QTM PRACTICAL QUESTIONS FOR MANAGERS BBA

QUANTITATIVE TECHNIQUE
FOR MANAGERS
PRACTICAL QUESTIONS

DETERMINATION OF SAMPLE SIZE
Q1. An economist is interested in estimating the average monthly household expenditure on food
items by the households of a tow. Based on past data, it is estimated that the standard deviation of the
population on the monthly expenditure on food item is Rs. 30. with allowable error set at Rs. 7,
estimate the sample size required at a 90 per cent confidence.
Q2. You are given a population with a standard deviation of 8.6. Determine the sample size needed
to estimate the mean of the population within +/- 0.5 with a 99 per cent confidence.
Q3. It is desired to estimate the mean life time of a certain kind of vacuum cleaner. Given that the
population standard deviation σ = 320 days, how large a sample is needed to be able to assert with a
confidence level of 96 per cent that the mean of the sample will differ from the population mean by
less than 45 days?

TESTS CONCERNING MEANS- CASE
OF SINGLE POPULATION
Q1. A sample with 200 bulbs made by a company give a lifetime mean of 1540 hours with a standard
deviation of 42 hours. Is it likely that the sample has been drawn from a population with a mean lifetime
of 1500 hours? You may use 5 per cent level of significance.
Q2. It is known from past studies that the monthly average household expenditure on the food items in a
locality is Rs. 2700 with a standard deviation of Rs. 160. An economist took a random sample of 25
households from the locality and found their monthly household expenditure on food items to be Rs.
2790. At 0.01 level of significance, can we conclude that the average household expenditure on the food
items is greater than Rs. 2700?
Q3. On a typing test, a random sample of 136 graduates of a secretarial school averaged 73.6 words with
a standard deviation of 8.10 words per minute. Test an employer’s claim that the school’s graduate
average less than 75 words per minute using the 5 per cent level of significance.
Q4.A sample of 16 graduating engineering students of a college was taken and the information was
obtained on their starting salary. The mean monthly starting salary was found to be Rs. 30,200 with a
standard deviation of Rs. 960. The past data on the starting salary has given a mean value of Rs. 30,000.
Using a 5 Per cent level of significance, can we conclude that the average starting salary is different from
Rs. 30,000?

Q5. Prices of share (in Rs.) of a company on the different days in a month were found to be 66, 65,
69, 70, 69, 71, 70, 63, 64 and 68. Examine whether the mean price of shares in the month is different
from 65. You may use 10 per cent level of significance.
Q6. The results of a household survey indicated that a sample of 20 households bought an average of
75 litres of milk per month with a standard deviation of 13 litres. Test the hypothesis that the value of
the population mean is 70 litres against the alternative that is more than 70 litres. Use 0.05 level of
significance.

TESTS FOR DIFFERENCE BETWEEN TWO
POPULATION MEANS
Q1. A study is carried out to examine whether the mean hourly wages of the
unskilled workers in the two cities- Ambala Cantt and Lucknow are the same. The
random sample of hourly earnings in both the cities is taken and the results are
presented in the Table:
Using a 5 per cent level of significance, test the hypothesis of no difference in the
average wages of unskilled workers in the two cities.
City Sample Mean
Hourly Earnings
Standard Deviation
of Sample
Sample Size
Ambala Cantt Rs. 8.95 0.40 200
Lucknow Rs. 9.10 0.60 175

Q2. Two drugs meant to provide relief to arthritis sufferers were produced in two
different laboratories. The first drug was administered to a group of 12 patients and
produced an average of 8.5 hours of relief with a standard deviation of 1.8 hours.
The second drug was tested on a sample of 8 patients and produced an average of
7.9 hours of relief with a standard deviation of 2.1 hours. Test the hypothesis that
the first drug provides a significantly higher period of relief. You may use 5 per cent
level of significance.
Q3. There were types of drugs (1 and 2) that were tried on some patients for
reducing weight. There were 8 adults who were subjected to drug 1 and seven adults
who were administered drug 2. the decrease in weight (in ponds) is given below:
Do the drugs differ significantly in their effect on decreasing weight? You may use 5
per cent level of significance. Assume that the variances of two populations are not
same.
Drug 1 10 8 12 14 7 15 13 11
Drug 2 12 10 7 6 12 11 12

TESTS CONCERNING POPULATION
PROPORTION
Q1. A food processing company wants to know whether the proportion of customers who prefer the
new packaging to the old one is 0.65. what can be concluded at the level of significance of 0.05 if 74
of the 100 randomly selected customers prefer the new kind of packaging and alternative hypothesis
is p ≠ 0.65.
Q2. A company is interested in considering two different television advertisements for the promotion
of a new product. The management believes that advertisement A is more effective than
advertisement B. Two test market areas with virtually identical customer characteristics are selected.
Advertisement A is used in one area and advertisement B in the other area. In a random sample of 60
consumers who saw advertisement A, 18 tried the product. In a random sample of 100 customers
who saw advertisement B, 22 tried the product. Does this indicate that advertisement A is more
effective than advertisement B, if a 5 per cent level of significance is used?

Q3. In a random sample of 100 persons taken from village A, 60 were found top be consuming tea.
In another sample of 200 persons taken from village B. 100 persons were found to be consuming tea.
Does the data reveal a significant difference between the two villages so far as the habit of taking tea
is concerned? You may use a 5 percent level of significance.

Chi- Square Test
Q1. A sample of 870 trainees was subjected to different types of training classified
as intensive, good and average and their performance was noted as above average,
average and poor. The resulting data is presented in the table below. Use a 5 per cent
level of significance to examine whether there is any relationship between the type
of training and performance.
Performance Training
Intensive Good Average Total
Above Average 100 150 40 290
Average 100 100 100 300
Poor 50 80 150 280
Total 250 330 290 870

Q2. The following table gives the number of good and defective parts produced by
each of the three shifts in a factory:
Is there any association between the shift and the equality of the parts produced?
Use a 0.05 level of significance.
Shift Good Defective Total
Day 900 130 1030
Evening 700 170 870
Night 400 200 800
Total 2000 500 2500

Q3. An accountant wants to test the hypothesis that the proportion of incorrect
transactions at four client accounts is about the same. A random sample of 80
transactions of one client reveals that 21 are incorrect; for the second client, the
number is 25 out of 100; for the third client, the number is 30 out of 90 sampled and
for the fourth, 40 are incorrect out of a sample of 110. Conduct the test at 0.05 level
of significance.
Q4.An insurance company provides auto insurance and is analysing the data
obtained from fatal crashes. A sample of motor vehicle deaths is randomly selected
for a two year period. The number of fatalities is listed below for the different days
of the week. At the 0.05 significance level, test the claim that accidents occur on
different days with equal frequency.
Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday
Number of
Fatalities
31 20 20 22 22 29 36

Q5. The manger of ABC ice-cream parlour has to take a decision regarding how
much of each flavour of ice cream he should stock so that the demands of the
customers are satisfied. The ice- cream supplier claims that among the four most
popular flavours, 62 per cent customers prefer vanilla, 18 per cent chocolate, 12 per
cent strawberry and 8 per cent mango. A random sample of 200 customers produces
the results below. At a 0.05 level of significance level, test the claim that
percentages given by the supplies are correct.
Flavour Vanilla Chocolate Strawberry Mango
Number of
Preferring
120 40 18 22

F Test or Variance Ratio Test
Q1. Two random samples were drawn from two normal population and their values
are:
Test whether the two population have the same variance at the 5% level of
significance.
Q2. In a sample of 8 observations, the sum of squared deviations of items from the
mean was 84.4. In another sample of 10 observations, the value was found to be
102.6. Test whether the difference is significant at 5% level.
You are given 5% level, critical value of F for v1 = 7 and v1 = 9 degrees of freedom
is 3.29.
A: 66 67 75 76 82 84 88 90 92
B: 64 66 74 78 82 85 87 92 93 95 97

Q3. Two samples are drawn from two normal population. From the following data
test whether the two samples have the same variance at 5% level:
Q4. The following data present the yields in quintals of common ten subdivisions of
equal area of two agricultural plots:
Test whether two samples taken from two random populations have the same
variance at 5 % level of significance.
Sample 1 60 65 71 74 76 82 85 87
Sample 2 61 66 67 85 78 63 85 86 88 91
Plot 1 6.2 5.7 6.5 6.0 6.3 5.8 5.7 6.0 6.0 5.8
Plot 2 5.6 5.9 5.6 5.7 5.8 5.7 6.0 5.5 5.7 5.5

ANOVA- One Way
Q1. to assess the significance of possible variation in performance in a certain test
between the convent schools of a city, a common test was given to a number of
students taken at random from the senior fifth class of each of the four schools
concerned. The results are given below. Make an analysis of variance of data.
Schools
A B C D
8 12 18 13
10 11 12 9
12 9 16 12
8 14 6 16
7 4 8 15

ANOVA- TWO WAY
Q1. A tea company appoints four salesman A, B, C and D and observes their sales in
three seasons – summer, winter and monsoon. The figures (in lakhs) are given in the
following table:
(i) Do the salesman significantly differ in performance?
(ii) Is there significant difference between the seasons?
Seasons Salesman Season’s
Total
A B C D
Summer 36 36 21 35 128
Winter 28 29 31 32 120
Monsoon 26 28 29 29 112
Salesmen’s
Total
90 93 81 96 360

Q2. The following data represent the number of units of production per day turned
out by 5 different workers using 4 different types of machines:
(a) Test whether the mean productivity is the same for different machine types.
(b) Test whether the 5 men differ with respect to mean productivity.
Machine Type
A B C D
Workers 1 44 38 47 36
2 46 40 52 43
3 34 36 44 32
4 43 38 46 33
5 38 42 49 39

CORRELATION AND REGRESSION
Q1. The following table gives indices of industrial production of registered
unemployed (in hundred thousand). Calculate the value of coefficient of correlation
Q2. From the following data obtain the two regression equation:
Index of
Production
100 102 104 107 105 112 103 99
No. of
Unemployed
15 12 13 11 12 12 19 26
X 6 2 10 4 8
Y 9 11 5 8 7

BAYES’ THEOREM
Q1. Assume that a factory has two machines. Past records show that machine 1 produces 30% of the
items of output and machine 2 produces 70% of the items. Further, 5% of the items produced by
machine 1 were defective and only 1% produced by machine 2 were defective. If a defective item is
drawn at random, what is the probability that the defective item was produced by machine 1 or
machine 2?
Q2. A manufacturing firm produces units of a product in four plants. Define event Ai : a unit is
produced in plant i, i = 1,2,3,4 and event B: a unit is defective. From the past records of the
proportions of defective produced at each plant the following conditional probabilities are set:
P(B/ A1 ) = 0.05 P(B/ A2 ) = 0.10 P(B/ A3 ) = 0.15 P(B/ A4 ) =
0.02
The first plant produces 30 per cent of the units of the product, the second plant 25 per cent, third
plant 40 per cent and the fourth plant 5 per cent. A unit of the product made at one of these plants is
tested and is found to be defective. What is the probability that the unit was produced in plant 3.

Q3. In a bolt factory machines A, B and C manufacture respectively
25%, 35% and 40%. Of the total of their output 5, 4 and 2 per cent are
defective bolts. A bolt is drawn at random from the product and is found
to be defective. What is the probability that it was manufactured by
machines A, B and C?

BINOMIAL DISTRIBUTION
Q1. A sample of 3 items is selected at random from a box containing 12 items of
which 3 are defective. Find the possible number of defective combinations of the
said 3 selected items along with the probability of defective combination.
Q2. Eight coins are tossed at a time 256 times. Number of heads observed at each
throw is recorded and the results are given below. Find the expected frequencies.
What are the theoretical value of mean and standard deviation
No. of
heads at a
throw
0 1 2 3 4 5 6 7 8
Frequency 2 6 30 52 67 56 32 10 1

POISSON DISTRIBUTION
Q1. Suppose on an average 1 house in 1000 in a certain district has a fire during a
year. If there are 2000 houses in that district, what is the probability that exactly 5
houses will have a fire during the year?
Q2. Ten Percent of the tools produced in a certain manufacturing process turns 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) the binomial distribution, (b) the poisson
approximation to binomial distribution

NORMAL DISTRIBUTION
Q1. The Mumbai Municipal corporation installed 2000 bulbs in the streets of
Mumbai. If these bulbs have an average life of 1000 burning hours, with a standard
deviation of 200 hours, what number of bulbs might be expected to fail in the first
700 burning hours?
Q2. In an intelligence test administered to 1000 students the average score was 42
and standard deviation 24. Find (a) the number of students, exceeding a score of 50,
(b) the number of students lying between 30 and 54.

STATISTICAL DECISION THEORY
Q1. A baker produces a certain type of special pastry at a total average cost of Rs. 3 and sells it at a
price of Rs. 5. This pastry is produced over the weekend and is sold during the following week; such
pastry being produced but not sold during a week’s time are totally spoiled and have to be thrown
away. According to past experience the weekly demand for the pastries is never less than 78 or
greater than 80. You are required to formulate action space, payoff table and loss table.
Q2. Under an employment promotion programme, it is proposed to allow sale of newspapers on the
buses during off peak hours. The vendor can purchase the newspapers at a special concessional rate
of 25 paise per copy against the selling price of 40 paise. Any unsold copies are, however, a dead
loss. A vendor has estimated the following probability distribution for the number of copies
demanded:
How many copies should he order so that his expected profits will be maximum?
No. of
copies
15 16 17 18 19 20
Probability 0.04 0.19 0.33 0.26 0.11 0.07

QTM PRACTICAL QUESTIONS FOR MANAGERS BBA

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