Question Bank for Statistics for Management

1. BA4101
STATISTICS FOR MANAGEMENT
QUESTION BANK
UNIT I INTRODUCTION
Basic definitions and rules for probability, conditional probability independence of events,
Bayes’ theorem, and random variables, Probability distributions: Binomial, Poisson, Uniform
and Normal distributions.
PART A
1. Define probability. (CO1, K1)
Probability is the branch of mathematics that studies the likelihood of occurrence of
random events in order to predict the behaviour of a defined system.
The probability of an event A,
P(A) =
Number of case𝑠𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑡𝑜 𝑡ℎ𝑒 𝑜𝑐𝑐𝑢𝑟𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒 𝑎𝑛𝑑 𝑒𝑥ℎ𝑎𝑢𝑠𝑡𝑒𝑑 𝑒𝑣𝑒𝑛𝑡𝑠
2. What is sample space? (CO1, K1)
A sample space is a collection of all possible outcomes of a random experiment.
Mathematically, the sample space is denoted by the symbol S.
e.g.) the sample space of tossing two coins is the set of all possible outcomes: HH,
HT, TH, and TT, where T for Tail and H for Head.
3. Define random variable. (CO1, K1)
When the numerical value of a variable is determined by a chance event i.e., by
conducting random experiment, that variable is called a random variable. The value of
a random variable will vary from trail to trail as the experiment is repeated.
4. Bring out the rules for probability. (CO1, K1)
1) The probability of occurrence of an event A should be: 0 ≤ P(A) ≤ 1 in a
sample space.
2) The related probability of sample space S is 1, i.e., P(S) = 1.
3) The related probability of occurring event A or event B or both events is equal
to the sum of the probabilities of the events individually, i.e., P(A or B) = P(A)
+ P(B).
4) If P(A) is the probability of occurring an event A then the probability of the
event which does not occur is: P(Ã) = 1 – P(A), where à is the non-
occurrence of P(A).

2. 5. Define mutually exclusive events. (CO1, K1)
Two or more events are said to be mutually exclusive events if it is not possible for
them to occur together at the same time.
e.g.) experiment of throwing a six-sided dice, A be the event of getting odd number A
= {1, 3, 5} whereas B be the event of getting even numbers B = {2, 4, 6}. Here, A and
B cannot occur together. So, it is said to be mutually exclusive events.
6. Define conditional probability. (CO1, K1)
Conditional probability involves estimating the probability of occurrence of a
particular event (A) as the conditional upon the occurrence of a particular event (B). It
is written as P(A/B)
P(A/B) =
P (A ∩ B)
𝑃(𝐴)
7. What do you mean by probability distribution of a discrete random variable?
(CO1, K1)
In a probability distribution, a random variable X can only take the value of discrete
integers i.e., countable values like 0, 1, 2, 3, 4, … And the probability distribution is
called discrete probability distribution.
8. State the Bayes’ theorem. (CO1, K1)
Bayes’ theorem states that the conditional probability of an event, based on the
occurrence of another event, is equal to likelihood of the second event given the first
event multiplied by the probability of the first event.
9. Suppose that X has a Poisson distribution with a parameter m = 2. Compute P[X≥1]
whereas e-2
= 0.1353. (CO1, K3)
Using the Poisson Distribution f(x) =
𝑒−𝑚𝑚𝑥
𝑥!
where x = 0, 1, 2, …
10. Write the Mean, Variance and Standard Deviation of uniform distribution. (CO1, K1)
Mean, µ =
𝑎+𝑏
2
, Variance, σ2
=
(𝑏−𝑎)2
12
and Standard Deviation, σ =
𝑏−𝑎
√12
PART B
1. Suppose that ¾ % of a population have a terminal disease and that the test to detect
this disease is 99% accurate in identifying those with the disease and 95% accurate in

3. identifying those without the disease. Compute the probability that one has the disease
given that the test so indicates. (CO1, K3)
2. The average number of defective chips manufactured daily at the plant is 5. Assume
the number of defects is a Poisson random variable X. compute mean and variance of
X if P[x = 0] = 0.0497. (CO1, K3)
3. A bag contains 5 balls and it is not known how many of them are white. Two balls are
drawn at random from the bag and they are noted to be white. What is the probability
that all the balls in the bag are white? (CO1, K3)
4. If the actual amount of instant coffee which filling puts into 6-ounce jar is a random
variable having a normal distribution with SD = 0.05 ounce and if only 3% of the jar
are to contain less than 6 ounces of coffee, what must be the mean fill of these jars?
(CO1, K3)
5. The incident of occupational disease in an industry is such that workers have a 20%
chance of suffering from it. What is the probability that out of six workers 3 or more
will contract the disease? (CO1, K3)
6. A Market analyst believes that the stock market has a 0.70 probability of going up in
the next year if the economy should do well, and a 0.20 probability of going up if the
economy should not do well during the year. The analyst believes that there is a 0.80
probability that the economy will do well in the coming year. What is the probability
that stock market will go up next year? (CO1, K3)
PART C
1. Sun Network took a survey of 500 Sun Tv viewers to determine people favourite film
to telecast on Diwali matni time show,
Male Female
Ayan 80 100
Sarkar 100 25
Other 50 125
a) Draw the probability distribution for the above given data.
b) What is the probability of favourite film being Ayan?
c) What is the probability of female?
d) What is the probability of a Sun Tv viewer being male and preferring Sarkar?
e) What is the probability of a Sun Tv viewer being male or preferring Sarkar?
f) Amala is a Sun Tv viewer, what is the chance that her favourite film will be
Ayan?
g) Find out whether gender influences the choices of films for the above given data.

4. (CO1, K3)
2. An aptitude test was conducted on 600 employees of the Provincial Life Care Limited
in which the mean score was found to be 60 with standard deviation of 25.
Find:
i. What is the number of employees whose mean score was less than 30?
ii. What was the number of employees whose mean score exceed 80?
iii. What was the number of employees whose mean score is between 30 and 90?
(CO1, K3)

5. UNIT II SAMPLING DISTRIBUTION AND ESTIMATION
Basic definitions and rules for probability, conditional probability independence of events,
Bayes’ theorem, and random variables, Probability distributions: Binomial, Poisson, Uniform
and Normal distributions.
PART A
1. What is the central limit theorem? (CO2, K1)
Central limit theorem states that when an infinite number of successive random
samples are taken from a population, the sampling distribution of the means of those
samples will become approximately normally distributed with mean μ and standard
deviation
𝜎
√𝑁
, irrespective of the form of the population distribution.
2. What is sampling distribution? (CO2, K1)
The probability distribution of statistic (mean, proportion, standard deviation) of
samples of size ‘n’ from a given population is called sampling distribution.
3. What is sampling method and its types? (CO2, K1)
A sampling method is a way of selecting a group of individuals from a population to
conduct research on. There are two types of sampling methods: probability sampling
and non-probability sampling.
4. What is the systematic sampling? (CO2, K1)
Systematic sampling is a type of probability sampling method in which sample
members from a larger population are selected according to a random starting point
but with a fixed, periodic interval. This interval, called the sampling interval, is
calculated by dividing the population size by the desired sample size.
5. What is the stratified sampling? (CO2, K1)
Stratified random sampling is a form of probability sampling that provides a
methodology for dividing a population into smaller subgroups as a means of ensuring
greater accuracy of your high-level survey results. The smaller subgroups are called
strata. Stratified random sampling is also called proportional or quota random
sampling.
6. Define the term simple random sampling. (CO2, K1)
Simple random sampling is a type of probability sampling in which the researcher
randomly selects a subset of participants from a population. Each member of the
population has an equal chance of being selected.
7. What is statistical inference and its applications? (CO2, K1)

6. Statistical inference is the process of analysing the result and making conclusions
from data subject to random variation. It is also called inferential statistics.
Hypothesis testing and confidence intervals are the applications of the statistical
inference.
8. What is statistical estimation? (CO2, K1)
Statistical estimation refers to the process by which one makes inferences about a
population, based on information obtained from a sample.
9. What is point estimation? (CO2, K1)
A point estimate of a population parameter is a single value of a statistic. For
example, the sample mean 𝑋
̅ is a point estimate of the population mean μ. Similarly,
the sample proportion p is a point estimate of the population proportion P.
10. What is interval estimation? (CO2, K1)
An interval estimate is defined by two numbers, between which a population
parameter is said to lie. For example, a < x < b is an interval estimate of the
population mean μ. It indicates that the population mean is greater than ‘a’ but less
than ‘b’
PART B
1. Illustrate sampling distribution. Explain its properties. (CO2, K2)
2. A random sample of 144 observations yields sample mean 𝑋
̅ = 160 and sample
variance s2
= 100. Compute a 95% confidence interval for the population mean.
(CO2, K3)
3. Explain the relationship between sample size and error. (CO2, K3)
4. Briefly explain the confidence interval for proportion and mean. (CO2, K2)
5. Explain statistical estimations with suitable examples. (CO2, K2)
6. How do you find confidential interval of population mean in both large sample and
small sample case? (CO2, K2)
PART C
1. The BOB (better of bests) departmental store wants to have an idea of the level of
satisfaction of its customers. The owner wants the exercise to be done on a Sunday
when about 2000 customers visits the store. The store is spread over two floors and
has ten cash counters. Discuss as to how a survey can be conducted for accessing the
overall level of satisfaction on a scale of 0 – 10. The management of the store feels
that a sample of 100 customers could be sufficient for the purpose. (CO2, K4)

7. 2. Discuss various techniques you would apply for sampling from a population.
(CO2, K2)

8. UNIT 3 TESTING OF HYPOTHESIS - PARAMETIRC TESTS
Hypothesis testing: one sample and two sample tests for means and proportions of large samples (z-
test), one sample and two sample tests for means of small samples (t-test), F-test for two sample
standard deviations. ANOVA one and two way
PART A
1. What is hypothesis testing? (CO3, K1)
Hypothesis testing is a statistical method used to determine if there is enough
evidence in a sample data to draw conclusions about a population
2. What is Type I and Type II error? (CO3, K1)
A type I error (false-positive) occurs if an investigator rejects a null hypothesis that is
actually true in the population; a type II error (false-negative) occurs if the
investigator fails to reject a null hypothesis that is actually false in the population.
H0
True False
Reject H0 Type I error -
Accept H0 - Type II error
3. What is null hypothesis? (CO3, K1)
A null hypothesis is a type of statistical hypothesis that proposes that no statistical
significance exists in a set of given observations. It is represented as H0.
4. What are the various elements in hypothesis testing? (CO3, K1)
The hypothesis test consists of several elements namely two statements (the null
hypothesis and the alternative hypothesis), the test statistic which in turn give us the
P-value and the critical value which in turn give us the rejection region.
5. Describe various stages involved in hypothesis testing. (CO3, K1)
The four steps of hypothesis testing include stating the hypotheses, formulating an
analysis plan, analysing the sample data, and analysing the result.
6. What do you mean by analysis of variance? (CO3, K1)
Analysis of Variance (ANOVA) is a statistical formula used to compare variances
across the means (or average) of different groups. It is used to determine if there is
any difference between the means of different groups.
7. What are the assumptions of ANOVA? (CO3, K1)
The three primary assumptions in ANOVA are
i. Population is normally distributed
ii. These distributions have the same variance
iii. The data are independent.

9. 8. When does the Z – Test apply? (CO3, K1)
Z test is used if there is a significant variation in the sample and population means. It
is used to test how an observed value of coefficient correlation r varies significantly
from the hypothetical value and to test whether two sample values of coefficient
correlation r vary significantly.
9. State Critical Value. (CO3, K1)
A critical value is the value of the test statistic which defines the upper and lower
bounds of a confidence interval, or which defines the threshold of statistical
significance in a statistical test.
10. Discuss the difference between parametric and non-parametric test. (CO3, K2)
In Statistics, a parametric test is a kind of hypothesis test which gives generalizations
for generating records regarding the mean of the primary/original population. The
non-parametric test does not require any population distribution, which is meant by
distinct parameters. It is also a kind of hypothesis test, which is not based on the
underlying hypothesis. In the case of the non-parametric test, the test is based on the
differences in the median. So, this kind of test is also called a distribution-free test.
PART B
1. Given a sample mean of 83, a sample standard deviation of 12.5 and the sample size
of 22, test the hypothesis that the value of the proportion mean is 70 against the
alternative that it is more than 70. Use the 0.025 significance level. (CO3, K3)
2. Two samples are drawn from two normal population. From the following data, test
whether the two samples have the same variance at 5% level of significance:
(CO3, K3)
Sample 1 60 65 71 74 76 82 85 87
Sample 2 61 66 67 85 78 63 85 86 88 91
3. Two independent samples of eight and seven items respectively following values of
the variable:
Sample 1 9 11 13 11 15 9 12 14
Sample 2 10 12 10 14 9 8 10
Do the two estimates of the population variance differ significantly at 5% level of
significant? (CO3, K3)
4. Four doctors each test four treatments for a certain disease and observe the number of
days each patient takes to recover. The results are as follows: (recovery time in days)
(CO3, K3)

10. Doctors
Treatment
1 2 3 4
A 10 14 19 20
B 11 15 17 21
C 9 12 16 19
D 8 13 17 20
Discuss the difference between (i) Doctors and (ii) treatments
5. A social experiment shows that in a group 20% people are ready to sell their votes for
money when they are offered a small amount. In another group, 40% people are ready
to sell their votes when they are offered huge sum money. In both the cases, 1000
members each were participated. Test at 5% level of significance (Two – sided) that
there is a difference two proportions. (CO3, K3)
6. A manufacturing company has purchased three new machines of different makes and
wishes to determine whether one of them is faster than the others in producing a
certain output. Five hourly production figures are observed at random from each
machine and the results are given below: (CO3, K3)
A1 25 30 36 38 31
A2 31 39 38 42 35
A3 24 30 28 25 28
Use analysis of variance and determine whether the machines are significantly different in
their mean speed of 5% level of significance.
PART C
1. In a feeding experiment of swine, three ratios R1, R2, R3 were tried. The animals were put
into three classes of three each according to litter and initial body weight. The following table
gives the gain in the body weight in Kg in a certain period. Analyse the data and state your
conclusion. (CO3, K3)
Class I Class II Class III
R1
R2
R3
4
14
3
16
18
14
10
19
7
2. At one of the management institutes, it is found that people come from diverse educational
background and from different cities across India, which could be Metro, Large, Medium, or
Small. To see if these two factors are dependent on each other, data about student having
different background such as B. Tech, B. Com, B.A, C. A and other with their corresponding
size of the cities is recorded. This data is shown below in table. At the 0.05 significant level,
does educational background differ according to the size of the cities to which these students
belong? (CO3, K3)
Educational
Background
Metro Large
Medium /
Small
Total
B. Tech 15 25 15 55

11. B. Com 35 20 15 70
B. Sc 10 10 5 25
B. A 15 10 20 45
C. A 10 5 4 20
Other 15 10 10 35

12. UNIT IV NON-PARAMETRIC TESTS
Chi-square test for single sample standard deviation. Chi-square tests for independence of attributes
and goodness of fit. Sign test for paired data. Rank sum test. Kolmogorov-Smirnov – test for goodness
of fit, comparing two populations. Mann – Whitney U test and Kruskal Wallis test. One sample run
test.
1. What is the chi-square test for a single variance? (CO4, K1)
The Chi-Square Test for One Variance is a statistical test used to compare the
variance of a sample to a known population variance. It is used to test a hypothesis
about the population variance.
2. Define rank sum test. (CO4, K1)
Rank sum test also known as Wilcoxon signed rank test is a non-parametric test used
for testing the difference of median in a paired data.
3. What is meant by paired data? (CO4, K1)
Paired data means that the two group’s value being compared are linked naturally and
rise from the individuals that are being measured more than one time usually.
4. State the working rule framed in Mann-Whitney test. (CO4, K1)
a) Set null hypothesis.
b) Combine all samples in array and arrange.
c) Find the ranks.
d) Calculation of U test.
5. What is Kruskal Wallis test and when it is used? (CO4, K1)
Kruskal Wallis test is a nonparametric (distribution free) test which assesses for
significant differences on a continuous dependent variable by a categorical
independent variable (with two or more groups). It is used when the assumptions of
one-way ANOVA are not met.
6. What are the two types of Pearson’s Chi-Square tests? (CO4, K1)
a) The Chi-Square goodness of fit test
b) The Chi-Square test of independence
7. State the uses of Chi-Square test. (CO4, K1)
The chi-square goodness of fit test is used to test whether the frequency distribution of
a categorical variable is different from the expectations. The chi-square test of
independence is used to test whether two categorical variables are related to each
other.
8. Write the formula for Kruskal Wallis test. (CO4, K1)

13. H =
12
𝑛(𝑛+1)
∑
𝑅𝑖
2
𝑛𝑖
− 3(𝑛 + 1)
𝑘
𝑖=1
Where, H = Kruskal Wallis test; n = Total number of observations in all
sample; k = Number of independent samples; ni = Number of cases in the ith
sample;
Ri = Rank of the sample
9. Write down the formula for Chi-Square test of standard deviation. (CO4, K1)
X2
=
(𝑛−1)𝑠2
𝜎2
for n – 1 degree of freedom
Where, n = Sample size; s = Standard Deviation; s2
= Variance sample; σ =
Expected standard deviation; σ2
= Expected variance
10. What is run test? (CO4, K1)
The runs test is a statistical test to determine whether random selection has been made
in the process of sample selection from an ordered population.
PART B
1. Discuss on One-Sample and Two Sample Sign Test. (CO4, K2)
2. The following is an arrangement of 25 men M and 15 women W lined up to purchase
tickets for a premier picture show: M WW MMM W MM W M W M WWW MMM
W MM WWW MMMMMM WWW MMMMMM Test for randomness at 5% level of
significance. (CO4, K3)
3. Verify whether Poisson distribution can be assumed from the data given below:
(CO4, K3)
No. of Defects 0 1 2 3 4 5
Observed Frequency 6 13 13 8 4 3
4. Melisa’s Boutique has three mall locations. Melisa keeps a daily record for each
location of the number of customers who actually make a purchase. A sample of those
data follows. Using the Kruskal Wallis test, can you say at the 0.05 level of
significance that her stores have the same number of customers who buy? (CO4, K3)
X mall 99 64 101 85 79 88 97 95 90 100
Y mall 83 102 125 61 91 96 94 89 93 75
Z mall 89 98 56 105 87 90 87 101 76 89
5. Fit a binomial distribution for the following data and also test the goodness of fit.
(CO4, K3)
x 0 1 2 3 4 5 6 Total
f 5 18 28 12 7 6 4 80
6. Use Mann Whitney U test to determine whether there is a difference at 5% level of
significance between cables made of Alloy I and Alloy II. (CO4, K3)

14. Alloy I
18.3 16.4 22.7 17.8
18.9 25.3 16.1 24.2
Alloy II
12.6 14.1 20.5 10.7 15.9
19.6 12.9 15.2 11.8 14.7
PART C
1. Use the run test to test the randomness of return on RIL, Infosys, NIFTY and Dollar, based on
the following data.
(CO4, K3)
Date Infosys Reliance Nifty Dollar Value
01-12-2018 2174.9 961.6 28559.6 62.05
02-12-2018 2126.6 962.7 28444 61.9
03-12-2018 2123.5 968.4 28442.7 61.87
04-12-2018 2101.8 958.6 28562.8 61.93
05-12-2018 2070.3 957.4 28458.1 61.83
08-12-2018 1970.2 944.6 28119.4 61.97
09-12-2018 1964.8 939.9 27791.1 61.88
10-12-2018 1963.8 932.6 27831.1 61.8
11-12-2018 1921.1 906.2 27602 62.04
12-12-2018 1938.7 882.4 27350.7 62.25
15-12-2023 1924.8 878.8 27319.6 62.2
2. Discuss on principles and techniques of various non-parametric tests. (CO4, K2)

15. UNIT V CORRELATION AND REGRESSION
Correlation – Coefficient of Determination – Rank Correlation – Regression – Estimation of
Regression line – Method of Least Squares – Standard Error of estimate.
PART A
1. What is Correlation? (CO5, K1)
Correlation is a statistical term describing the degree to which two variables move in
coordination with one another. If the two variables move in the same direction, then
those variables are said to have a positive correlation. If they move in opposite
directions, then they have a negative correlation.
2. What is the use of correlation and regression? (CO5, K2)
Correlation and Regression are the most commonly used techniques for investigating
the relationship between two quantitative variables. Correlation quantifies the strength
of the linear relationship between a pair of variables, whereas regression expresses the
relationship in the form of an equation.
3. What can be the values for correlation coefficient? (CO5, K1)
The value of a correlation coefficient an vary from -1 to 1. -1 indicates a perfect
negative correlation and +1 indicates a perfect positive correlation. A correlation
coefficient of zero means there is no relationship between the two variables.
4. What is the interpretation of the correlation coefficient values? (CO5, K2)
When there is a negative correlation between two variables as the value of one
variable increases, the value of the other variable decreases and vice versa. In other
words, for a negative correlation the variables work apposite each other. When there
is a positive correlation between two variables as the value of one variable increases
the value of the other variable also increases. The variables move together.
5. What is simple regression? (CO5, K1)
Simple linear regression is a regression model that estimates the relationship between
one independent variable and one dependent variable using a straight line. Both
variables should be quantitative.
6. What is the use of regression? (CO5, K2)
The regression statistics can be used to predict the dependent variable when the
independent variable is known. Regression goes beyond correlation by adding
prediction capabilities.
7. Define rank correlation. (CO5, K1)

16. It is a measure of correlation which is used when quantitative measures for certain
factors. It can be arranging in serial order.
8. Define least square method. (CO5, K1)
The least square method is the process of finding the best-fitting curve or line of best
fit for a set of data points by reducing the sum of the squares of the offsets (residual
part) of the points from the curve.
9. List advantages of least square. (CO5, K1)
a) Objective method.
b) Easy calculation.
c) Determines trend values.
d) Flexible method.
10. What is coefficient of determination? (CO5, K1)
The coefficient of determination (R²) measures how well a statistical model predicts
an outcome. The outcome is represented by the model's dependent variable. The
lowest possible value of R² is 0 and the highest possible value is 1.
PART B
1. Explain the Karl Pearson’s coefficient of correlation and Spearman’s rank correlation.
(CO5, K2)
2. Explain regression line and why are there two regression lines? When do we use one
in the preference to the other? (CO5, K2)
3. Study the correlation from the given data of industrial city. (CO5, K3)
Sales (‘000 in Rs.) 125 170 175 180 190 210 250 300 320 400
Profit (‘000 in Rs.) 20 29 32 35 34 41 55 60 64 70
4. The following table gives the age of bike and annual maintenance cost. Obtain the
regression equation. Also find the maintenance cost of the bike whose age is 12 years
old. (CO5, K3)
Age of Bike (years) 2 5 7 11 15
Maintenance Cost (in Thousands) 1 3 5 8 10
5. Cost accounts often estimate overhead based on the level of production. At the
Standard Knitting Co., they have collected information on the overhead expenses and
units produced at different plants and wants to estimate a regression equation to
predict future overheads. (CO5, K3)
Overheads 191 170 272 155 280 173 234 116 153 178
Units 40 42 53 35 56 39 48 30 37 40
i. Develop the regression equation for the cost accountants

17. ii. Predict overheads when 50 units are produced
6. Find the coefficient of correlation between X and Y using the following data:
(CO5, K3)
X 65 67 66 71 67 70 68 69
Y 67 68 68 70 64 67 72 70
PART C
1. Obtain the equation of regression lines from the following data using method of least
square. Hence find the co-efficient of correlation between x and y. also estimate the
value of: (CO5, K3)
i. Y when x = 38
ii. X when y = 18
X 20 26 29 30 31 31 34 35
Y 20 20 21 29 27 24 27 31
2. Find the standard error of estimate of y on x and x on y from the following data:
(CO5, K3)
x 1 2 3 4 5
y 2 5 9 13 14