Download Business Matheamatics and Statistics

INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
MODULE I – MEASURES OF CENTRAL TENDENCY
1. Calculate the simple arithmetic average of the following items of
sizes.
20, 50, 72, 28, 53, 74, 34, 54, 75, 39, 59, 78, 42, 64, 79
2. The following table gives the number of children born per family in
735 families. Calculate the average number of children born per
family.
Number of
Number of
Number of
Number of
Children born
families
children born
families
per family
per family
0
96
7
20
1
108
8
11
2
154
9
6
3
126
10
5
4
95
11
5
5
62
12
1
6
45
13
1
3. The following data give the sizes of shoes sold by a store during a
given week. Find the average size by the short cut method.
Size of Shoes
No. of pairs
Size of Shoes
No. of pairs
4.5
1
8
95
5
2
8.5
82
5.5
4
9
75
6
5
9.5
44
6.5
15
10
25
7
30
10.5
15
7.5
60
11
4
4. The following table gives the heights of 350 mean. Calculate the
mean height of the group.
Height (in cms)
159 161 163 165 167 169 171 17
(x)
3
No. of persons
1
2
9
48 131 102 40
17
(f)
5. Calculate the arithmetic average of the following by the Direct
Method.
Weekly wages (Rs.)
11-13
13-15
15-17
17-19
19-21
21-23
23-25
No. of labourers
3
4
5
6
5
4
3
6. The following table gives the marks obtained by a set of students in
a certain examination. Calculate the average marks per student.
Marks
No. of students
Marks
No. of students
10-20
1
60-70
12
20-30
2
70-80
16
30-40
3
80-90
10
40-50
5
90-100
4
50-60
7
7. Find arithmetic mean for the following data.
Class
50-59
40-49
30-39
20-29
10-9
0-9
interval
Frequency 1
3
8
10
15
3
8. Given below is the distribution of 140 students obtaining marks or
higher in a certain examination (all marks are given in whole
numbers):
X
10
20
30
40
50
60
70
80
90
100
C.F
140 133 118 100 75
45
25
9
2
0
Calculate the arithmetic mean marks obtained by the candidates.
9. The following table gives the life time in hours of 400 radio tubes of
a certain make. Fine the mean lifetime of the radio tubes.
Life time (in
Number of
Life time (in
Number of
hours)
tubes
hours)
tubes
Less than 300
0
Less than 800
265
Less than 400
20
Less than 900
324
Less than 500
60
Less than 1000 374
Less than 600
116
Less than 1100 392
Less than 700
194
Less than 1200 400
10.
A market with 168 operating firms has the following
distribution of average number of workers in various income groups.
Income
150-300
300-500
500-800
800-1200 1200-1800
groups
No. of
40
32
26
28
42
firms
Average
8
12
7.5
8.5
4
No. of
workers
Find the average salary paid in the whole market.
11.
Find the value of median of the following items:
5,7,9,12,10,8,7,15,21
12. Find the value of the median from the following data.
10, 18, 9, 17, 15, 24, 30, 11
13. Find the value of the median, Q1, Q3 from the following data.
Daily wages (Rs.)
10
5
7
11
8
Number of workers
15
20
15
18
12
14. Find the median, Q1, Q3 from the following distribution.
Class intervals
Frequencies
Class intervals
Frequencies
(Rs.)
1-3
6
11-13
16
3-5
53
13-15
4
5-7
85
15-17
4
7-9
56
TOTAL
245
9-11
21
15. Calculate median, Q1, Q3 from the following data.
Value
Frequency
Value
0-4
328
30-39
5-9
350
40-49
10-19
720
50-59
20-29
664
60-69
Frequency
598
524
378
244
16. Calculate median, Q1, Q3 from the following data.
Age
Number of
Age
persons
55-60
7
35-40
50-55
13
30-35
45-50
15
25-30
40-45
20
20-25
Total
Number of
persons
30
33
28
14
160
17. Calculate median, Q1, Q3 from the following data.
Value
Frequency
Value
Less than 10
4
Less than 50
Less than 20
16
Less than 60
Less than 30
40
Less than 70
Less than 40
76
Less than 80
18. Calculate median, Q1, Q3 from the following data.
Size
Frequency
Size
More than 50
0
More than 20
More than 40
40
More than 10
More than 30
98
Frequency
96
112
120
125
Frequency
123
165
19. Compute median, Q1, Q3 from the following data.
Mid values
115 125
135
145
155
165
175
Frequency
6
25
48
72
116
60
38
185
22
195
3
20. Find the mode of the following data relating to the weight of 10
students.
120, 130, 135, 130, 140, 130, 132, 132, 135, 141
21. Calculate the value of mode from the following data.
Class intervals
Frequency
Class intervals
10-20
4
60-70
20-30
6
70-80
30-40
5
80-90
40-50
10
90-100
50-60
20
100-110
Frequency
22
24
6
2
1
22. Find the value of Mode from the following data.
Size of the items
Frequency
Size of the items
Frequency
100-110
4
140-150
33
110-120
6
150-160
17
120-130
20
160-170
8
130-140
32
170-180
2
23. Given Mean = 20; Mode = 15; Find the value of median
24. Given Mode = 25; Median = 20; Find the value of mean.
25. Calculate simple Geometric mean from the following data
133, 141, 125, 173, 182
26. The following data related to the distance traveled by 520 villagers to
buy their weekly requirements.
Miles
2
4
6
8
10
12
14
16
18
20
traveled
No. of
38
104 140
78
48
42
28
24
16
2
villagers
Calculate the arithmetic average.
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
MODULE I – MEASURES OF CENTRAL TENDENCY AND
MEASURES OF DISPERSION
1. The profits of the company for the last 8 years are given below.
Calcultate the Range and its co-efficient.
Year
Profits (in
‘000 Rs.)
1975 1976
40
30
1977
80
1978
100
1979
120
1980
90
1981
200
1982
230
2. Calculate co-efficient of Range from the following data.
Weekly
wages (Rs.)
No. of
labourers
50-60
60-70
70-80
80-90
90-100
50
45
45
40
35
100-110 110-120
30
30
3. Calculate Quartile Deviation and its co-efficient from the following
data.
Weight (in
pounds)
Frequency
Cumulative
Frequency
120
122
124
126
130
140
150
160
1
1
3
4
5
9
7
16
10
26
3
29
1
30
1
31
4. Calculate Semi-Inter Quartile Range and its co-efficient from the
following data.
Marks
No. of
Students
0-10
11
10-20
18
20-30
25
30-40
28
40-50
30
50-60
33
5. Calculate Quartile Deviation and its relative measure
60-70
22
70-80
15
80-90
22
Variable
20-29
30-39
40-49
Frequency
306
182
144
Variable
50-59
60-69
70-79
Frequency
96
42
34
6. Estimate an appropriate measure of dispersion for the following
data:
Income (Rs.)
Less than 50
50-70
70- 90
90-110
No. of persons
54
100
140
300
Income (Rs)
110-130
130-150
Above 150
Total
No. of persons
230
125
51
1000
7. The following are the marks obtained by a batch of 9 students in a
certain test.
S.No. Marks (out of 100)
S.No. Marks (out of 100)
1
68
6
38
2
49
7
59
3
32
8
66
4
21
9
41
5
54
8. Calculate Mean Deviation (from arithmetic average) for the following
values. Also calculate its co-efficient.
4800, 4600, 4400, 4200, 4000
9. Find mean deviation from the following data.
No. of accidents Persons having said
No. of
number of accidents accidents
0
15
7
1
16
8
2
21
9
3
10
10
4
17
11
5
8
12
6
4
Total
10.
Marks
No. of
students
11.
Persons having said
number of accidents
2
1
2
2
0
2
100
Calculate mean deviation from the following data.
0-10
10-20
20-30
30-40
40-50
50-60
6
5
8
15
7
6
60-70
3
Calculate the standard deviation from the following data.
160, 160, 161, 162, 163, 163, 163, 164, 164, 170
12. The mean of 200 items is 48 and then Standard deviation is 3. Find
(i) the sum of all the items; (ii) the sum of squares of all the items.
13. For two groups of observations of the following results are available.
Group I Total of (X-5) = 3; Total of squares of (X-5) = 43; N1 = 18
Group II Total of (X-8) = - 11; Total of squares of (X-8) = 76; N2 = 17
14.
Calculate the standard deviation from the following data.
Size of the items
6
7
8
9
Frequency
3
6
9
13
Size of the items
10
11
12
Frequency
8
5
4
15.
Calculate standard deviation for the following distribution.
Values
10
20
30
40
50
60
70
Frequency
1
5
12
22
17
9
4
16.
The following data relate to the age of a group of Government
employees. Calculate the arithmetic mean and standard deviation.
Age
50-55
45-50
40-45
35-40
30-35
25-30
20-25
No. of
25
30
40
45
80
110
170
employees
17.
Calculate the mean wages and standard deviation of all the
workers taken together.
Section
No. of workers
Mean wages
Standard Deviation
A
50
113
6
B
60
120
7
C
90
115
8
18.
Calculate the combined mean and standard deviation of the
two series.
Series A
Series B
Mean
50
40
Standard Deviation
5
6
No. of items
100
150
19.
An analysis of the monthly wages paid to the workers in two
firms A and B, belonging to the same industry, gives the following
results :
Particulars
Firm A
Firm B
Number of wage earners
586
648
Average monthly wages (Rs.) 52.5
47.5
Standard Deviation of the
Square root of 100
Square root of 121
distribution of wages (Rs.)
20.
The mean and standard deviation of the 20 items was found to
be 10 and 2 respectively. Later, it was found the item 12 was
misread as 8. Calculate correct mean and standard deviation.
21.
The arithmetic mean and standard deviation of a series of 20
items were calculated by a student as 20 cms and 5 cms
respectively. But, while calculating them, an item 13 was misread
as 30. Find the correct arithmetic mean and standard deviation.
22.
In a frequency distribution of 100 persons grouped in class
intervals of Rs. 10-12, 12-14, 14-16 etc., revealed the mean wage to
Rs. 32.02 and its standard deviation a Rs. 13.18. Later, it was
found that the wage of a labourer who was getting Rs. 57 was
misread as Rs. 27. Calculate the correct mean and standard
deviation.
SKEWNESS, MOMENTS AND KURTOSIS
1. Calculate the co-efficient of skewness from the following data.
Value
10
20
30
40
50
60
70
Frequency 1
5
12
22
17
9
4
2. Assume that a firm has selected a random sample of 100 from its
production line and has obtained the data shown in the table below.
Class interval
Frequency
Class interval
Frequency
130-134
3
150-154
19
135-139
12
155-159
12
140-144
21
160-164
5
145-149
28
Total
100
3. Calculate the co-efficient of skewness based upon Mean and Median
from the following distribution.
Class
0-10
10-20 20-30 30-40 40-50 50-60 60-70 70-80
interval
Frequency 6
12
22
48
56
32
18
6
4. Calculate Karl Pearson’s Measure of Skewness on the basis of Mean,
Mode and Standard Deviation.
X
14.5
15.5
16.5
17.5
18.5
19.5
20.5
21.5
F
35
40
48
100
125
87
43
22
5. Calculate the Karl Pearson’s co-efficient of skewness from the
following data.
Marks
0
10
20
30
40
50
60
70
80
(above )
No. of
150 140
100
80
80
70
30
14
0
students
6. Consider the following distributions:
Particulars
Distribution A
Distribution B
Mean
100
90
Median
90
80
Standard Deviation
10
10
(i)
The Distribution A has the same degree of the variation as the
distribution B.
(ii)
Both the distributi8ons have the same degree of Skewness – State
whether it is True / False.
7. From the data given below, calculate the co-efficient of variation.
Pearson’s measure of skewness
= 0.42
Arithmetic mean
= 86
Median
= 80
8. Given, Mean = 30; Standard Deviation = 8; Karl Pearson’s Coefficient of Skewness = +0.40; Find the Median and Mode.
9. Calculate Bowley’s co-efficient of skewness.
Wages (Rs.)
30-40
40-50
50-60
60-70 70-80
No. of persons 1
3
11
21
43
80-90
32
90-100
9
10.
Calculate the Bowley’s co-efficient of skewness from the
following data.
Class intervals 10-19 20-29 30-39
40-49 50-59 60-69 70-79
Frequency
5
9
14
20
25
15
8
11.
The measure of skewness for a certain distribution is – 0.8. If
the lower and upper quartiles 44.1 and 56.6 respectively. Find the
median.
12.
In a frequency distribution, the co-efficient os skewness based
on quartiles is 0.6. If the sum of the upper and lower quartiles is
100 and median is 38, find the value of upper quartile.
13.
From the data given below, calculate Karl Pearson’s, Bowley’s
co-efficients of skewness. Mean = 150; Median = 142; Q1 = 62;
Q3 = 195; Standard Deviation = 30.
80-89
4
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
KARL PEARSON’S CORRELATION CO-EFFICIENT
1. Find Out the correlation co-efficient
of the following 10 firms.
Firms
1
2
3
4
5
Sales
50
50
55
60
65
Expenses 11
13
14
16
16
between the sales and expenses
6
65
15
7
65
15
8
60
14
9
60
13
10
50
13
2. Calculate the Karl pearson’s co-efficient of correlation from the
following data.
X
65
66
67
67
668
69
70
72
Y
67
68
65
68
72
72
69
71
3. Calculate the Karl Pearson’s co-efficient of correlation between the
values of X and Y from the data given below.
Values of X 15
16
17
18
19
20
Values of Y 80
75
60
40
30
20
4. With the help of following data in 6 cities, calculate the co-efficient
of correlation by Karl Pearson method between the density of
population and death rate.
Cities Area in Sq. miles Population in ‘000
Number of deaths
A
150
30
300
B
180
90
1440
C
100
40
56
D
60
42
840
E
120
72
1224
F
80
24
312
5. Calculate the co-efficient of correlation between the corresponding
values of x and y in the following table.
Values of X
2
4
5
6
8
11
Values of Y
18
12
10
8
7
5
6. Calculate the co-efficient of correlation between the variables X and
Y.
X
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Y
30,000 50,000 60,000 80,000 1,00,000 1,10,000 1,30,000
X
Y
7. Calculate the co-efficient of correlation between X and Y.
2
4
6
8
5
9
13
17
8. Given the following information. Calculate the correlation coefficient by Karl Pearson’s method.
Series Assumed Mean Total deviations from
Sum of squares of
assumed mean
deviations from assumed
mean
X
41
- 170
8180
Y
32
- 20
2290
Sum of products of deviations of X and Y from their respective
assumed mean = 2880. Number of pairs of observations = 10.
9. Calculate co-efficient of correlation from the following results.
N = 10; Sum of X = 100; Sum of Y = 150; Sum of squares of (X – 10);
Sum of squres of (Y – 15) = 215; Sum of product of (X – 10) (Y – 15) =
60.
10. Calculate the Karl Pearson’s co-efficient of correlation between X
and Y from the following information.
N = 12; Sum of X = 120; Sum of Y = 130; Sum of squares of (X – 8) =
150; Sum of Squares of (Y – 10) = 200; Sum of product of (X – 8) (Y –
10) = 50.
11. Test the significance of correlation for the following values based
upon the number of observations (i) 10 and (ii) 100. The co-efficient
of correlation + 0.4 and + 0.9.
12. Find the co-efficient of correlation from the following data by the
method of Karl pearson.
X
1
2
3
4
5
Y
166
184
142
180
338
13. Calculate the correlation co-efficient from the following 15 pairs of
students in Projects A and Project B.
(1,10) (2,7) (3,2) (4,6) (5,4) (6,8) (7,3) (8,1) (9,11) (10,15) (11,9)
(12, 5) (13,14) (14,12) (15,13). Use Spearman’s formula to find the
rank correlation co-efficient.
14. Ten students were given tests in English and Mathematics. Their
marks are given below.
Student No.
1
2
3
4
5
6
7
Marks in
78
40
50
55
52
49
60
English
Marks in
70
60
60
75
69
55
70
Mathematics
Determine Spearman’s rank correlation co-efficient.
8
54
9
59
10
58
65
65
60
14.
Ten competitors in a beauty contest were ranked by three
judges in the following orders. Use the Spearman’s rank correlation
method.
First
1
6
5
10
3
2
4
9
7
8
Judge
Second
3
5
8
4
7
10
2
1
6
9
Judge
Third
6
4
9
8
1
2
3
10
5
7
Judge
15.
Calculate the rank correlation co-efficient from the following
data.
X
60
34
40
50
45
41
22
43
42
66
64
46
Y
75
32
34
40
45
33
12
30
36
72
41
57
16.
Calculate the co-efficient of correlation from the following
data by Spearman’s Rank Difference method.
Prices of Tea (Rs)
75
88
95
70
60
80
81
Prices of Coffee (Rs.)
120 134 150 115
110
140
142
50
100
17.
The co-efficient of rank correlation between the debenture
prices and share prices of a company was + 0.8. If the sum of the
squares of the difference in ranks was 33, find the value of n.
18.
The co-efficient of rank correlation of marks obtained by 10
students in Statistics and Accountancy was found to be 0.2. It was
later discovered that the difference in the ranks in the two subjects
obtained by one of the students was wrongly taken as 9 instead of 7.
Find the correct value of co-efficient of rank correlation.
19.
Calculate the co-efficient of rank correlation from the
following data.
X
48
33
40
9
16
16
65
24
46
Y
13
13
24
6
15
4
20
9
6
57
19
20.
Calculate concurrent deviations from the data given below.
No. of workers
300 350 400
450
500
550
No. of bales consumed
30
32
33
35
40
50
21.
If there is a lag of one year between price and supply (so that
the supply increases after one year of price change), calculate the
co-efficient of correlation between the price and supply from the
following data.
Year
1973
1974
1975
1976
1977
1978
1979
1980
1981
Price
78
89
97
69
59
79
68
61
60
Supply
115
125
137
156
112
107
136
123
108
22.
From the following data, obtain two regression equations using
the method of Least Squares.
X
Y
2
5
4
7
6
9
8
8
10
11
23.
From the following data, calculate two regression equations
using the method of Least Squares.
X
Y
10
100
15
90
35
110
40
80
50
120
24.
From the following data, obtain two regression equations using
the method of Least Squares.
Sales
91
Purchasaes 71
97
75
108
69
121
97
67
70
124
91
51
39
73
61
111
80
57
47
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
DECISION TREE / DECISION THEORY
1. A company has to choose one of the three types of Biscuits, Cream,
Coconut and Glucose. Sales expected during the next year are
highly uncertain. Marketing Department estimates the profits
considering manufacturing cost, promotional efforts and
distribution set up etc., as given in the table below.
Types of Biscuits
Cream (C)
Coconut (Co)
Glucose (G)
Profits on estimated level of sales ( in lakhs) for quantities
5,000
15
20
25
10,000
25
55
40
20,000
45
65
70
2. A trading company of Delhi is considering expansion of its activities
and planning to open a marketing office at Kanpur to boost up the
sales in North- East UP. It is to be decided whether to operate from
the existing office at Delhi and cover the area by frequent traveling
or else establishing the office at Kanpur. The connected pay-offs
and probabilities are as under:
Alternatives
A. Operate from
Delhi
B. Open office at
Kanpur
States of Nature
(i) increase in
demand by 30%
(ii) no appreciable
change
(i) increase in
demand by 30%
(ii) no appreciable
change
Probability
0.60
Pay-off (Rs. In
lakhs)
50
0.40
5
0.70
40
0.30
- 10
3. Under an employment promotion programme, it is proposed to allow
sales of newspapers on the buses during off-peak hours. The vendor
can purchase the newspaper at a special discounted rate of 25 paise
per copy against the selling price of 40 paise. Anny unsold copies
are, however a dead loss. A vendor has estimated the following
probability distribution for the number of copies demanded.
No. of copies to be demanded
Probability
15
0.04
16
0.19
17
0.33
18
0.26
19
0.11
20
0.07
How many copies should he order so that his expected profir will be
maximum ?
4. A manufacturer finds the opportunity to increase his business
beyond his present on existing production capacity. In order to
decide hether to increase the production capacity, he would need a
reliable information about increase in demand of the product, based
on which only, he can commit his resources. He has two choices
open to him, firstly, the expansion of the existing capacity with a
cost of Rs. 8 lakhs. Or the modernization of the plant at a cost of
Rs. 5 laksh. The time required for implementation of both the
options is expected to be the same. While considering the demand
pattern, he estimates the high demand situation at a probability of
0.35 as compared to the modern rise in demand at 0.65 probability.
He also estimates that he would be spending an additional amount
of Rs. 12 lakhs for expansion against Rs. 6 lakhs for modernization,
if the demand rise is high, whereas in the case of moderate demand
increase, the expenditure involved would be Rs. 7 lakhs for the
expansioin or Rs. 5 laksh for modernization process.
(a) Calculate the conditional profits under various
combibinations.
(b) Establish expansion or modernization so as to maximize its
expected monetary value
(c) Work out EPPI, EVPI and EOL.
5. A manufacturer has faced with a problem of fast change in
technology and hence, fast change in production line. At this point
of time, the research and development wing of the organization has
suggested an improved new product with easy acceptance. It will
cost the manufacturer Rs. 60,000 for the pilot testing and
development testing before establishing the product in the market.
The organization has 100 customers and each customer, might
purchase, at the most, one unit of the product, due to cost and
newness. The selling price suggested is Rs. 6,000 for each unit.
The probability distribution for proportion of customers buying the
product is estimated as follows.
Proportions of customers
Probability
0.04
0.1
0.08
0.1
0.12
0.2
0.16
0.4
0.20
0.2
Work out the expected opportunity, losses and suggest whether the
manufacturer should develop the product or not.
6. A food product company is contemplating the introduction of a
revolutionary new product with new packaging to replace the
existing product at the same price (S1) or a moderate change in the
composition of the existing product with a new packaging at a small
increase in price (S2) or a small change in the composition of the
existing except the “New” with a negligible increases in price (S3).
The three possible states of nature of events are (i) high increases
in sales (N1), (ii) no change in sales (N2) and (iii) decreases in sales
(N3). The marketing department of the company worked out the
pay-offs in terms of yearly net profits for each course of action for
these events (expected sales). This is represented in the following
table.
States of Nature
N1
N2
N3
S1
7,00,000
3,00,000
1,50,000
Courses of action
S2
5,00,000
4,50,000
0
S3
3,00,000
3,00,000
3,00,000
Which strategy should he choose on the basis of (a) Maximum
criterion (b) Maximax criterion (c) Minimax Regret criterion (d) Laplace
criterion.
7. An investor has 3 options to invest, but he can invest in only one
option at a time. He can invest either in a departmental store, a
cold storage or in a car maintenance shop. If he invests in a
departmental stores and succeeds, he can invest in the car
maintenance shop. If he invests in the cold storage and succeeds,
he can invest in the departmental stores. If he invests in the car
maintenance shop, and succeeds, he can invest in the cold storage
and if he succeeds, he can invest in the departmental store. Based
on the data given below, draw a decision tree and advice the
investor on the best decision to take.
Particulars
Probability of
Investment
Loss on failure
success
Departmental
0.65
8,00,000
45,000
stores
Cold storage
0.60
7,50,000
40,000
Car maintenance
0.70
7,40,000
3,50,000
shop
8. A business man has two options to sell his products. He can set up
a show room in the city or can sell from his factory outlet. Setting
up a show room will cost Rs. 3,00,000 with 60% probability of
success. If the showroom succeeds, he can earn gross profits of Rs.
8,00,000 per year. If it fails, he can close the show room or rent it
out for an annual rent of Rs. 2,40,000 ( for the rest of the year).
The probability of getting rent is 80%. If he sells from the factory
outlets, he has to incur Rs. 50,000 as renovating charges. The
chances of successful selling here is 40%, with a net profit of Rs.
4,00,000 per year.
(a) What would you advise the business man to do ?
(b) Advise the business man on how a decision tree helps him to
make decisions.
9. A business man from Chennai wishes to sell his products in
Bangalore. He can set up a show room will entail costs of Rs.
6,00,000 with a 55% probability of success. If the showroom
succeeds, he can gain a net profit of Rs. 10,00,000 per year. If it
fails, he can either shutdown the show room or rent it out for an
annual rent of Rs. 3,60,000 (for the rest of the year). The
probability that he gets rent for the show room is 40%. If he sells
through a wholesaler, he incurs Rs. 3,00,000 initial costs. The
chances of selling successfully are 45% with a net profit of Rs.
5,50,000 per year.
(a) Advise the business man on the best decisions.
(b) How is the decision tree analysis useful in business decision ?
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
THEORY QUESTIONS (UNIVERSITY QUESTIONS)
SECTION A
MODULE I
1.
2.
3.
4.
5.
Define Statistics.
Define Matrix
What is a function ?
What do you mean by Quadratic function ?
Give one example for each of the following :
(a) Linear function
(b) Geometric progression
6. What is meant by regret criterion ?
7. What is differentiatioin ?
8. Define Marginal Revenue. Express this concept as a
derivative.
9. What is maxima and minima ?
10. Define marginal cost. Express this concept as a derivative.
11. Define Arithmetic Progression.
12. Find the sum of the first 20 terms of an Arithmetic
Progression, 2,5,8,……….. ?
13. What is Integration ?
14. What is ratio ?
15. What is Proportion ?
16. Find the sum of the first ten terms of an Arithmetic
Progressiion, 1,3,5,7…………..
17. Find the three numbers in Arithmetic Progression whose sum
is 21 and product is 280 ?
18. the fourth term of a GP is ½ and the seventh term is 4/27.
Find the 12th term.
19. Find the sum of the first 8 terms of a Geometric Progression,
4,8,16……….
20. Show that 1 + 3 + 5 + ……….. to n terms = n^2.
21. Find the sum of the series 3 + 7 + 11 ……. To 40 terms.
22. The third term of a Geometric Progression is 3/4 , and the
seventh term is 4/27. Find the fourth term.
23. How many terms of Geometric Progression, 4,2,1 amount to
127/16 ?
24. The first term and the last term of a Geometric Progression
are 4 and 324. The sum of the series is 384. Find C.R.
25. A firm produces a single product and it can market as many
units as it is able to produce at a price of Rs. 1.75. Its plant
and equipment can produce as many units as 5000 units a
day. The total fixed cost is Rs. 2000 daily. Unit variable cost
is Rs. 0.50. How many units per day must be produced in
order that the firm breaks even ?
MODULE II
1. What is the importance of statistics in managerial decision
making ?
2. Define Business Statistics.
3. What do you mean by Primary and Secondary Data ?
4. What purpose does a measure of central tendency serve ?
5. What are averages ? How are they useful ?
6. What are the desirable properties of an average ?
7. Define Geometric mean and Harmonic zmean.
8. Define a data array. Obtain the median of the data set: 3,0,7,6, -10.
9. Define Arithmetic mean and median for the raw data.
10.
Give any 4 merits of the Arithmetic mean.
11.
What do you understand by Dispersion ? Define
Dispersion and give the properties of a good measure of
Dispersion.
12.
What is co-efficient of variation ? How is this measure
useful ?
13.
What are Ogives ?
MODULE III
1.
2.
3.
4.
5.
What
What
What
What
What
is
is
is
is
is
Probability ?
Poisson Distribution ?
Binomial Distribution ?
Normal Distribution ?
standard normal variate ?
MODULE IV
1. What is Decision theory ?
2. Briefly explain decision trees.
3. What is Expected Pay-off in decision tree ?
4. What is Risk and Certainty & uncertainty ?
5. What is Administrative Decision ?
6. What is Certainty Equivalent ?
7. What is conditional pay-off ?
8. What is Decision Table ?
9. What is Decision tree ?
10.
What is Expected Monetary Value ?
11.
What is Expected Opportunity Loss ?
12.
What is Expected Value Perfect Information /
13.
What is Laplace criterion ?
14.
What is Maximax criterion ?
15.
What is Minimax criterion ?
16.
What is Maximin criterion ?
17.
What is operating decision ?
18.
What is Risk Premium ?
MODULE V
1.
2.
3.
4.
5.
6.
7.
What is sampling ?
Define Stratified Random Sampling. Illustrate.
Explain and illustrate Systematic Sampling.
What is meant by Cluster sampling ?
What is the significance of testing hypothesis ?
What is meant by Type I & Type II error ?
What are the three basic assumptions considered in the
analysis of variance ?
8. What is the sum of the square ? What is the mean square ?
9. What is Chi-square test and for what it is used ?
10.
What is Quota Sampling ?
MODULE VI
1. What is correlation ? What is its significance ?
2. What is co-efficient of Determination ? How will you interpret its
value ?
3. What do you mean by lag in correlation ?
4. Explain Regression and its uses.
5. What do you mean by regression analysis ? Give its uses.
6. What is Multiple Regression ?
7. Explain the difference between correlation and regression analysis.
Correlation and Regression analyses are constructed under different
assumptions. They furnish different types of information.
8. What are the components of Time series ?
9. List the different methods that can be used for determining trend.
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
THEORY QUESTIONS (UNIVERSITY QUESTIONS)
SECTION B & C
MODULE I
1. Define Arithmetic and Geometric Progression and give in each case
(i) the expression for the sum of first n terms and (ii) a business
application.
2. Define the concept of the derivative of a function. Give the
business applications of this concept.
3. Define Marginal Cost, Average Cost and Total Cost. What is the
relationship between the three ?
MODULE II
1. “It is never safe to take the published statistics at their face value
without knowing their meaning and limitations” - Elucidate this
statement.
2. What are the various types of measurement of averages ? Write the
relationship between them.
3. Write short notes on Skewness and Kurtosis.
4. What are the properties of a good measure of dispersion ?
MODULE III
1. Define and illustrate the concept of probability.
2. Describe the features of the following probability models (i) Poisson
model (ii) Normal model
3. How is a normal curve ?
4. Describe a binomial distribution. Illustrate the applications of this
distribution in Business Management.
5. Briefly explain the use of the Baye’s formula
6. Explain and illustrate : (i) Relative frequency approach and (ii)
Subjective approach for evaluating probabilities.
7. Write the features of the Exponential Distribution.
MODULE IV
1. What is sampling ? Explain the various methods of sampling.
2. Graphs and Diagrams have an advantage over written reports.
Comment briefly.
3. What is hypothesis ? Explain the different types of hypothesis.
4. Explain how to set up and test a hypothesis.
5. Explain and point out the difference between one tailed and two
tailed tests.
6. What is (i) Chi-square test (ii) Sign test (iii) Median test ?
MODULE V
1. What are the various decision criteria followed in Decision Theory ?
MODULE VI
1. What is an Index Number ? How do Laspeyres method and Paasches
method differ ?
2. Explain the method of construction used and interpretation of the
Fisher’s Ideal price Index number. Why his Index number method is
considered as the best method ?
3. What is correlation ? Distinguish between positive and negative
correlation. What is the significance of the co-efficient of
correlation ? How IS correlation ueful in dealing with business
problems ?
4. Explain the method of construction used and interpretation of the
Pearsonian correlation co-efficient.
5. “The value of the co-efficient of correlation lies beween + 1 to – 1.
Elaborate. Give the example for the following diagram.
6. List the methods of determining correlation between two variables.
7. What is business forecasting ? Mention its methods. Distinguish
between variance and its co-efficient of variance.
8. Distinguish between cyclical variation and seasonal variation.
9. Write the differences between correlation and regression.
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
PROBABILITY & SAMPLING DISTRIBUTION
SECTION B & C
1. A print shop has three presses; P1, P2 and P3, each press is shut
down 10% of the time. The probability of any press being shut
down is not influenced by the operating conditions of the other two.
Determine the probability that (a) all the three presses are shut
down; (b) atleast one press is shut down; (c) P1 and P2, both are
shut down.
2. In a Post Office, three clerks are assigned to process incoming mail.
The first clerk processes 40%, second 35% and the third 25% of the
mail. The first clerk has an error rate of 0.04, the second ha 0.06
and the third has 0.03. A mail selected at random from a day’s
output is found to have an error. The Post Master wishes to know
the probability that the mail was processed by the first, second or
third clerk respectively.
3. A sample of 100 dry battery cells tested to find the length of the life
produced the following results : Arithmetic mean = 12 hours,
Standard Deviation = 3 hours. Assuming that the data are
normally distributed, what percentage of battery cells are expected
to have life (i) more than 15 hours; (ii) less than 6 hours; (iii)
between 10 and 14 hours ?
Z
2.5
2.0
1.0
0.67
Area
0.4938
0.4772
0.3413
0.2487
4. In an Intelligence test administered to 500 students, the average
score was 42 and standard deviation was 24. Find (a) the number
of students whose score exceeded 50 (b) the number of students
who got between 30 and 40 (c) the number of students who got
score above 60.
5. A survey conducted by a detergent soap manufacturer of Brand P
indicates that when men go shopping, they are likely to buy brand P
3 out of 4 times. When women of shopping, they are likely to buy
brand P 2 out of 3 times. Find the probability of Brand P being
bought when men and women shop together.
6. 1000 light bulbs with a mean life of 120 days are installed in a firm.
Their length of life is normally distributed with a standard deviation
of 20 days. How many bulbs will expire in less than 90 days ?
7. The average monthly sales of 700 branch offices is 42. (Rupees in
lakhs). The standard deviation of sales is 18 (Rupees in lakhs).
Find the number of branches whose sales exceed 50 (Rupees in
lakhs) assuming normal distribution.
8. An MBA graduate is selected for an interview in 3 different firms.
For the first post, there are 5 candidates, for the second post, there
are 10 candidates and for the third post, there are 9 candidates.
What are his chances of getting atleast one post ?
9. Ten percent of the tools produced in a certain manufacturing
process turn out to be defective. Find the probability that in a
sample of 10 tools chosen at random, exactly two will be defective
by using (a) Binomial Distribution (b) Poisson Distribution
10.
The number of machine malfunctions per shift at a factory is
recorded for 180 shifts and the following data are obtained :
No. of
0
1
2
3
4
5
6
malfunctions
No. of shifts
82
42
31
12
8
3
2
11.
A Philanthropist wants to build a hospital in a rural area. He
can build 100, 200 or 300 bed hospital, depending upon whether
anticipated demand is low, medium or high. The expected net
profits and the prior distribution regarding the states of nature is
given below.
State of Nature
Action A1 (Build Action A2 (Build (Action A3 (Build
100 bed hospital) 200 bed hospital) 300 bed hospital)
Low Demand
20,000
- 10,000
- 30,000
Medium Demand
25,000
30,000
- 5,000
High Demand
30,000
50,000
60,000
12.
A company has to choose between two products, any one for
manufacturing. A market research was conducted and the
probabilities and the profits at different levels of market
acceptability are as follows:
Acts
State of Nature
Good
Fair
Bad
Products “A”
0.75
0.15
0.10
Products “B”
0.60
0.30
0.10
Calculate the Expected Value of the choice of alternatives and advise
the company.
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
SAMPLING DISTRIBUTION
SECTION B & C
1. 1. A simple random sample size is 36, is drawn from a finite
population consisting of 101 units. If the population standard
deviation = 12.6, find the standard error of sample mean when the
sample is drawn (a) with the replacement (b) without replacement.
2. The mean lifetime of a sample of 100 fluorescent light tubes
produced by a company is computed to be 1570 hours, with a
standard deviation of 120 hours. The company claims that the
average life of the tubes produced by a company is 1600 hours.
Using a level of significance at 5 % level, is the claim acceptable /
3. A sample of 400 men managers is found to have a mean height of
171.39 cms. Can it be reasonably regarded as a sample from a large
population of men managers of mean height 171.17 cms and
standard deviation of 3.30 cms ? Justify your answer statistically.
4. Intelligence test given to a group of boys and another of girls
provided the following summarized information:
Gender
Mean Score
Standard Deviation
Numbers
Girls
75
10
50
Boys
70
12
100
Is the difference in the mean scores of boys and girls statistically
significant ? Show clearly the steps leading to your conclusion.
5. For a population with known variance 185, a sample of 64
individuals leads to 2217 as an estimate of mean. Compute the
standard error of the mean and also construct an interval that
would include the population mean 95% of the time. What will be
the width of this interval if the sample size were to be only 16.
6. A finite population of 1000 units has been stratified into three
groups having 200, 500 and 300 units respectively. Allocate a
sample of size 100 units proportionately to the strata. What are the
chief advantages of stratifying a population before drawing
samples ?
7. The standard deviation of the wages of 400 textile workers is Rs.
12.8. Another sample of 600 textile workers gives the standard
deviation at Rs. 15.7. Find out if the standard deviation of the first
sample significantly differs from the combined deviation of the two
samples which is 14.
8. A factory is producing 50,000 pairs of shoes daily. From a sample
of 500 pairs, 2% were found to be sub-standard quality. Estimate
the number of pairs that can be reasonably expected to be spoiled in
the daily production and assign the limits at 95% level of
confidence.
9. A psychologist claims lady students are smarter than men students.
The IQ tests of 40 females had a mean of 131 and Standard
deviation of 15. A sample of 36 males had a mean of 122 and
Standard Deviation of 15. Do the data support psychologist’s
claim ? Use the 5% level of significance.
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
CHI SQUARE TEST
1. One hundred students appeared for an examination and results were
categorized as follows depending upon whether they have received
special training:
Special Training
Result
Pass
Fail
Yes
36
12
No
30
22
Test whether the special training was useful to the candidates.
Clearly state the steps in your analysis. (You may use the
information that the relevant table value at 5% level is 3.84).
2. A Manufacturing Limited launched a new car and then interviewed
500 people. Find out whether people consider the new model of the
car as superior in style compared to the old model of car, using Chisquare test.
Superior
Not Superior
Total
New Model
280
60
340
Old Model
120
40
160
Total
400
100
500
3. Using Chi-square test, determine the incidence of accidents caused
by technically skilled and unskilled personnel.
Personnel
Accidents
No accidents
Total
Skilled
10
90
100
Unskilled
15
65
80
Total
25
155
180
4. 2000 families of a city were selected at random to test belief that
families with higher income bought the sedan type of car and
families with lower income bought the small car. Given the
following results, use the chi-square test to find out if the belief is
true.
Income
Sedan type of car
Small car
Total
High
594
606
1200
Low
262
538
800
Total
856
1144
2000
5. Using chi-square analysis, determine the incidence of job related
conflicts caused by trained and untrained personnel.
Personnel
Conflicts
No conflicts
Total
Trained
20
180
200
Untrained
30
130
160
Total
50
310
360
6. The table below shows cross classification of 500 individuals by
income level and preference for ice creams of Type A or Type B.
Test whether the two attributes are associated. Draw suitable
conclusion at 5% level. (Use table value of 9.4877).
Income level
Preference
Total
A
Indifference
B
Low
170
30
80
280
Medium
50
25
60
135
High
20
10
55
85
Total
240
65
195
500
7. The following figures show the distribution of digits in numbers
chosen at random from a telephone directory.
Digit
0
1
2
3
4
5
6
7
8
9
Frequency 1026 1107 997 966 1075 933 1107 972 964 853
Test at 5% level whether the digits may be taken to occur equally
frequently in the directory. (The table value for chi-square test for 9
degrees of freedom at 5% level of significance = 16.919).
8. A political party claims that men and women voters support it
equally. In a sample survey, out of 360 men voters, 120 favoured
the party, while 170 out of 490 women preferred it. Do the survey
results support the claim ? Clearly display the main steps in your
anlysis.
Men
Women
Total
Support
120
170
290
Do not Support
240
320
560
Total
360
490
850
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
ANALYSIS OF VARIANCE TEST – F TEST
(ONE WAY CLASSIFICATION AND TWO WAY
CLASSIFICATION TESTS)
1. The following data represent the number of units of a product
produced by 3 different workers using 3 different machines.
Workers
MACHINES
X
Y
Z
A
8
32
20
B
28
36
38
C
6
28
14
2. A new brand of soap was introduced in some targeted shops in 4
cities. The following table gives the sales.
City
Sales in thousands per month
A
24
26
29
25
B
22
21
25
14
C
21
19
23
20
D
26
28
31
28
Use Analysis of Variance to test the significance of difference between
the sales of soap in the four cities.
3. A new brand of tooth paste was introduced in some targeted sales
points in 4 metros. The following table gives the sales.
Metro cities
Sales in thousands per month
P
14
16
17
16
Q
12
11
13
9
R
11
9
11
11
S
16
18
20
15
Using ANOVA, test the significance of difference between the sales of
the tooth paste in the four metro cities.
4. A company has 5 show rooms in 5 cities selling the same model of
car. The number of cars sold over 4 months is given below. Using
ANOVA, advice the company whether there is a significant
difference in the sales among the different show rooms.
Months
Show Rooms
A
B
C
D
E
September
8
9
7
6
9
October
10
11
8
9
9
November
8
10
9
11
10
December
7
10
8
9
9
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
INDEX NUMBERS
1. Calculate Fisher’s ideal index and test for the factor reversal test
and the time reversal test for the following data:
Commodity
A
B
C
D
E
Base year Price
30
32
30
31
32
Base year Quantity
95
115
120
125
125
Current year Price
22
24
25
27
28
Current year Quantity
215
220
219
222
224
2. Calculate Fisher’s ideal index number from the following data and
prove that it satisfies the time reversal test and factor reversal test.
Commodity
A
B
C
D
E
Base year
Price
10
12
8
16
14
Base year
Expenditure
55
65
40
80
70
Current year
Price
14
16
10
18
12
Current year
Expenditure
80
50
60
70
95
3. Given, the following data prices and average monthly quantities of
four items purchased by children, compute price index for the year
2000 with the year 1995 as base by Fisher’s Method :
ITEMS
1995
2000
Price
Quantity
Price
Quantity
Comic Books
8
1
10
2
Toffees
1
30
2
25
Ice Cream
5
5
6
10
Play Articles
10
1
15
1
4. Compute Fisher’s index number for prices, from the following
information.
Items
Price
Base year
Current year
Rice
400
850
Wheat
320
690
Sugar
720
1600
Dhaal
720
2100
Interpret the results.
Quantity
Base year
Current year
100
120
20
60
10
10
10
20
5. Calculate Index number based on Bowley’s method and Fisher’s
method for the following data:
Commodity
2007
2008
Price
Quantity
Price
Quantity
A
2
8
4
6
B
5
10
6
5
C
4
14
5
10
D
2
19
2
13
6. Calculate Fisher’s ideal index from the following data and prove
that it satisfies the time reversal test and factor reversal test:
Commodities
2007
2008
Price
Quantity
Price
Quantity
P
14
70
18
95
Q
16
80
20
50
R
12
55
14
75
S
20
95
22
60
T
18
85
16
85
7. A ready made Garment Industry has worked out its seasonal index
as given below :
Quarter
Seasonal Index
January to March
98
April to June
90
July to September
82
October to December
130
If the total sales of garments in the first quarter is 1,00,000,
determine how much worth of garments of this type should be kept in
stock to meet the demand in each of the remaining quarters.
8. Calculate seasonal indices from the following data.
Year
QUARTER
I
II
III
2000
68
62
61
2001
65
58
66
2002
68
63
63
IV
63
61
67
9. Construct Laspeyre’s, Paasche’s and Fisher’s ideal index for the
following data.
Commodity
Base Year
Current Year
Price
Quantity
Price
Quantity
A
2
8
4
6
B
5
10
6
5
C
4
14
5
10
D
2
19
2
13
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
CORRELATION AND REGRESSION ANALYSIS
1. Calculate whether there is any correlation between the salaries and
the amount spent on car maintenance. Use Karl Pearson’s method
for correlation, determine the probable error and the comment on
the significance of correlation.
Average salary (in Rs.)
10,000 12,000 15,000
18,000
20,000
Maintenance (in Rs.)
750
900
1,200
1,500
2,000
2. Obtain two regression equations for the following data.
X
30
50
20
80
10
20
20
Y
50
80
30
110
20
20
40
40
50
3. For the data below on the related variables X and Y, (i) Draw the
scatter diagram (ii) Compute the co-efficient of correlation (iii)
Interpret the results.
X
110
100
140
120
80
90
Y
70
60
80
60
10
20
4. Calculate the Pearson’s co-efficient of correlation between
Advertisement Cost and the Sales as per the data given below :
Advertisement (in ‘000 Rs.)
39 65 62 90 82 75 25 98 36 78
Sales (in Rs. Lakhs)
47 53 58 86 62 68 60 91 51 84
5. In a singing competition, 7 candidates were ranked by Judges A and
B as follows.
Candidate
I
II
III
IV
V
VI
VII
Rank by A
3
4
5
6
1
2
7
Rank by B
2
1
7
5
3
4
6
6. Four students of a class, the regression equation of marks in
Statistics (X) on the marks in Accountancy (Y) was 3 y – 5 x + 180 =
0. The mean marks in Accountancy was 44 and the variance of
marks in Statistics was 9/16th of the variance of marks in
Accountancy. Find the mean marks of Statistics and the coefficient of correlation between marks in two subjects.
7. The data below show the experience (X years) of machine operators
and their performance ratings as given by the number of good parts
turned out (Y) per 100 pieces by the operator.
Operator
A
B
C
D
E
F
G
H
X
16
12
18
4
3
10
5
12
Y
87
88
89
68
78
80
75
83
8. A term lending institution has the following data relating to 9 units
in Textile industry.
Fixed Assets as a percentage of
65 66 67 67 68 69 71 72 73
total investment
Long term debt as a percentage of
65 66 69 67 70 71 76 66 70
total investment
9. Obtain the regression equations for
Operator
A
Experience in years (X)
17
Production in units daily (Y)
86
the following data :
B
C
D E
F G H
13
19
5
4 11 6 14
87
88 67 66 69 74 82
10.
The following data gives the experience of lathe operators in a
factory and the number of units of production turned out per day.
Lathe Operator
1
2
3
4
5
6
7
Experience in years (X)
14
10 16 2
1
8
3
Units of production (Y)
84
80 87 60 54 78 56
Calculate the regression lines and (i) Estimate the probable units of
production of a lathe operator with an experience of 15 years (ii)
Estimate the years of experience of a lathe operator with a daily
production of 96 units.
11.
The manager of a company is interested in studying the effect
of price (X) on the sales (Y) of a new product. He has selected four
similar stores in the chain and measured demand in a two week
period at each of the four prices, Rs. 20, Rs. 30, Rs. 40 and Rs. 50.
The total demand at each of these prices is given in the following
table.
Price (Rs.)
20
30
40
50
Demand (Qtty)
138
111
84
47
(a) Find the least square regression line for this problem.
(b) Find the co-efficient of determination.
12.
The following data gives the work experience of machine
operators in a factory and the number of units of production turned
out per day :
Machine Operator
1
2
3
4
5
6
7
8
9
Work Experience (X)
6
8
7
5
2
1
3
9 10
Work Experience (Y)
50 60 54 47 25 20 41 62 70
(a) Calculate the regression lines and estimate the probable units of
production of a machine operator with an experience of 12 years.
(b) Estimate the probable years of experience of a machine operator
whose daily production is 85 units.
13.
From the following data, obtain the regression equations of X
and Y and predict X, when Y = 20.
X
1
2
3
4
5
6
7
8
9
Y
9
8
10
12
11
13
14
16
15
INDIAN ACADEMY SCHOOL OF MANAGEMENT STUDIES
BANGALORE
BUSINESS STATISTICS
TREND ANALYSIS
1. Fit a straight line trend by the method of least squares to the data
given below and project the probable sales for the next two years. A
graph is not necessary.
Year
2001
2002 2003 2004 2005 2006
Sales (in ‘000 Rs.)
164
180
186
187
190
192
2. Calculate the trend values by the method of least squares from the
following data :
Year
2001
2002
2003
2004
2005
Values
75
68
50
67
64
3. Fit a straight line trend by the method of least squares and plot on
a graph for the following data about the sales of a trading firm.
Year
2001 2002 2003 2004 2005
Sales (in lakhs of rupees)
75
90
91
95
98
4. Calculate the trend values by the method of least squares from the
data given below :
Year
2001
2002
2003
2004
2005
Value
78
72
54
70
67