BUSINESS STATISTICS THEORY-SEMESTER 2

CA PANKAJ GOEL-9811860116 Page 1

CA PANKAJ GOEL

BUSINESS STATISTICS THEORY-SEMESTER 2

BY CA PANKAJ GOEL

CLASSES FOR BCOM/CA/CPT-ACCOUNTANCY/MATHEMATICS/STATISTICS/AUDITING

Measures of Central Tendency

Ques 1. Which type of Average should be used in following cases:

i) Size of shoes sold in large no: in a shop.

ii) Marks obtained in an exam.

iii) Average change in Cost of living of workers.

iv) When distribution has open- ended classes and wide variations. (B.Com (P) 99)

v) When Quantities are in ratio.

vi) A stockiest of readymade garments (B.Com (P) 2001)

vii) When average deprecation rate is to be calculated and depreciation is charged as per W.D.V. (B.Com (P) 99)

viii) When speed is changing but distance is constant and average speed is to be calculated. (B.Com (P) 99)

Ans. i) Mode ii) Mean iii) Geometric Mean iv) Mode or Median

v) Geometric Mean vi) Mode vii) Geometric Mean viii) Harmonic Mean

Ques 4. Fill in the blanks

a) Sum of deviation from mean is __________________

b) Absolute Sum of deviation is minimum from _______________

c) The Geometric Mean of a set of values lies between Arithmetic Mean and __________________

d) Median is same as ________________ quartile.

e) Mean & Mode are called______________ averages.

f) A distribution may have many__________.

Ans. a) Zero b) Media c) Harmonic Mean HMGMX

d) Second Quartiles Q2 because it covers 50% of item to its right and to its left.

e) Positional average f) Mode (like in Bimodal Series)

Comparison Between Different types of Averages

Mathematical Average Positional Average

Basis Mean GM HM Median Mode

1) Meaning

It is the figure obtained by dividing total values of various item by their number.

It is nth root of product of n item of a series

It is reciprocal of the arithmetic average of the reciprocals of the values of its various items.

It is value of middle item of a series arranged in ascending or descending order of

magnitude

It is the value

Which has the

greatest

frequency

classify in its

immediate neighborhood

2) Algebraic Manipulations

It is capable of algebraic manipulation i.e. we can calculate Combined Mean

Same Not capable of algebraic manipulation

Not capable of algebraic manipulation

No

3) Weighted Average

Weighted AM

= ΣW

ΣWX

Weighted

Geometric Mean = AL

ΣW

X log W Σ

Weighted

Harmonic Mean

=

X

W Σ

ΣW

No No

4) Properties

1) Sum of deviation from mean is zero i.e.

0X-X Σ

2) Sum of square of deviation from mean is minimum i.e.

2X-X Σ is

min

1) In case of GM, the product of item remains unchanged if each item is replaced by GM

2) In case of GM, product of corresponding ratio on either side is always equal.

___________

Absolute sum of deviation From median is minimum.

__________

5) Applicable to Open ended Classes

Indeterminable Indetermin-able

Indetermin-able

Determinable Determinable

6) Effect of extreme items

Yes

(greatest)

Yes

(less than AM)

Yes

(less than GM)

No No

Measures of Dispersion

Ques 1. a) Fill in the blank

a) The income of a person in a particular week is Rs 20 per day. The mean deviation of his income for 7 days would be __________ (B.Com (H) 89)

b) Absolute Sum of deviation is minimum from _____________ (B.Com (H) 95)

c) In any distribution, standard deviation is always ______________ the mean deviation.

d) All measure of relative dispersion are ____________ from unit employed

e) _________ method is most affected by extreme items while measuring dispersion by this method.

f) Under Normal Curve X ± 3σ covers _________ area.

g) The Quartile deviation includes ___________ of items.

h) In normal distribution QD is equal to ___________ SD & MD is equal to ___________ SD.

i) The standard deviation, is ____________ measure of dispersion.

Answers

a) As income for each day is Rs 20, so sum of deviation from X i.e. 7 will be zero, so even mean deviation will also, be zero.

b) Median i.e. | X – median | is min

c) greater or equal to i.e. ≥

d) free

e) Range

f) 99.73%

g) Central 50%

h) QD = 2/3 σ & MD = 4/5 σ

or .6748 σ = .7979 σ

i) absolute

Ques 4. State Which of the following statement is correct.

i) Mean, standard deviation and variance have the same unit.

ii) For calculating standard deviation, deviation can be taken from median also.

iii) Coefficient of variation is expressed in same units as original class.

iv) Since 0X - X Σ , then 2X - X Σ is also zero.

v) The standard deviation of a set of 50 items is 6.5. If every item is increased by 5, standard deviation is 11.5 (CA (P.E. -1) 98 N, 99 N)

vi) If standard deviation of a set of 50 item is 8. Then standard will be 16 if each item is multiplied by 2 (CA ( P.E. -1) 96M)

Answer

i) Incorrect, Mean & standard deviation have same unit but variance being square of standard deviation has different unit.

ii) Incorrect, For calculating SD, deviation can be taken only from mean.

iii) Incorrect C.V. is always expressed in percentage

iv) Incorrect 2Nσ

2X - X Σ , & is zero only when all y’s are equal.

v) Incorrect standard deviation remains unaffected by adding or subtracting a constant item.

vi) Correct.

Short Answer Questions

Ques 1. What is the difference between Mean deviation (Average Deviation) & Standard deviation? (B.Com (H) 89, 93, 2001, B.Com (P) 2000)

Ans. Difference between mean deviation & Standard deviation can be discussed as follows:

Basis Mean deviation Standard deviation

a) Meaning

b) Basis of calculation

c) Algebraic Sign

d) Algebraic Manipulation

It is the arithmetic mean of the absolute deviations of individual values from average of given data.

Mean deviation can be calculated from either mean, median or mode i.e. either of 3 average are necessary for its calculation

It ignores algebraic signs by taking modulus.

It is not capable of algebraic manipulation i.e. we cannot calculate combined MD.

It can be calculated as square root of arithmetic mean of square of deviation from arithmetic mean.

It can be calculated only from arithmetic mean only so, mean is the basis of its computation.

It doesn’t ignore the algebraic signs as here deviations are squared.

It is capable of Algebraic Manipulation as we can calculate Combined SD.

Ques 2. Why Standard deviation is considered as best measure of dispersion?

(B.Com (H) 80, 88, 89, 93, B.Com (P) 95, 98)

Ans. Standard deviation is square root of arithmetic mean of square of deviation from Arithmetic Mean i.e.

SD (σ) =

N

2X - X Σ

Standard deviation is the only measure of dispersion which satisfies most of the properties of good measure of dispersion & due to which it is considered as best measure of dispersion.

It rectifies the short comings of the other measure of dispersion & to prove this, it is compared with other measure of dispersion as done below.

a) Range & Quartile deviation are not based on each every item in the series because range only taken into account largest & smallest item whereas Quartile deviation is concerned with only first & third Quartile whereas standard deviation is based on each & every item of series.

b) Mean deviation ignore (+) & (-) signs which is not sound mathematically, & therefore in standard deviation we square the deviation from mean & in this way it satisfies important property of mean that sum of square of deviation from mean is minimum.

Besides above, it also satisfies several other properties which are discussed in Ques - 3.

Ques 3. Explain important properties of standard deviation.

(B.Com (H) 2002, B.Com (P) 2000, C.A. (P.E.-1) 97 M)

Ans. Important properties of standard deviation can be discussed as follows:

i) The value of standard deviation is independent of change of origin but not of change of scale. i.e. its value remain same if each item of series is increase or decreased by constant say K but its value will be affected similarly if each item is multiplied or divided by a constant K.

ii) It is capable of algebraic manipulation i.e. we can calculate combined standard deviation & variance.

Variance = σ2

&

Combined Standard deviation

2n

1n

22

σ2

n2

1σ

1n

22

d2

n2

1d

1n

12σ

d1 = X1 – X 12

d2 = X 2 – X 12

iii) For a symmetrical distribution

X ± 1 σ, Covers 68.27% of items

X ± 2 σ, Covers 95.45% of items

X ± 3 σ, Covers 99.73% of items

iv) The standard deviation of ‘n’ natural numbers will be

1-2

N 12

1σ

Where N = no. of items.

SKEWNESS

Ques 1. Identify the Correct Statements

a) Both Skewness & Kurtosis are indicative of the shape of the distribution.

b) For any symmetrical distribution

1/3 (Mean – Mode) = Mean – Median

c) In a highly skewed distribution, value of mean varies a great deal from that of median.

d) Two distributions with the same mean, standard deviation & coefficient of Skewness, have same peakedness.

e) β2 must always be positive.

f) For symmetrical distribution, coefficient Skewness is zero.

g) Variance is equal to second Central Moment.

h) Arithmetic Mean is equal to first moment about origin.

i) Series representing U-shape curve is symmetrical. (B.Com (P) 2001)

Answer a) Correct

b) Incorrect, in a symmetrical distribution mode = mean = median.

c) Correct

d) Incorrect because peakedness concept is related to Kurtosis which is measured by value of β2.

e) Incorrect, in can be negative in case of platykurtic distribution.

f) Correct

g) Correct

h) Correct

i) Incorrect because in such a series mean ≠ mode ≠ median

Ques 3. Fill in the blanks.

1) If mean & mode of a given distribution are equal, then its coefficient of Skewness is ________________.

2) Skewness is positive when mean _______________ mode.

3) In asymmetrical distribution, the distance between the _________ & the __________ is about ______________ of distance between the ____________ and the __________.

4) In case of symmetrical distribution, β1 = _______________.

5) In case of symmetrical distribution, quartiles are at equal distance from __________.

Answer i) zero ii) Greater than iii) Mean, Median, 1/3, of mean & mode

iv) zero v) Median i.e. Q3 – Median = Median – Q1

Short Answer Question

Ques 1. Differentiate between Dispersion & Skewness?

(B.Com. (H) 81, 2000, B.Com. (P) 2000)

Ans. The difference between Dispersion & Skewness can be summed up in the following table:

Difference between Dispersion & Skewness

Basis Dispersion Skewness

1) Meaning

2) Purpose

It is a measure of the extent variation in the individual items

e.g.

Series X Series Y

6 4

2 4

4 4

X = 4 4

but different variation in data or series y is more consistent as compared to series X.

When a distribution is not symmetrical, it is said to be Skewed i.e. absence of symmetry denotes presence of Skewness.

The following three figures would give an idea of an absence or presence of Skewness.

a) perfectly Symmetry Curve (Lack of (Skewness)

X = Median = Mode

b) Asymmetrical Curve

i) Right Tail (Positively Skewed)

Mode < Median < X

ii) Left Tail (Negatively Skewed)

X < Median < Mode

3) What it deals with?

4) Methods of Measuring

Its purpose is to identity amount of variation.

It deals with variability in general & spread of values around central value.

Dispersion can be measured by Range, QD, MD, SD

It identifies the direction of the variation, the extent to which they depart from symmetry.

It deals with symmetry of distribution of central value & with nature of variation on either side of central value.

Skewness can be measured by

→ Bowley Coeff of Skewness

→ Kelly Coefficient Skewness

→ Moments

→ Karl peasson Method.

Ques 2. What do you mean by Kurtosis? What purpose it serves?

(B.Com (H) 91, 95)

Ans. Kurtosis is one of the measure which tells us about the form of distribution. It tells us whether the distribution if plotted on a graph paper would give us normal curve or a curve more peaked or more flat than Normal Curve.

Basically “word kurtosis is a greek term which means ‘bulginess.”

Measure of Kurtosis

Kurtosis measured by Coefficient β2 or its derivative

i.e. .2γ

&

If β2 > 3, & 2γ is positive, distribution is Lepto Kurtic

If β2 = 3, & 2γ = 0, distribution is Meso Kutric

If β2 < 3, & 2γ is negative, distribution is Platy Kurtic

Ques 3. Differentiate between Skewness & Kurtosis? (B.Com (H) 99)

Basis Skewness Kurtosis

1) Meaning

2) Purpose

When a distribution is not Symmetric, it is said to be said to be skewed i.e. absence of symmetry which denotes the presence of Skewness.

The following three figures would give an idea of an absence or presence of Skewness.

It refers to the degree of peakedness of the hump of the distribution.

When we compare two or more

22μ

4μ

2β

32β

2γ

It reveals whether distribution is Symmetric or not i.e. it reveals the absence or presence of symmetry.

symmetrical distribution, the difference in heights of symmetrical curves will be called Kurtosis.

Note: - Besides above points, you can also write formulas, & for formulae see end of the chapter.

Ques 4. Explain in brief Sheppard Correction method applied to moments?

(B.Com (H) 2002)

Ans. As per W.F. Sheppard, the effect due to grouping at mid points of intervals can be corrected by formula given below:

2μ (Corrected) =

2μ (Uncorrected)

12

2h

3μ (Corrected) =

3μ (Corrected)

4μ

(Corrected) = 4μ

(Uncorrected) = 4

h240

7

2μ

2h

2

1

Where h is the width of Class Interval.

Purpose of Moments

i) The first central moment is always zero i.e. μ1 = 0.

ii) The Second Central moment about mean indicate variance i.e. μ2 = σ2

iii) The first moment about origin (zero), indicates Arithmetic Mean i.e.

V1 = A + μ11

iv) The third & second central moment is used to measure Skewness.

32μ

23μ

1β

or as per Fisher

2μ

2μ

3μ

1β

1γ

v) The fourth & second central moment in used to measure Kurtosis

2

2μ

4μ

2β

or as per Fisher, 32β

2γ

Statistical Decision Theory

Ques 1. What do you mean by Statistical decision theory. (B.Com (H) 99)

Ans. In simple words, statistical decision theory can be defined as theory which deals with decision-making using statistical tools & such theory can be applied only when there are several alternatives for a particular objective.

Ques 2. Fill in the blanks (CA (P.E. -1) 93 Dec.)

1) E.M.V.& EOL criterion _____________ the knowledge of probabilities of states of nature.

2) Maximax & Minimax criterion _____________ the knowledge of probabilities of status of natures.

3) While making decision using EMV criterion, out of several EMV for several action, EMV with _______________ value is taken.

4) While making decision using EOL criterion, out of several EOL for several action EOL with ____________ is taken.

5) Under Laplace criterion ________________ probability are assigned to various states of nature.

6) Events beyond the control of the decision maker called _____________ or _____________ of nature.

7) The maximum amount that a retailer will be willing to pay for a perfect predictor is called the _____________.

8) There are two types of losses in a stocking operation: ___________ losses and _________ losses.

9) The pleasure or displeasure one receives from certain outcomes is one’s _____________.

Answer

1) Require 2) does not require 3) maximum value

4) Minimum value 5) equal 6) outcomes, states

7) EVPI 8) obsolescence, opportunity 9) utility

Ques 3. State which of the following statement are correct or incorrect.

i) A person can have one utility for one situation and quite a different one of the next situation.

ii) It is always difficult to make use of other people’s knowledge about a situation without explaining statistical techniques to them.

iii) With perfect information, a retailer would consistently make the maximum profit possible.

iv) One advantage of using decision trees is that every outcome, desirable or undesirable, must be investigated.

v) On a decision tree, a circle represents a decision point.

vi) If a retailer can earn $100 per day with perfect information, then EVPI = $100.

vii) A businessman with a linear utility curve can effectively use expected monetary value as his decision criterion.

viii) A decision that maximizes expected profits will also minimize expected losses.

Answer

1) Correct 2) Incorrect 3) Correct 4) Correct

5) Incorrect 6) Incorrect 7) Correct 8) Correct

Short Answer Questions

Ques 1. Define following terms (B.Com (H) 2000) (CA (P.E. -1) 94 J, 97 N)

a) EVPI

b) EOL

c) EPPI

d) EMV

e) Actions

f) States of nature

Ans. a) EVPI: → The expected value of perfect information is the maximum amount of money a decision maker can spend to get additional information about the states of nature.

EVPI = EPPI – EMV

b) EOL: → The difference between profit actually derived from a certain decision and that would have been derived if decision had been the best one for the event actually occurred, it is known as opportunity loss.

The expected opportunity loss is the expected less incurred because of failure to take a specific action & it is derived from loss table.

EOL = EVPI

c) EPPI: → The expected pay off of perfect information is the maximum expected profit decision maker can make if perfect predictor is available & thus all the information about the states of nature is available to you.

d) EMV: → The expected monetary value reveals expected profit decision maker can hope to make on the basis of available information about status of nature.

EMV = i

Xi

Σp

i

X → Pay off for each action for several state of nature.

i

P → Probabilities of several states of nature.

e) Actions: → To make any decision, several alternatives are available to a decision maker, all such relevant alternatives are termed as action in the statistical decision theory.

f) States of Nature: → These are those possible events which are uncertain but are vital for a choice of any one of the alternative course of action & Therefore such events are considered while making decision.

Ques 2. Differentiate between pay off Table & Regret (Loss) Table?

(B.Com (H) 96, 97) (CA (P.E. -1) 94 N)

Ans. Difference between pay off Table & Loss Table can be discussed in the following table.

Basis Pay off Table Loss Table

(1) Meaning

(2) Criterion

It is a table which reveals the values of actual pay off i.e. value of a consequences expressed in terms of gain which is expressed is money terms.

Here EMV criterion is used for the purpose of selecting best course of action.

It is a table which reveals the difference between profit actually derived from a certain decision & that would have been derived if the decision had been the best one for event actually occurred.

Here EOL is the criterion used for the purpose of selecting best course of action.

Correlation

Ques 1. Fill in the blanks

i) Value of Correlation Coefficient lies between ___________ & ______________

ii) Value of Correlation Coefficient is independent of ____________ & ___________

iii) Co variation implies that two variables should vary in the ______________ direction.

iv) When value of r = 1, Correlation is ____________ & when value of r = -1, Correlation is ____________ & when r = 0 _____________

v) The Coefficient of Concurrent deviation can have both ___________ & ____________ values.

vi) Rank Correlation can be applied in ______________ data.

Answer

i) – 1 & 1 – 1 ≤ r ≤ 1

ii) Change of Scale & Change of Origin

iii) Same

iv) Positive, negative, no correlation i.e. variables are independent.

v) Positive & negative.

vi) Qualitative.

Ques 2. Identify the Correct Statement.

i) Correlation always reveal cause and effect relationship.

ii) Coefficient of correlation is a relative measure of relation between two or more variables

iii) The coefficient of correlation have both positive & negative values

iv) In a Scatter diagram the independent variable is shown on X axis & dependent variable on Y-axis

v) If r2 = 0, this implies there is no association between the variables

vi) Coefficient of correlation must be in the same units as original data

Answer

i) In correct, Not always because we may have chance Correlation [See Long Answer Question]

ii) Correct iii) Correct iv) Correct v) Correct

vi) Incorrect, it has no units because it is a relative measure

Ques 3. State nature of the following Correlation. (B.Com (H) 2000)

i) Sale of Woolen garments & change in temperature

ii) Rainfall & Crop Yield

iii) Colour of Shirt & Weight of person wearing it

iv) Production of wheat & use of fertilizers

v) Age of applicant for Life insurance & premium of insurance

Answer

i) Negative ii) Positive iii) No iv) Positive v) Negative

Explain in brief the properties of Coefficient of Correlation.

Ans. → The Properties Coefficient of Correlation can be discussed as follows:

i) Its values lies between – 1 & 1 i.e. – 1 ≤ r ≤ 1

ii) It is independent of Change of scale & Change of origin i.e. its value remains unaffected even if each value of data is increased, decreased, multiplied or divided by same number.

iii) It is a pure number & is independent of the unit of measurement.

Example. Comment on the following statements:

(a) “If the coefficient of correlation between two variable is + .5 it means 50% of variation are explained.”

(b) “If the coefficient of correlation between two variables of the 1st series is + .2 and between

two variable of other series is + .4 it means the degree of relationship in second series is twice as compared to that of the 1

st series.”

Solution:

(a) Coefficient of determination explains the degree of relationship between two variables. Therefore, if r = .5 then explained variation is r

2, i.e., (. 5)

2 or 25%. Therefore, explained

variation is not 50%, but 25%.

(b) Explained variation in the 1st series is (. 2)

2 4% and Explained variation in second series is (.

4)2 16%. Therefore, degree relationship is ‘Not Twice’ but ‘Four Times’ in the second series.

a) Probable Error (P.E.) → It should be used for inter pretation only when N is very large otherwise may give misleading conclusion.

P.E. = .6745n

2r1

Where: r = coefficient of correlation

n = number of items.

Interpretation of r on the bases of Probable error can be expressed as follows:

(a) if r > r6PE r is significant

(b) if r < r6PE r is insignificant.

Ques 1. What do you mean by Correlation & what are its various types? Does it always imply cause and effect relationship?

Ans. “Correlation analysis attempts to determine the degree of relationship between variables”

Ya Lun Chow

→ Thus following important elements of Correlation can be identified on the basis of above definition.

i) There should be two or more variables.

ii) There should be some relationship between them.

iii) The Change in value of one may affect another also.

TYPES OF CORRELATION

Correlation can be:

(i) Positive or Negative;

(ii) Simple, Multiple or Partial;

(iii) Linear or Non-linear.

(i) Positive and Negative Correlation

Correlation can be either Positive or negative. When the values of two variables move in the same direction i.e. when an increase in the value of one variable is associated with an increase in the value of other variable, and a decrease in the value of one variable is associated with the decrease in the value of the other variable, correlation is to be positive.

If, on the other hand, the value of two variables move in opposite directions, so that with an increase in the values of one variable the value of the other variable decrease, and with a decrease in the values of one variable the values of the other variable increase, correlation is said to be negative.

Thus generally price and supply are positively correlated. When prices go up supply also increases and with the fall in prices supply also decreases. The correlation between price and demand is generally negative. With an increase in price the demand goes down and with a decrease in price the demand generally goes up. Demand curve is downward sloping where as supply curve is upward sloping.

Negative Correlation Positive Correlation

Price Demand Price Supply

10 100

12 80

10 10

18 20

(ii) Simple, Multiple and Partial Correlation

In simple correlation we study only two variables – say price and demand. In multiple correlation we study together the relationship between three or more factors like production, rainfall and use of fertilizers. In partial correlation though more than two factors are involved but correlation is studied only between two factors and the other factors are assumed to be constant.

(iii) Linear and Non-linear (Curvi-linear) Correlation (C.A. (P.E. -1) 92 N)

The correlation between two variables is said to be linear if corresponding to a unit change in the value of one variable there is a constant change in the value of the other variable i.e. incase of linear correlation, the relation between the variables X and Y is of the type

y = a + bx.

In such cases, the values of the variables are in constant ratio. The correlation between two variables is said to be non-linear or curvilinear if corresponding to a unit change in the value of one variable the other variable does not change at a constant rate e.g.

linear correlation

X Y Ratio between

X and Y

10

20

30

5

10

15

2: 1

2: 1

2: 1

Non-linear correlation

X Y Ratio between X and Y

10

20

30

4

10

12

10: 4 or 5: 2

20: 10 or 2: 1

30: 12 or 5:2

CORRELATION – CAUSE AND EFFECT RELATIONSHIP:

Though the word correlation is used in the sense of mutual dependence of two or more variables yet it is not at all necessary that it should always be so. Even a very high degree of correlation between two variables does not necessarily indicate a cause and effect relationship between them. There can be correlation between two variables due to any one or more of the following reasons:

(1) Both the correlated variables are being affected by a third variable or by more than one variable. For example we may find a high degree of correlation between the in reality it may be found that tie is due to good fertilizer etc.

(2) Related variables might be mutually affecting each other so that neither of them could be designated as a curve or effect. This situation particularly holds good in the field of economics and business. For example the demand of a commodity may go down as a result of rise in prices. One would normally presume that price is the cause and demand is the effect. However it may be, that the demand of the commodity has gone up due to anticipated shortage in future and has resulted in the price rise. Now demand would be the cause and price would be the effect.

(3) The correlation may be due to random or chance factors. Many times correlation is noticed between two variables without any real relationship between them. It may happen due to chance. This generally happens when a very small sample is chosen from a large universe.

(B.Com (p) 97)

Income (Rs) Weight (Kg.)

200

300

400

40 Kg.

50 Kg.

60 Kg.

The above points make it clear that correlation is only a mathematical relationship and it does not necessarily signify a cause and effect relationship between the variables.

Regression

Ques 1. Define Regression

Ans. Regression analysis attempts to establish the nature of the relationship between variables i.e. to study the functional relationship between the variables & there by help in prediction.

Ques 2. What do you mean by Standard Error of Estimate

Ans. Standard Error of Estimate help in finding the likely error in estimated values of Y or X.

N

2 Yc-Y Σ

yS

N

2Xc - X Σ

xS

Where Sy → Standard error of the estimate of y values

Sx → Standard error of the estimate mode x values

Xe → estimated values of x

Ye → estimated values of y

Y → original values of y

X → original values of x

Ques 3. Fill in the blanks

a) The regression lines cut each other at the point of ____________

b) The regression lines of Y on X ____________ the total of square of horizontal deviations

c) If r = +1 or -1, those will be only ______________ regression line

d) The greater the distance at which regression lines cut each other i.e. greater the angle formed at their point of intersection, the degree of correlation will be _____________

e) When r = 0, the regression lines cut each other at angle of _____________

f) When one of the regression coefficient is greater than one, other will be ____________ less than one

g) When one regression coefficient is negative, other would be _________

h) Lines of regression are ____________ if r = 0 & they are ____________ if r = ± 1

i) The purpose of regression is to study ___________ between variables

j) The sign of regression coefficient is ____________ as that of correlation coefficient

k) Regression coefficient is independent of change of ___________ but not change of __________

l) An association between two variables that is described by a curved line is a ___________ one.

m) Every straight line has a _____________, which represents how much each change of the independent variable changes the dependent variable.

n) The extent to which observed values differ from their predicted values on the regression line is measured by the _________________.

o) ________________ is a measured of the proportion of variation in the dependent variable that is explained by the regression line.

Answers

a) Average of X & Y b) Minimizes c) One d) Less

e) 90 f) Less g) Negative h) Separate,

Same

i) Dependence j) Same k) Origin, scale l) Curvilinear

m) Slope n) Standard Error of Estimate

o) Coefficient of determination

Ques 4. State which of the following statement are ‘Correct or Incorrect’.

1. Regression analysis is used to described how well an estimating equation describes the relationship being studied.

2. Given that the equation for a line is Y = 26 – 24X, we may say that the relationship of Y to X is direct linear.

3. An r2 value close to 0 indicates a strong correlation between X and Y.

4. Regression and correlation analyses are used to determine cause and effect relationships.

5. The sample coefficient of correlation, r, is nothing more than r2, and we cannot

interpret its meaning directly as a percentage of some kind.

6. The standard error of estimate measures the variability of the observed values around the regression equation.

7. The regression line is derived from a sample and not the entire population.

8. We may interpret the sample coefficient of determination as the amount of the variation in Y that is explained by the regression line.

9. Lines drawn on either side of the regression line at ± 1, ± 2 and ± 3 times the value of the Standard error of estimate are called confidence lines.

10. The estimating equation is valid over only the same range as that given by the original sample data upon which it was developed.

11. In the equation y = a + bx for dependent variable y and independent variable X, the Y is intercept is b.

12. If a line is fitted to a set of points by the method of least squares, the individual positive and negative errors from the line sum to zero.

13. If eS = 0 for an estimating equation, it must perfectly estimate the dependent

variable at the observed points.

14. Suppose the slope of an estimating equation is positive. Then the value of r must be the positive square root of r

2.

15. If r = .8, then the regression equation explains 80 percent of the total variation in the dependent variable.

16. The coefficient of correlation explains the percentage of the total variation of the dependent variable.

17. The standard error of estimates is measured perpendicularly from the regression line rather than on the Y-axis.

Answers

1) Incorrect 2) Incorrect 3) Incorrect 4) Incorrect 5) Correct

6) Correct 7) Correct 8) Correct 9) Incorrect 10) Correct

11) Incorrect 12) Correct 13) Correct 14) Correct 15) Incorrect

16) Incorrect 17) Correct

Different between Correlation & Regression?

Ans.

Basis Correlation Regression

1) Meaning

2) Functional Relationship

3) Change of scale & origin

4) Nature

“Correlation analysis attempts to determine the degree of relationship between variables”

Ya Lun Chow

Thus following elements of Correlation can be identified on the basis of above definition.

i) There should be two or more variables.

ii) There should be some relationship between them.

iii) The change in the value of one may affect another also.

Here y= f(x) or X = f(y) is irrelevant i.e. here cause a effect relationship cannot be studied.

Correlation coefficient is independent of change of scale of change of origin

It is a relative measure showing association between variables.

Regression analysis attempts to establish the nature of the relationship between variable i.e. to study the functional relationship between the variables & there by help in prediction.

There y = f(x) or X = f(y) are not some because regression analysis establishes functional relationship between the variables.

Regression coefficient are independent of change of origin but not change of scale

It is an absolute measure of relationship.

Ques 4. Explain important properties of Regression Coefficient?

(B.Com (H) 94, 98, C.A. (P.E. -1) 97 N, 99 N)

Ans. Important of Regression Coefficient are as follows:

i) Regression Coefficient are independent of Change of Origin but not Change of Scale.

ii) Coefficient of Correlation is GM between two regression coefficient i.e.

r = ± bxy . byx

iii) Regression lines intersect at average values of X α Y

iv) Both regression coefficient will have same algebraic sign i.e. either both of them would be positive or both of them would be negative.

v) If one of the regression coefficient is more than 1, other has to be less than 1 because value of coefficient of correlation cannot exceed one i.e.

-1 ≤ r ≤ 1

Probability

Ques 1. Define following terms.

Mutually exclusive events

a) Independent events

b) Dependent events

c) Simple events

d) Compound events

e) Exhaustive Cases

f) Equally likely

g) Joint Probability

Ans. a) Mutually Exclusive Events. Two or more events are said to be mutually exclusive if the happening of any one of them excludes the happening of any one of them excludes the happening of all others in a single (i.e. same) experiment. Thus in the throw of a single dice the event 5 and 6 are mutually exclusive because if the event 5 happens no other event is possible in the same experiment. Here one and only one of the events can take place at a time excluding others.

b) Independent Events. An event is said to be independent when occurrence of one does not affect occurrence of other, events.

c) Dependent Events. An event is said to be dependent when occurrence of one affect occurrence of other, dependent events.

d) Simple Events. Explanation in compound events.

e) Compound Events. An event is called Simple if it corresponds to a single possible outcome. Thus in tossing a dice, the chance of getting 3 is a simple event (because 3 occurs in the dice only once). However the chance of getting an odd number is a compound event (because odd numbers are more than one i.e. 1, 3 and 5).

f) Exhaustive Cases. All possible outcomes of an event are known as exhaustive cases. In the throw of a single dice the exhaustive cases are 6 as the dice has only six faces each marked with a different number.

g) Equally Likely Cases. Two or more events are said to be equally likely if the chance of their happening is equal I.e., there is no preference of any one event over the other. Thus in a throw of an unbiased die, the coming up 1, 2, 3, 4, 5 or 6 is equally likely. In the throw of an unbiased coin the coming up of head or tail is equally likely.

h) Joint Probability. → are arrived at by multiplying two or more probabilities depending on no. of events involved.

i) Marginal probabilities → are sum of probabilities of two or more events.

Ques 1. Explain in brief (B. Com (H) 98, CA (P.E. – I) 94N, 95M, 95 N, 97M)

a) Addition Theorem of Probability

b) Multiplication Theorem of Probability

Ans. a) Addition Theorem

If A & B are two events, then probability of at Ieast one of them occurs is denoted by P (A U B) & given by.

P (A B) = P (A) + (B) – P (A B)

Where P (A) → Probabilities of the occurrence of event A

P (B) → Probabilities of the occurrence of event B

P (A B) → Probabilities of simultaneous occurrence of event A & B.

If events are Mutually Exclusives, then

P (A B) = P (A) + P (B),

Because P (A B) = 0

& in case of finite no. say n Mutually Exclusive Events

P (A1 A2 ---- An) = P (A1) + --- P (An)

a) Where events are not Mutually Exclusive

b) When events are Mutually Exclusive

b) Multiplication Theorem

The probabilities of the simultaneously occurrence of the events A & B is denoted by P (AB) or P (A B) & given by

a) If events are independent

P (A B) = P (A) . P (B)

b) If events are dependent

P (A B) = P (A) . P

A

B

= P (B) P

B

A

Ques 2. Explain in brief Conditional probability. (B.Com (H) 2000)

Ans. → Two events A & B are said to be dependent when B can occur only when A is known to have occurred or vice versa. The probabilities associated with such events are called Conditional Probabilities.

P

B

A=

B P

B A P

Where P

B

A → Probabilities of occurrence of event A when B has occurred

P A P

A B P

A

B

Where P

A

B → Probabilities of occurrence of event B when A has occurred

Ques 5. States & illustrate Baye’s Theorem. (B.Com (H) 2001)

Ans. Bayes’, Theorem

This Theorem is based on revision of priori probabilities, it is basically an extension of conditional probability.

Imaging a situation where two uncertain events (A) and (not A) are possible. Suppose we know their probability i.e., we know the probability of A’s happening and also the probability of A’s not happening. These probabilities are prior probabilities because they are probabilities before any further information is available. Suppose an investigation is conducted. The investigation may have several outcomes which would be dependent on event A. For any particular outcome (which may be called B) the conditional probabilities P (B/A) and P (B or A) are available.

The result itself serves to revise the probabilities for event (A) and event (not A). The resulting values would be the posterior probabilities since they have been obtained after the results of the investigation.

Thus according to Bayes theorem the posterior probability of event (A) for a particular result of an investigation (B) may be found from.

P (A/B) =

B/Not A P Not A P B/A P A P

B/AP AP

Theoretical Distribution

Properties of Normal Distribution

1. It is perfectly symmetrical about the mean (μ ) and is bell shaped. This means that if we

fold the curve along the vertical line at the centre. The two halves of the curve would coincide.

2. Mean = Median = Mode.

3. It has one mode, it is unimodal.

4. The ordinate at the mean of distribution divides the total area under the normal curve into two equal parts.

5. The following are the descriptive measures of the normal distribution:

Mean = X or μ (Standard form: X = 0)

Standard deviation = σ (Standard form: σ2 = 1).

Variance or μ2 = σ2

Third central moment, μ2 = 0

Fourth central moment, μ4 = 3σ

4 = 3μ2

2

Moment coefficient of skewness, 0

2μ

2μ

3μ

1β

Moment coefficient of Kurtosis 32

2μ

4μ

2β

0σ

Hence, it is a meso kurtic curve.

6. The normal curve is concave near the mean value, while near ± 3σ it is convex to the horizontal axis. The points of inflexion, i.e., the points where the change in curvature occurs are ± σ.

7. The quartiles Q1 and Q3 are equidistant from the median.

8. The mean deviation about mean is 4/5 σ or 0.7979σ.

9. The Standard deviation distributes the area under the normal curve as given below:

(i) Mean ± 1σ covers 68.268% area, 34.134% area will lie on either side of the mean.

(ii) Mean ± 2σ covers 95.45% area, 47.725% area will lie on either side of the mean.

(iii) Mean ± 3σ covers 99.73% area, 49.865% area will lie on the either side of the mean. Thus mean ± 3σ covers nearly the whole area, leaving only .27% area.

Ques 1. What do you mean by Time series?

Ans. A time series consist of data which is arranged in a series chronologically.

Component of time series →

Long term Short term

Trend a) Seasonal variation

b) Cyclical

c) Irregular variation

EXAMPLES

(i) Recording of daily temperature for a month or a year

(ii) Recording of weekly sales of a shop taken over a year

(iii) Recording of bi-monthly telephone bills of a consumer taken over a period, say 5 years

(iv) Recording of imports at the end of each year for a period of 10 years, etc.

These are all examples of time series.

In fact, time series occur in every walk of life, be it finance, commerce, industry, agriculture, medicine, education or even the domestic life of an individual.

TIME SERIES ANALYSIS By a time series analysis, we mean

(i) To study the behavior of the phenomenon over a period of time, and

(ii) To determine the various forces or influences which produce the variations in time series.

It is done mainly for the purpose of making forecasts for future & also for the purpose of equally past performances.

Ques 2. With which Component of time series. Would you mainly associate the following:

(B.Com (P) 2001)

a) Heavy Sales on the occasion of Deepawali

b) Price hike in petroleum products due to Arab-Israel war

c) Increase in garment sales in October

d) Decline in sale of ice-cream during winter season

e) An era of prosperity

f) A fire in factory delaying production for 3 weeks

g) The annual stock taking in a departmental store

h) General increase in the demand for TV sets.

Answer

a) Seasonal, b) Irregular, c) Seasonal, d) Seasonal,

e) Cyclic, f) Irregular, g) Seasonal, h) Trend

Ques 3. What is the difference between Additive model & Multiplicative model use for decomposition of time series.

Ans. Difference Additive model & Multiplicative model can be discussed as follows:

Basis Additive model Multiplicative model

1) Meaning

2) Formula

3) Nature of S, C & I

4) Assumption

Under this model decomposition of time series is done on the assumptions that effects of various components are additive in nature.

Y = T + S + C + I

Y → Time series Value

Here S.C & 1 are quantitative deviation from trend

This model assumes that four components of time series are independent of each other & none has any effect of remaining three components.

Under this model, decomposition of time series is done on the assumption that effects of various components are multiplicative in nature.

Y = T * S * C * I

Here S, C & I are expressed as rates percentages.

This model assumes that effect of Four components of time series are not necessarily independent of each other i.e. their effects are inter dependent.

Index Numbers

Ques 1. What do you mean by Index Numbers. (B.Com (P) 82)

Ans. Index Number is a statistical measure which is used to measure change over time in magnitude which are not capable of direct measurement.

Ques2. Comment on the following statement:

i) During a certain period, cost of living index measured from 110 to 200 & salary of worker increased from Rs 325 to Rs 500. Does the worker really gain?

ii) The average salary paid to worker in year 2000 is double that of 1990. So workers enjoy a 100% higher standard of living in 2000 as compared to 1990.

iii) “For constructing index number, the best method on the Theoretical ground is not the best method from practical point of view.”

iv) “Laspeyer method is best as compared to Paasches.”

Answers

i) Amount worker should have got in current year is 325 * 100

200 Rsa 591 but he is getting

Rs 500. So worker is not fully compensated

ii) No, because increase in money wages is not an important criteria for standard of living rather than it is real wages which should be the criteria for judging the standard of living of workers.

iii) For construction of index no. best average on theoretical ground is Geometric mean because of following reasons:

i) Geometric mean is best to measure ratio or percentage.

ii) Fisher ideal index no. is nothing but Geometric mean of Laspeyer & Paasches index no.

but Geometric Mean because of time involved & complex calculation is not preferred from practical point of view.

iv) It is true to some extent as Laspeyers methods weights are constant while in Paasche weights are to be determined every time index no. is constructed. However, Laspeyeres method has on upward bias whereas Paasches index no. has on downward bias

Short Answer Question

Ques 1. Define the following terms.

Base Shifting

i) Splicing

ii) Deflating

Ans. i) Base Shifting –

It mean charging of the given base year of an index number & forming a new series based on some recent new base year & generally base shifting is resorted because of following reasons:

Reasons for Base Shifting

1) Distant Base Period

When base year is too old to be of any use for meaningful comparison e.g. if prices of year 2000 are compared with price of 1950, comparison will be useless because of changed conditions.

2) Comparison

When two series of index number with different base are to be compared if then such comparison will be useful only if they are can vested so as to have common base.

Formula for Base Shifting

New Index of One Year = 100base real of No. Index

yearthe of Index Old

Year Prices Index (Year 1 base) Index (Year 2 base)

e.g. 1 20 100 50

2 40 200 100

3 20 100 50

ii) Splicing

It means combining two or more series of over lapping index no. to obtain a single index number on a common base.

For Splicing of index no. the important condition is that index no. are constructed with same item & have an over lapping year.

Reason for Splicing of index no:

Splicing is done when an old index no: with an old base is being discontinued & a new index with a new base is being started. To have continuity of comparison, the new index no: is spliced to the old index number in the over lapping year or vice versa.

Series A (Year 3) as base)

e.g. Year Price Index (Year 1 are base) Index B

1 20 100

2 15 75

3 25 125 100

4 30 120

5 60 240

6 40 160

Index B Spliced to index A Index A Spliced to index B

(i.e. to make A Continuous Series) (to make B Continuous Series)

Year

4 150 Year

5 300 1 80

6 200 2 60

iii) Deflating

In simple words, it means making allowance for change in purchasing power of money due to change in general price level.

e.g.

Year Price of rice (Rs per/kg)

1980 20

1990 40

Assume income Mr. X has is Rs 1,000 in 1980 & 1990, Then Quantities of rice that can be purchased.

1980 50 Kg

1990 25 Kg

So, purchasing power has decreased by 50%.

Purchasing Power = Index Price

I

Let us illustrate this why,

e.g. Price index = 2 (of 50% rise in price)

50%2

1

Because of change in purchasing power of money due to change in general price level, one is more interested in real wages than money wage.

Real wage = 100index Price

wageMoney

Real wage index = 100Year Prev ious of wageReal

YearCurrent of wageReal

Ques 2. Differentiate between Chain Base Index method and Fixed Base Method to constructed index no? (B.Com (P) 93)

Ans. Fixed Base Method Vs Chain Base Method

Basis Chain Base Method Fixed Base Method

(1) Meaning

(2) Adjustment of weights

(3) Suitability

(4) Formula

Under this method, base period immediately precedes the period for which index is sought.

Weights can be adjusted as frequency as possible

There are suitable for short periods only.

CBI of CY = 100

CY of LR *PY of CBI

Under this method, any Base period is arbitrary chosen & it is kept fixed.

Weights cannot be adjusted so frequently.

These are money suitable for long periods.

FBI of CY = 100

CY of CBI *PY of FBI

Ques 2. What are various test of adequacy of index numbers? Explain in brief limitation of index numbers.

(B.Com (H) 95, 98, 2001, B.Com (P) 86, 99, C.A. (P.E. – I) 95 M, 95 N, 97 M, 97 N, 99 M)

Ans. From a statistical point of view, the system of calculation used for current year index numbers should be such that it satisfies certain mathematical tests. A number of tests have been developed for this purpose. These include (a) the unit test, (b) the time reversal test, (c) the factor reversal test and (d) the circular test.

The credit for proposing the time reversal test and factor reversal test goes to Professor living Fisher.

(a) Unit Test. This requires the index numbers to be independent of the units in which prices and quantities of various commodities are quoted. This test is satisfied by all the formulae except the simple aggregative index.

(b) Time Reversal Test. According the Prof. Fisher “the formula for calculating an index number should be such that it gives the same ratio between one point of comparison and the other, no matter which of the two is taken as the base or putting it another way, the index number reckoned forward should be reciprocal of the one reckoned backward.”

In symbols

P01 * P10 = 1 (Omitting the factor 100 from each index)

Where P01 denotes the index for current period 1 based on the base period 0 and P10 that is for period 0 based on the base period 1.

Time reversal test is based on following analogy: If the price of a commodity increased from Rs 4 per unit in 1980 to Rs 6 in 1990 the price in 1990 is 150% i.e., 1.5 times the price in 1975, and the price in 1970 is 66.67% i.e., 67 times the price in 1990. The product of these two price ratios is 1.5 * 0.67 = 1.

(c) Factor Reversal Test. In the words of Fisher, “Just as our formula should permit the interchange of two times without giving inconsistent results, so it ought to permit interchanging the price and quantities without giving inconsistent results – i.e., the two results multiplied together should give the true value ratio, except for a constant of proportionality.

Analytically if

P01 is a price index for given year with reference to base year,

Q01 is the quantity index for the current year with reference to base year for the same coverage of commodities.

P01 * Q01 =

0q

0p Σ

1q

1p Σ

Factor reversal test is based on the following analogy: If the price per unit of a commodity increases from Rs 4 in 1980 to Rs 6 in 1998 and the quantity of consumption changes from 100 units to 140 units during the same period, then the price and quantity in 1980 are 150% and 140% respectively of the corresponding figures in 1980. The values (price * quantity) of consumption were Rs 400 in 1990 and Rs 800 1978, so that the value ratio is thus we find that the product of price ratio and quantity ratio equals the value ratio:

Value ratio = 400

800 = 2:1

1.5 * 1.4 = 2.1. This is the true for each commodity.

Circular Test. This is another test for the adequacy of an index number. This test was first suggested by Westergaard and highly favoured by C.M. Walsch who gave it the name of ‘circular test’. It is base on the shift ability of the base and is merely an extension of the time reversal test. According to this test, the index should work in a circular fashion, i.e., if an index number is computed for the period 1 on the base period 0, another index number is computed for period 3 on the base period 2, and then the product should be equal to 1.

Symbolically

P01 * P12 * P23 * … * Pn-1,n * Pn,0 = 1

Limitation of Index Numbers

The limitation of index number can be discussed as follows:

(1) Limited Case: - Indices constructed for one purpose cannot be used for another purpose. Every index number is constructed by a technique which is appropriate for the objective with which the index is constructed. It cannot be used to serve a different objective. A wholesale price index number cannot measure cost of living.

(2) Likely to be misused if based on using data: - Index numbers are liable to be misused. If a wrong base has been chosen, or if weights are not assigned rationally or if the appropriate formula for the construction of the indices has not been chosen the results could be highly misleading and mischievous. Index Numbers use only limited number of items in their calculation and may not reflect the true picture of the problem under study, if the items chosen are not representative of the universe.