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STATISTICS FOR MANAGEMENT
STATISTICS FOR MANAGEMENT
UNIT I: INTRODUCTION
1. What do you meant by probability?
The word "probability" that denotes that there is an uncertainty about the happening of
event. It can also be defined as an expression of likelihood or chance of occurrence of an
event. The probability is a number, which ranges from 0 to 1.
2. Define Event?
An outcome or a combination of outcomes of a random experiment is called an event.
Events are generally denoted by capital letters A, B, C, etc. Example: If a coin is tossed,
getting a head or tail is an event.
3. Define the term random experiment?
The term experiment is used to describe an act, which can be repeated under some
conditions. Random experiment is one which result depends on chance. That is the results
can never be predicted. Example: Tossing a coin, throwing a die.
4. Define Mutually Exclusive events:
Two events if the occurrence completely excludes the occurrence of the other. That is the
two events cannot occur simultaneously. Example: Tossing a coin, either head or tail will
turn up.
5. Define Equally likely events:
Events are said to be equally likely when one does not occur more often than others.
Example: If an unbiased coin or die is thrown, each face may be expected to be observed
approximately the same number of times in the long run.
6. Define compound Events:
The joint occurrence of two or more events is called compound events.
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7. Define sample space:
The set of all possible outcomes of a given experiment is called its sample space. Each
element of the sample space is called a sample point.
Example: When a die is thrown, the sample space S= 1, 2, 3, 4, 5, 6
8. State the statistical definition of probability?
If the probability of an event can be determined only after the actual happening of an
event, is called statistical probability. Statistical probability is also called Empirical
Probability or A posterior probability.
9. Define Independent Events?
Two events are said to be independent if the occurrence of one does not affect the
occurrence or non-occurrence of the other.
10. State the three axioms of probability?
a) The probability of any event ranges from zero to one. That is O ≤ P (A) ≤1
b) The probability of the entire space is 1 That is P (S) = 1
c) If A and B are mutually exclusive events, then the probability of occurrence of either A
or B denoted by P (A ∪ B). That is P (A ∪B) = P (A)+ P (B)
11. Define non-independent events?
If the occurrence of one event is affected by the occurrence of the other event, then the
two events are said to be non-independent Events.
12. State the mathematical definition of probability?
If the probability of an event can be calculated event before the actual happening of an
event that is even before conducting the experiment it is called mathematical probability.
P (A) =M = Number of Favorable Cases N
Total Number of Exhaustive Events.
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13. What do you meant by exhaustive events?
Events are said to be exhaustive when their totality includes all the possible outcomes of
a random experiment.
14. What do you meant by conditional probability?
Two events A and B are said to be dependent when B can occur only when A is known to
have occurred. The probability attached to such an event is called conditional probability
(A/B) =P (AB)/P(B) For three events A, B and C.
P (ABC) = P (A) x P (B/A) x p (C/AB).
15. Write short notes on Baye's Theorem?
Baye's Theorem is based on the concept that probabilities should be revised when some
new information is available and it reduces the element of risk involved in decision
making. It is given by British mathematician Thomas bayes (1702-61). His theorem was
known as Baye's Theorem or Rule.
16. What is meant by prior probability and posterior probability?
a. Prior:
Probabilities before revision, by Baye's rule, are called prior or simply Priori
because they are determined before the sample information is taken into account.
b. Posterior:
A probability, which has been revised in the light of sample information, is called
posterior probability or revised probability.
17. Define Theoretical discrete distribution?
Distributions, which are not obtained by actual observation or experiment but are
mathematically deducted on certain assumptions, are called theoretical Distribution.
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18. Explain Binomial distribution?
If 'P' be the probability of success for an event and 'q' be the probability of its failure in a
single trial, then the probability of exactly 'X' success and (n-x) failures in a series of 'n'
independent trials is P (x) = nCx px qn-x
19. State the assumptions of Binomial Distribution?
 Trials are repeated under identical conditions for a fixed number of times.
 There are only two mutually exclusive outcomes (ie) successes or failure.
 The probability of success in each trial remains constant and does not change from trial to
trial.
20. What are the characteristics of a Binomial distribution?

It is a discrete distribution.

Mean = np, Variance = npq

The mode of binomial distribution is that value of the variable, which occurs with the
S.D = √npq
largest probability. It may be either one or two modes.

Skewness = p-q / √npq
21. Explain Poisson distribution?
If the number of trials 'n' is very large and the probability of success P is very small so
that the product np always exists and is finite. It relates to rare events.
22. What are conditions of Poisson distribution?
 The number of trials 'n' is indefinitely large n →α
 np= m (finite)
 The prob. of success p for each trial is very small, ie P →O
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23. State the characteristics of Poisson distribution?
 It is a discrete distribution, where random variable assumes a countable infinite number
of values 0, 1, 2. . . . . .
 Mean = m= variance, S.D = √ m np = m remains constant
24. Write down the density function of normal distribution?
It is defined by the probability density function. Y =1/ σ√2π exp [-(x-µ)2 / 2 σ2 ] π =
3.14; e = 2.71828 Ranges from (-α + α ) σ→ S.D
25. State the properties of N.D?
1. The normal curve is perfectly symmetrical about the mean (µ) and is bell shaped
2. It has only one mode.
3. The mean and median coincide with mode
4. The points of inflexion are at x = µ ±σ
5. The maximum ordinate is at x = µ Its value is1/ σ√2π
26. Define statistic?
Statistic is the science which makes use of the methods for collecting classifies,
presenting, comparing, and interpreting numerical data which are collector to make some
impairing about the data.
27. Types of statistics?
 Descriptive statistic
 Analytic statistic
 Inductive statistic
 Impressive statistic
 Applied statistic
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28. What are methods of collecting the primary data?
 Direct personal investigation
 Indirect oral investigation
 By local report
 By schedule and questionnaire
29. Write about the classification of data?
 Qualitative data
 Quantitative data
 Chronological data
 Geographical data
30. What are the types of variables?
 Discrete variables
 Continuous variables
31. Define the measure of the central tendency?
Quantitative data shows a tendency to concentrate at certain values usually somewhere in
the center of the distribution. Measure of this tendency are called measure of Centre
tendency the average of statistical series is the value of the variable which is
representative of the entire distribution.
32. Define arithmetic mean?
Arithmetic mean is a set of observation is the sum divided by the number of observation.
33. Define median?

Median of distribution is a value of the variable which divided it into two equal parts.
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
Median is the value of the middle item when the items are arranged is ascending or
descending order of magnitude. It is the value which exceeds and it is exceed by the same
number of observation. It is positional average.
34. Define the mode?
Mode is the value which excess most frequency is a set of observation. It is the value of
variable which predominate in the series.
35. Define the mean deviation?
Mean deviation is the arithmetic mean of the absolute value of deviation of observation
from mean or median or mode.
36. What is standard deviation?
Standard deviation is the positive square root of arithmetic mean of the square of the
deviation of observation from the arithmetic mean.
37. Define the co-efficient of variation?
When we want to compare variability of two series which defer widely in the averages or
which are measure in different unit.
38. Define the skewness?
Skewness refers to lack of symmetric in the shape of a frequency distribution.
39. What are the measures of the skewness?
Skewness indicates the direction and extends of skewness. The study of skewness is the
study of distribution of item around the Centre tendency. The distribution or either side of
the mode helps in deciding the direction of skewness.
40. Define the symmetrical distribution?
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1. It is the set of symmetrical if the frequency is equally distributing both side of the Centre
value.
2. A symmetrical distribution may be either bell shape or U shaped.
41. Define skewed distribution?
A frequency distribution said to be skewed. A skewed distribution may be positive
skewed or negative skewed or L shaped skewed or J shaped skewed.
42. What is bowley’s co-efficient of skewness?
In a symmetric the two quartile are in equal distribute from median so that the difference
between the median and lower quartile Q1and Q2 is equal to difference between the upper
quartile and median.
M-Q1=Q3-M=0
43. What is absolute value of distribution?
It is an asymmetric distribution difference between the median and upper quartile &
median quartile. Their value fines absolute value distribution.
M-Q1≠Q3-M≠0
44. Defines harmonic mean?
Harmonic mean is the reciprocal of the arithmetic mean of the reciprocal of the values of
the item in series.
45. Measures of dispersion?

Range

Quartile deviation

Standard deviation

Mean deviation
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UNIT - II: SAMPLING DISTRIBUTION AND ESTIMATION
1. Define population?
The word 'population' or Universe denotes aggregate or group of individual
objects of any nature whose general characteristics are studied by a statistical
investigation. The population may finite or infinite.
2. Define sample?
Sample is a finite sub set of the population and the number of items in a sample is
called size of a sample. It may be large or small sample.
3. Define standard Error?
The standard deviation of sampling distribution of statistic is known as standard
error.
4. What is a point estimate?
An estimate of a population parameter given by a single number is called a point
estimate of a parameter.
5. What is interval estimates?
An interval estimate would always be specified by two values. That is the lower
one and the upper one. In simple words, interval estimation refers to the estimation of a
parameter by a random interval is called confidential interval.
6. Who is an estimator and estimate?
The numerical value of the sample mean is said to be an estimate of the
population. In statistical measure used that is the method of estimation is referred to as an
estimate.
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7. Define parameters and statistic?
Statistical constants of population namely mean ( µ) and variance ( σ2) etc, which
are usually referred as parameter .The statistical measures from sample observation are
known as mean (x) and S.D (S), variable (S2).
8. Explain the central limit theorem?
According to this theorem, if we selected larger number of simple random
samples, say from any population and determine the mean of each sample, the
distribution of these sample means will tend to be described by the normal probability
distribution with a mean µ and variance σ2/n.
9. What is Small Sample?
If number of observations less than 30 is called small sample.
10. List the criteria of a good estimator?
Unbiasedness, Consistency, Efficiency, Sufficiency.
11. What is a Sampling distribution?
From a population a number of samples are drawn of equal size n. find out the mean
of each sample. The means of the sample are not equal. The means with their respective
frequencies are grouped. The frequency distribution so formed as sampling distribution.
12. What are the elements and variables in a data set?
Frequency and Random variables.
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UNIT – III: TESTING OF HYPOTHESIS - PARAMETIRC TESTS
1. Define parameters and statistic?
Statistical constants of population namely mean ( µ) and variance ( σ2) etc, which
are usually referred as parameter. The statistical measures from sample observation are
known as mean (x) and S.D (S), variable (S2).
2. Explain the central limit theorem?
According to this theorem, if we selected larger number of simple random
samples, say from any population and determine the mean of each sample, the
distribution of these sample means will tend to be described by the normal probability
distribution with a mean µ and variance σ2/n
3. What is hypothesis?
"A hypothesis in statistics is simply a quantitative statement about a population".
It is based on assumptions.
4. What is Null hypothesis?
Null hypothesis is the hypothesis, which is tested for possible rejection under the
assumption that it is true and is denoted as H0.
5. What is Alternative hypothesis?
It is the statement about the population, which gives an alternative to the null
hypothesis and is denoted by H1.
6. Define Type I and Type II error?
Rejection of the hypothesis when it should be accepted is known as Type I error.
It is denoted as α. Acceptance of a hypothesis when it should be rejected is known as Type II
error. It is denoted as β.
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7. What do you meant by level of significance?
In testing a given hypothesis, the maximum probability with which we could be
willing to risk is called level of significance of the test. It is usually denoted as α٪
8. What is critical value?
The value of the test statistic, which separates the sample space into rejection region
and the acceptance region, is called the critical value.
9. Explain the procedure for testing of hypothesis?
a) Set up the Null hypothesis: H0
b) Set up the alternative hypothesis: H1
c) Choose an appropriate level of significance
d) Calculate the test statistic Z.
e) Compare the computed value with the table value. if Z > table value → Reject the Null
hypothesis, Z < table value → Accept the Null hypothesis.
10.Explain one-tailed test and two-tailed test?
One-tailed test:
In any test, the critical region is represented by a portion of the area under the
probability curve of the sampling distribution of the test statistic. A test of any statistical
Correct decision Type I error Ho is true Ho is false Type II error Correct decision hypothesis
where the alternative hypothesis is one tailed (right or left tailed) is called a one-tailed test.
Two-tailed test:
A test of statistical hypothesis where the alternative hypothesis is two tails. H0: µ =
µ0 against the alternative hypothesis H1: µ >µ0 and H1: µ< µ0 is known as two tailed test
in
such case the critical region is given by the portion of the area lie in both the tails of the
probability curve of the test statistic.
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11.Define sampling Distribution?
"The distribution of all possible values which can be assumed by some statistic computed
from samples of same size rand only drawn from the same population is called the sampling
distribution of the statistic".
12. Errors in Hypothesis Testing:
13. The F-Distribution (SNEDECOR’S)
F distribution is defined as the ratio off two dependent chi-square variates divided by
their respective degree of freedom and it follows snedcor’s F-distribution with (v1 and v2)
with the probability density function
14. ANOVA (ANALYSIS OF VARIANCE)
Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality
of two or more population (or treatment) means by examining the variances of samples that
are taken.
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ANOVA allows one to determine whether the differences between the samples are
simply due to random error (sampling errors) or whether there are systematic treatment
effects that cause the mean in one group to differ from the mean in another.
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UNIT – IV: NON-PARAMETRIC TESTS
1. A non-parametric statistical test
A non-parametric statistical test is a test whose model does NOT specify conditions about
the parameters of the population from which the sample was drawn.
Do not require
measurement as strong as that required for the parametric tests. Most non-parametric tests apply
to data in an ordinal scale, and some apply to data in nominal scale.
2. Three types of tests
Three types of tests

A one-sample test analyzes one field.

A test for related samples compares two or more fields for the same set of cases.

An independent-samples test analyzes one field that is grouped by categories of another
field.
3. one-sample test
Automatically compare observed data to hypothesize. Binomial test: categorical fields
with only two categories; Chi-Square test: all other categorical fields; and Kolmogorov-Smirnov
test: continuous fields. Test sequence for randomness. Runs test: test the observed sequence of
data values for randomness.
4. Independent sample nonparametric tests
Independent sample nonparametric tests identify differences between two or more groups
using one or more nonparametric tests. Nonparametric tests do not assume your data follow the
normal distribution. 2 Independent samples: grouping variable has two categories. K
Independent samples: grouping variable has more than two (k) categories
5. Nonparametric Tests
Recall, nonparametric tests are considered “distribution-free” methods because they do
not rely on any underlying mathematical distribution. They do, however, have various
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assumptions that must be met. Do not be confused by not having the need to meet an assumption
of “normality” with the notion of “assumption less.”
6. Sign Test

Consider a simple example where 20 farmers are given two fertilizers (A & B) by an
extension agent and asked to evaluate which one was “better”.

In the end, 16 farmers reported that fertilizer A was the better of the two (based on
qualitative estimates of visual quality, greenness, yield, etc.).

If there were no difference between fertilizers, we would expect fertilizer A to be
binomially distributed with P = 0.05 and N = 20.
7. Mann-Whitney U-Test

This is the nonparametric analog to the two-sample t-test with equal variances.

It is used primarily when the data have not met the assumption of normality (or
should be used when there is sufficient doubt).

Assumptions: Independent samples Continuous variable Equal variances Identical
(non-normal) distributions
8. Kolmogorov-Smirnov Test

This is the nonparametric analog to the two-sample t- test with unequal variances.

It is often used when the data have not met either the assumption of normality or the
assumption of equal variances.

Assumptions: Variable at least ordinal Two samples are independent both simple
random samples Identical distributions.
9. Wilcoxon Signed-Ranks Test

This is the nonparametric analog to the paired two-sample t-test.

It is used in those situations in which the observations are paired and you have not
met the assumption of normality.

Assumptions: Differences are continuous Distribution of differences is symmetric
Differences are mutually independent Differences all having the same median.
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10. Choosing the Appropriate Two-Sample Statistical Test
Independent Variants: Normal Data, Equal Variances Equal Variance t-test, Normal
Data, Unequal Variances Unequal Variance t-test, on-normal Data, Equal Variances MannWhitney U-test (Wilcoxon), Non-normal Data, Unequal Variances Kolmogorov-Smirnov test.
Paired Variants:
Normal Data, continuous variable Paired t-test,Non-normal Data,
ranked variable Wilcoxon Signed-Rank test.
11. Chi Square Tests
The χ2 statistic appears quite different from the other statistics which have been used in
the previous hypotheses tests. It also appears to bear little resemblance to the theoretical chi
square distribution just described. χ2 =sum of (Oi − Ei)2/ Ei
12. Chi Square Calculation
Each entry in the summation can be referred to as “The observed minus the expected,
squared, and divided by the expected.” The chi square value for the test as a whole is “The sum
of the observed minus the expected, squared, and divided by the expected.”
13. Chi Square Test for Independence
The chi square test for independence of two variables is a test which uses a cross
classification table to examine the nature of the relationship between these variables. These
tables are sometimes referred to as contingency tables or cross classification tables
14. Assumptions about Ei
No expected case Ei should be less than 1. No more than 20% of the cells should have
less than 5 expected cases.
15. What is a non-parametric test?
The tests, which do not depend upon the population parameters such as mean and the
variance, they are called non-parametric tests.
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16. What do you meant by non-parametric statistics?
Non-parametric statistics is a collection of tools for data analysis that offers a Different
approach to many of the decision problems.
17. What are the merits of non-parametric tests?
Non-parametric tests are distribution free. That is they do not require any assumption to
be made about population. They are simple to understand and easy to apply when the sample
sizes are small. Non-parametric test make fewer and less stringent assumptions than do the
classical procedures. It is less time consuming.
18. What are the limitations of non-parametric test?
As the sample size gets larger, data manipulations required for non-parametric procedures
are sometimes laborious unless appropriate computer software is applicable. A collection of
tabulated critical values for a variety of non-parametric tests under situations dealing with small
and large 'n' is not readily available.
19. What are the methods used in non-parametric tests?
The following are the methods used in non-parametric tests. They are:
1. The sign test
2. A Rank sum test
3. The one sample Runs Test
4. The kruskal wallis or H test
5. The spearman's Rank correlation procedure
20. What do you meant by the sign test?
The sign test is the simplest of the non-parametric tests. It is based on the direction of a
pair of observations and not on their numerical magnitude. It can be used as: we count:
Number of + signs
Number of – signs
Number of 0's cannot be included either as positive or negative.
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21. What do you meant by a rank sum test?
It is also known as Mann-Whitney U test. It is used only when two population are
involved. It is also based on Ranks (R 1, R 2)
22. What do you meant by one sample runs test?
The Run test is based on order in which the sample observations are obtained. It is
the useful technique for testing the null hypothesis that the observations have been indeed
to be dream.
23. Define the team "Run"?
Run is defined as, "set of identical or related symbols contained between two
different symbols are no symbol".
24. What is Kruskal Wallis test?
It is also known as H – test, It is the extension of Mann Whitney U – test. It is
used when more than two populations are involved.
25. Define correlation?
Correlation analysis deals with the association between two or more variables.
26. What is the significance of correlation?
The following are the significances of correlation:

There are some kinds of relationship between variables. For example Relationship
between price and supply, income and expenditure etc.

The two variables are closely related. That is the estimate the value of one Variable
given the value of another.

The effect of correlation is to reduce the range of uncertainty.
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27. What is positive correlation?
It is the two variables tend to move together in the same direction. That is an increase in
the value of one variable is accompanied by an increase in the value of other variable.
28. What is negative correlation?
If two variables tend to move together in opposite directions so that an increase or
decrease in the value of one variable is accompanied by a decrease or increase in the
value of other variable then the correlation is called negative or inverse correlation.
29. What do you meant by linear correlation?
If the amount of change in one variable bears constant ration to the amount of
change in the other variable then the correlation is said to be linear.
30. What do you meant by non-linear correlation?
If the amount of change in one variable does not bear constant ratio to the amount
of change in other variable, it is also known as curvilinear correlation.
31. What are the methods used in correlation?
The following are the various methods used in correlation.

Scatter diagram method.

Graphic Method.

karl pearson's coefficient of correlation.

Concurrent Deviation Method.

Method of least squares.
32. What do you meant by scatter diagram?
This is a simple method of diagrammatic representation of a bivariate distribution
for ascertaining the nature of correlation between two variables.The 'n' pairs of values (x1,
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y2), (x2, y2) and ……. (xn, yn) of two variables x and y can be plotted as dots in the xy –
plane. This diagram of dots so obtained is called as scatter diagram.
33. What do you meant by Karl Pearson's coefficient of correlation?
Karl Pearson, a great Biometrician and statistician suggested a mathematical
method for measuring the magnitude of linear relationship between two variables. This
method is widely used and is known as pearsonian coefficient of correlation. It is denoted
by the symbol 'r'.
34. State the assumptions of the pearsonian coefficient?
There is linear relationship between the variables. That is when the two variables
are plotted on a scatter diagram a straight line will be formed by the points so plotted.
There is a cause and effect relationship between the forces affecting the distribution of
the items in the two series. Variables like height, such forces affect weight, price,
demand, supply etc that a normal distribution is formed.
35. What are the limitations of Pearson’s coefficient?
The correlation coefficient always assumes linear relationship regardless of the
fact

whether that assumption is correct or not.

This method takes more time to computer the value of correlation coefficient.

The value of the coefficient is unduly affected by the extreme items

Great care must be exercised in interpreting the value of this coefficient as very often the
coefficient is misinterpreted.
37. State the properties of correlation coefficient?
1. The coefficient of correlation of correlation lies between – 1 and + 1.
2. The coefficient of correlation is independent of change of scale and origin of the
variable x and y.
3. The coefficient of correlation is the geometric mean of two regression coefficients.
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4. The degree of relationship between the two variables is symmetrical.
38. What do you meant by Rank correlation coefficient?
In 1904, Charles Edwin Spearman a British psychologist found out the method by
determining the coefficient of correlation by ranks. This measure is useful in dealing with
qualitative characteristics such as intelligence, beauty, morality, character etc.
39. State the features of spearman's correlation coefficient?
1. The sum of the difference of ranks between two variables shall be zero That is
Σd = o
2. Spearman's correlation coefficient is distribution free.
40. What are the Merits of Rank Method?
1. It is simple to understand and easy when compared to Kart pearson's Method.
2. Where the data are of a qualitative nature like honesty, efficiency, intelligence
etc.
41. State the limitations of Rank Method?
1. This method cannot be used for finding out correlation in a grouped frequency
distribution.
2. Where the numbers of items exceeds 30, the calculations become quite tedious
and require a lot of time.
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STATISTICS FOR MANAGEMENT
UNIT – V: CORRELATION, REGRESSION AND TIME SERIES
ANALYSIS
1. Define Time series?
A time series is a set of observations arranged in chronological order, a set of data
depending on the time is called time series.
2. What are the objectives of time series?
o study of the past behavior of the data
o To forecast future behavior
o Comparison with other series
o study of present fluctuations
o Estimation of trade cycles
3. What are the models of time series?
a. Additive model
b. Multiplicative model
c. Mixed model
4. What are the types of variations in time series?
1. Long term
a. secular or trend,
b. Cyclical
2. Short term
a. seasonal, b. Irregular or erratic
Both cyclical and seasonal are regular.
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STATISTICS FOR MANAGEMENT
5. What is meant by secular or trend series?
The general tendency of the time series data to increase or decrease or stagnate during a
long period of time is called the secular trend or trend analysis.
6. What is meant by seasonal variations?
Seasonal variations refer to rhythmic forces of a change inherent in most time series
showing a regular or a periodic pattern of movement over a span of less than a year and have the
same or almost the same pattern year after year.
7. What is meant by cyclical variations?
As the economy expands during a period of boom, we would expect to find that such data
as sales, output or customer expenditure also show a rising trend; and during a period of slump,
we would expect them to show a downward trend. Thus wavelike motion may be observed in the
pattern of our data.
8. What is meant by trend analysis?
Trend analysis uses a technique called least squares to fit a trend line to a set of time
series data and then project the line into the future for a forecast. It is a special case of regression
analysis where the dependent variable is the variable is the variable to be forecasted and the
independent variable is a time.
9. What are the uses of time series analysis?
a. It is used to forecast future behavior
b. It is used to compare with other series
c. It is used to study present fluctuations
d. It is used to estimate trade cycles.
(FORMULAS - REFER AT END OF THE NOTES)
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