
x1,x2 ≥ 0
15 An animal feed company must produce 200 kg of mixture of consisting of
ingredients X1 & X2 . X1 cost Rs 3 per kg & X2 cost Rs 8 per kg. No more than 80 kg
of X1 can be used and atleast 60 kg of X2 must be used. Formulate LPP.
16 An electric Co. produces products P1 & P2 . Products are produced & sold on a
weekly basis. The weekly production cannot exceed 25 for products P1 & 35 for
prodct P2 because of limited available facilities. The company employs total of 60
workers. Product P1 requires 2 man weeks of labour, while P2 requires one man
week of labour. Profit margin on P1 is Rs 60 & on P2 is Rs 40. Form LPP.
17A caterers knows that he will need 80 napkins on a given day and 140 napkins the day
after. He can purchase napkins @ Rs 8 each and after they are purchased, he can
have dirty napkins laundered @ Rs 2 for the use the next day. In order to minimize
his cost how many dirty napkins should he have laundered.
18 company manufactures two products X & Y using four major departments Q,R,S & T.
The capacity limit of those departments are given in the table below:
Department
Capacity for the production of
X
Y
Q
4000
NIL
R
5000
5000
S
7000
4000
T
8000
3000
Solve graphically for the optimal production level if both the products sell at Rs 40
per unit & the average costs of two products X & Y are Rs14 & Rs 20 respectively.
19 A company produces two types of products say type A & B. Product B is of superior
quality & product A is a lower quality. Profit on the two types of products are Rs 30
& Rs 40 respectively. The data about resources required and availability of
resources are given below.
Requirement
Product A
Product B capacity available
Per month
Raw material (kgs)
60
120
12000
Machine hrs (per piece)
8
5
630
Assembly
3
4
500
How should the company manufacture the two types of products in order to get at
maximum overall profits.
Page 33
HOMEWORK PROBLEMS
1. A firm manufactures two products A and B by using two machines M1 and M2. One
unit of A require 1 hour at machine M1 and 3 hours at machine M2, one unit of B
requires 2 hours at each of the two machines. If the profit contribution from each
unit of A and B are Rs. 60 and Rs. 50 respectively and the number of hours available
per week on machines M1 and M2 are 40 and 60 respectively. Find the number of
units of product A and B to be manufactured to get maximum profit and solve it
graphically.
2. A manufacture of leather belts makes two types of belts A and B which are
processed on two machinesM1 and M2. Belt A require 2 hours on machine M1 and 3
hours on machine M2. There are 15 hours of time per day available on machine M1
and 18 hours of time per day available on machine M2. Profit gained from belt A is
Rs. 3 per unit from belt B is Rs. 5 per unit. What should be the daily production of
each type of belts so that the profit is maximum (use graphical method)
3. Solve graphically the following linear programming problem. Mark the feasible
region represented by constraints equation and find out the optimal product mix.
Objective function –
Maximize profit Z = 3x + 4 y
8x + 3y ≤ 46
X≤0
Y≤0
4. Solve graphically the following linear programming problem. Mark the feasible
region represented by constraints equation and find out the optimal product mix.
Objective function –
Maximize profit Z = 4x + 3 y
3x + 7y ≤ 42
X≤4
X≥0
Y≥0
5. Solve graphically the following linear programming problem.
Maximize Z = 9x + 10 y
11x + 9y ≤ 9900
7x + 12y ≤ 8400
6x + 16y ≤ 9600
Where x ≥ 0; y ≥ 0
6. Solve graphically the following linear programming problem.
Objective function
Minimize cost Z = 2x + 3y
3x + 7y ≥ 42
x≥4
x≥0
y≥0
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QUANTATIVE TECHNIQUES WORKSHEET SEMESTER -I

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