Fundamental of Stat by S C GUPTA

NORTH MAHARASHTRA UNIVERSITY JALGAON

SYLLABUS UNDER SCIENCE FACULTY

FOR F.Y.B.Sc. STATISTICS (Semester I and II)

WITH EFFECT FROM ACADEMIC YEAR 2009-2010

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

NORTH MAHARASHTRA UNIVERSITY, JALGAON SYLLABUS FOR F.Y.B.Sc. (Semester-I and Semester-II) SUBJECT: STATISTICS With effect from June 2009 The Board of Studies in Statistics in its meeting held on 14/5/09 resolved to accept the revised syllabus for F.Y.B.Sc.(Statistics) in view of semester pattern. The course codes and titles for the courses offered at F.Y.B.Sc. (Statistics) are as below.

F.Y.B.Sc.( Statistics) Course Structure Course Title of Course Semester Code ST-111 DESCRIPTIVE STATISTICS-I I ST-112 PROBABILITY AND PROBABILITY I DISTRIBUTIONS -I ST-121 DESCRIPTIVE STATISTICS-II II ST-122 PROBABILITY AND PROBABILITY II DISTRIBUTIONS –II ST-103 STATISTICS PRACTICALS I and II

Marks Ext Int 40 10 40 10

Periods/ week 03 03

Total Periods 40 40

03 03

40 40

40 40

10 10

04/Batch

104

80

20

Where, course code ST 1ab means 1 for F.Y.B.Sc. a for semester number (a=1 for semester I a=2 for semester II and a=0 for whole year for practical course) and b for course number. Note: Distribution of external examination marks (80) for Statistics Practicals will be as below: Practical Exam Paper- 60 marks. Journal - 10 marks. Viva-voce - 10 marks. 1

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

ST-111 DESCRIPTIVE STATISTICS-I 1. INTRODUCTION TO BASIC CONCEPTS

(6L,6M)

1.1 Meaning of Statistics: numerical information, science, decision making science, general definition of Statistics as science. 1.2 Scope of Statistics: In the field of Industry, Biological Sciences, Medical Sciences, Economics Sciences, Social, Sciences, Management Sciences, Agriculture, Insurance, Information Technology, Education and Psychology., Importance of quantification scope of statistical methods. 1.3 Statistical institutes and organizations: ISI, NSS, Beauro of Economics and Statistics in States. 1.4 Limitations of statistics 1.5 Population, statistical population, census, sample, sampling, 1.6 Objectives of sampling. Advantages of sampling over census 1.7 Methods of sampling; Simple random sampling with without replacement, Stratified sampling and Systematic sampling 1.8 Illustrations from real life situations 2. PRESANTATION OF DATA

(10L,10M)

2.1 Meaning of data, Raw data, Qualitative and Quantitative data 2.2 Attributes and Variables, continuous and discrete variables 2.3 Primary data and Secondary data 2.4 Sources of secondary data 2.5 Measurement scales: nominal, ordinal, ratio and interval scales 2.6 Illustrations from real world situations. 2.7 Tabular presentation of data :- Meaning of table , Parts of table , and construction of table(up to three factors of classification ) 2.8 Diagrammatic representation of data: simple, Multiple and subdivided bar diagrams, pie diagram. 2.9 Frequency distribution: - Meaning of frequency, class, exclusive and inclusive classes, Open-end classes, class width, mid-value, class boundaries and limit, relative frequency, 2.10 Cumulative frequency distribution: less than, more than , 2.11 Guidelines for construction classes, Struges formula. 2

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

2.12 Graphical representation of data:-Histogram(equal and unequal classes), Frequency curve, Frequency Polygon , ogives, stem & leaf chart. 2.13 Examples and problems. 3. MEASURES OF CENTRAL TENDENCY(LOCATION):

(12L,12M)

3.1 Meaning of central tendency of data, objectives, requirements of a good measure of Central Tendency. 3.2 Arithmetic mean(A.M.): Definition, effect of change of origin and scale, sum of deviations from A.M., combined mean for k groups, merits and demerits. 3.3 Geometric mean (G.M.): Definition, merits and demerits, uses 3.4 Harmonic Mean (H.M.): Definition, merits and demerits, uses 3.5 Median: Definition, computation formula(without derivation),graphical method of determining median, merits and demerits, 3.6 Mode: Definition, computation formula(without derivation),graphical method of determining median, merits and demerits, 3.7 Weighted Means: A.M.,G.M.,H.M. 3.8 A.M.>G.M.>H.M.(for 2 and 3 values) 3.9 Trimmed mean 3.10 Use of appropriate measure of central tenancy in different situation. 3.11 Empirical relation among mean, median and mode. 3.12 Partition values:-Quartiles, deciles, & percentiles ( Definition and Computation for ungrouped and grouped data).Box plot. 3.13 Examples and problems. 4. MEASURES OF DISPERSION:-

(12L,12M)

4.1 Meaning of Dispersion of data and objective. Requirements of a good measure of dispersion. 4.2 Range .Definition, Merits and Demerits ,uses. 4.3 Quartile Deviation (Q.D.) : Definition,computation, merits and demerits 4.4 Mean deviation (M.D.) , Definition, computation, merits and demerits Minimal Property of Mean Deviation without proof . 4.5 Mean Squared Deviation.: Definition, Minimal property with proof ,Variance and Standard deviation,

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North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

4.6 Properties of variance and Standard Deviation:i)combined Variance and Standard deviation for two groups (with proof) and its extension for k groups.ii).Effect of change of origin and scale iii) S.D. ≥ M.D. 4.7 Absolute and relative measures of dispersion :Coefficient of range, Coefficient of Q.D., Coefficient of M.D., Coefficient of variation (C.V.),Uses Of C.V. 4.8 Examples and problems . BOOKS FOR REFERENCE.:1 Statistics Volume-I by B.R.Bhat, T.Srivenkatraman,K.S. Madhava Rao 2.Fundamentals of Mathematical Statistics by S.C.Gupta V.K.Kapoor 3.Applied General statistics by Croxton F.E.Cowden D.J and Klein s. 4 Statistical Method by Snedecor G.W and Cochran W.G (Iowa state university press) 5 Fundamental of statistics Vol.1 by A.M.Goon,M.K.Gupta and B.Dasgupta. 6 Theory and problems of statistics , Schaum’s Publishing series by Spiegel M.R. 7.Textbook of statistics-Paper I by P.G.Dixit

4

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

ST-112 PROBABILITY AND PROBABILITY DISTRIBUTIONS -I 1. SAMPLE SPACE AND EVENTS

(8L,8M)

1.1 Meaning of experiment, random experiment, deterministic and non-deterministic models 1.2 Definitions of the following terms:- Outcome , sample space (finite and infinite), discete sample space, Event, Elementary event, Compound event, Complementary event, Favorable event, Equally-likely events, Sure event, Impossible event. 1.3 Concept of occurrence of an event 1.4 Union and intersection of two or more events 1.5 Exhaustive events, Mutually exclusive events 1.6 Representation of sample space and events by Venn diagram 1.7 Occurrence of (i) at least one of the given events (ii)all of the given events (iii) none of the given events 1.8 Examples and problems

2 PROBABILITY

(10L,10M)

2.1 Classical definition of probability, Relative frequency approach of probability,. 2.2 Equiprobable sample space , probability of an event 2.3 Axioms of probability 2.4 Addition theorem of probability up to three events and extension for k events (with proof using Axioms of probability only.) 2.5 Following results with proof (i) P(A’) =1- P(A) (ii) If A ⊆ B, then P(A) ≤ P(B) (iii) P( ∪ Ai ) ≤ ΣP(Ai) 2.6 Examples and problems 3. CONDITIONAL PROBABILITY AND INDEPENDENCE

(10L,10M)

3.1 Independence of events, pairwise and mutual independence 3.2 Conditional probability of an event 3.3 Multiplication theorem of probability (with proof) 3.4 Partition of sample space, 3.5 Baye’s theorem (with proof) 3.6 Examples and problems. 4. UNIVARIATE PROBABILITY DISTRIBUTION. 4.1 Concepts of random variable, discrete random variable 5

(12L,12M)

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

4.2 Function of random variable. 4.3 Probability mass function of a discrete random variable. 4.4 Distribution function of a discrete random variable. 4.5 Statement of properties of a distribution function. 4.6 Concept of symmetric random variable. 4.7 Median and mode of a discrete random variable. 4.8 Introduction of some standard probability distributions 4.9 Examples and problems BOOKS FOR REFERENCE.:(1) Elementary Probability Theory with stochastic Processes By Chung K.L Springer International Student Edition. (2) Elementary Probability Theory By David striazaker Cambridge University Press (3) An introduction Probability Theory and it’s applications By feeler W, Wiley (4) Probability By Pitman, Jim Narosa Publishing house. (5) Statistics Volume-II by B.R.Bhat, T.Srivenkatraman,K.S. Madhava Rao (6) Textbook of statistics-Paper II by P.G.Dixit

6

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

ST-121 DESCRIPTIVE STATISTICS-II 1. MOMENTS,SKEWNESS AND KURTOSIS:

(10L,10M)

1.1 Raw & central moments; Effect of change of origin and scale on central moments. 1.2 Expression for central moments in terms of raw moments (with proof). 1.3 Concept of Skewness of a frequency distribution; Positive and negative skewness, symmetric frequency distribution 1.4 Bowley’s coefficient of skewness Limits of Bowley’s coefficient of skewness 1.5 Karl Pearson’s coefficient of skewness. 1.6 Kurtosis: Meaning, Types of Kurtosis :-leptokurtic , mesokurtic & platykurtic. 1.7 Measures of skewness and kurtosis based on moments. 1.8 Examples and Problems. 2. CORRELATION

(10L,10M)

2.1 Bivariate data. Ungrouped and grouped. 2.2 Meaning of correlation between two variables, positive & negative correlation, 2.3 Scatter diagram, Construction of scatter diagram and interpretation. 2.4 Covariance between two variables: Definition, Effect of change of origin and scale 2.5 Product moment correlation (Karl Pearson’s correlation coefficient) and its properties, interpretation 2.6 Rank correlation: Spearman’s rank correlation coefficient, derivation of the formula of rank correlation coefficient (without ties).Rank correlation with ties 2.7 Simple numerical Examples and Problems. 3.REGRESSION:

(12L,12M)

3.1 Meaning of regression, concept of linear and non-linear regression. 3.2 Concept of method of least squares. 3.3 linear regression: Fitting of lines of regression by method of least squares. 3.4 Regression coefficients and their properties (statement and proof). 3.5 Angle between the two lines of regression. 3.6 Standard error of regression estimate, 3.7Explained and unexplained variation and coefficient of determination 3.8 Non-linear regression : Fitting of non-linear curves of the type 7

North Maharashtra University, Jalgaon

2

b

(i) y=a+bx+cx (ii) y=ax (iii) y=ab

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

x

3.9 Examples and problems. 4. THEORY OF ATTRIBUTES

(8L,8M)

4.1 Concept of attribute, , dichotomy, manifold classification, Notations 4.2 Class frequency , order of class, positive class frequency ,negative class frequency, contra class frequency, ultimate class frequency 4.3 Relation between class frequencies, 4.4 Method of dot operator to express any class frequency in terms of positive class frequencies . 4.5 Fundamental set of class frequencies: Definition, determination whether a set of frequencies is fundamental set of or not (two attributes). 4.6 Independence and association of two attributes 4.7 Yule’s coefficient of association(Q) . 4.8 Properties of Q and interpretation of Q 4.9 Examples and Problems. BOOKS FOR REFERENCE.:1 Statistics Volume-I by B.R.Bhat, T.Srivenkatraman,K.S. Madhava Rao 2.Fundamentals of Mathematical Statistics by S.C.Gupta V.K.Kapoor 3.Applied General statistics by Croxton F.E.Cowden D.J and Klein s. 4 Statistical Method by Snedecor G.W and Cochran W.G (Iowa state university press) 5 Fundamental of statistics Vol.1 by A.M.Goon,M.K.Gupta and B.Dasgupta. 6 Theory and problems of statistics , Schaum’s Publishing series by Spiegel M.R. 7. Textbook of statistics-Paper I by P.G.Dixit

8

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

ST-122 PROBABILITY AND PROBABILITY DISTRIBUTIONS –II 1. MATHEMATICAL EXPECTATION

(8L,8M)

1.1 Definition of expected value of a discrete random variable, expectation of function of random variable 1.2 Discussion of the following results with proof:(i) E(K)=K,where K is constant , (ii)E(aX+b)=aE(X)+b,where a&b constants , (iii) E(E(X)) = E(X) 1.3 Definition of mean, variance and standard deviation of a random variable, 1.4 Effect of change of origin and scale on mean, variance and standard deviation 1.5 Concept of standardized random variable 1.6 Raw, central and factorial moments of probability distribution 1.7 Examples and problems 2. BIVARIATE PROBABILITY DISTRIBUTION.

(10L,10M)

2.1 Definition of two dimensional random variable. 2.2 Joint probability mass function of two dimensional random variable. 2.3 Joint distribution function of two dimensional variable. 2.4 Statement of properties of joint distribution function. 2.5 Marginal probability mass function. 2.6 Conditional probability mass function. 2.7 Independence of two random variables. 2.8 Examples and problems. 3. EXPECTATION OF FUNCTIONS TWO DIMENSIONAL RANDOM VARIABLE (8L,8M) 3.1 Expected value of function two-dimensional random variable 3.2 Discussion of the following results with proof:(i) E(aX+bY)= aE(X)+bE(Y), where a and b are constants (ii) E(aX,bY)= abE(X)E(Y),where X&Y are independent random variables. 2

2

2

(iii) [E(XY)] ≤ [E(X) .E(Y) ] 3.3 Conditional expectation, conditional mean and variance. 3.4 Raw and central moments 3.5 Definition of covariance of a random variable. 3.6 Properties of covariance (with proof). 9

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

3.7 correlation coefficient 3.8 Variance of linear combinations of two random variables with proof. 3.9 Examples and problems. 4. SOME STANDARD DISCRETE DISTRIBUTIONS.

(14L,14M)

4.1 DISCETE UNIFORM DISTRIBUTION: Statement of probability mass function, mean and variance, applications. 4.2 BERNOULI DISTRIBUTION : Bernoulli trial, Statement of probability mass function, mean and variance , applications.. 4.3. BINOMIAL DISTRIBUTION : Binomial situation ,statement of probability mass function, mean ,variance and mode,statement and proof of reproductive (additive) property of binomial distribution,recurrence relation of central moments, applications. 4.4 HYPERGEOMETRIC DISTRIBUTION: statement of probability mass function, mean and variance, applications. 4.5 Examples and problems. BOOKS FOR REFERENCE. :(1) Elementary Probability Theory with stochastic Processes By Chung K.L Springer International Student Edition. (2) Elementary Probability Theory By David striazaker Cambridge University Press (3) An introduction Probability Theory and it’s applications By feeler W, Wiley (4) Probability By Pitman, Jim Narosa Publishing house. (5) Statistics Volume-II by B.R.Bhat, T.Srivenkatraman,K.S. Madhava Rao (6) Textbook of statistics-Paper II by P.G.Dixit

10

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

ST-103: STATISTICS PRATICAL INSTRUCTIONS:(1) The total duration of practical examination shall be 5(five)hours. (2) The examination shall be executed into two parts .Duration of each part shall be TWO & HALF HOURS. (3) Each part will carry maximum 30 marks. A student will have to solve two questions out of four given questions Each question carries 15 marks (4) Student must complete all the practicals to the satisfaction of concerned teacher. (5) Students must produce at the time of the practical examination the laboratory journal of practical completed along with the completion certificate signed by the concerned teacher and the Head of the department. INSTUCTIONS TO TEACHERS (1) Practicals be carried out by using calculators. (2) Encourage students to collect live data from real life situations. Such data be used for practicals (3) Demonstrate MS EXCEL commands to draw diagrams and graphs. (4) Demonstrate MS EXCEL commands to calculate various statistical measures

DETAILED CONTENTS OF SYLLABUS OF PRACTICALS PART - 1

(1) Construction of tables from given raw information (up to 3 factors of classification is expected Students should be able to calculate and supply missing information in the tables)

(1P)

(2) Construction of diagrams(simple, multiple , subdivided bar diagrams and pie diagrams are expected). (1P)

(3) Construction of frequency and cumulative frequency distribution from raw data (students should be able to decide the number of classes limits and class Width). Construction of histogram , frequency curve , frequency polygon ,orgies form the frequency distribution. (2P)

11

North Maharashtra University, Jalgaon

F.Y.B.Sc.(Statistics) Syllabus – w.e.f. 2009-2010

(4) Computation of A.M , G.M & H.M for ungrouped and grouped data(problems based on only raw data and frequency distribution are expected ) (1P)

(5) Computation of median , mode & partition values (quartiles , deciles & percentiles) based on algebraic and graphical methods(problems based on only raw data and frequency are expected). (2P)

(6) Computation of range , quartile deviation, mean deviation and standard deviation and their coefficients for ungrouped and grouped data(problems on only raw data and frequency distribution are expected). (2P)

(7) Computation of raw and central ,moments ,Karl Pearson’s & Bow ley’s coefficient of skew ness, coefficient of skew ness and kurtosis based on moments for ungrouped and grouped data(problems and frequency distribution are expected). (2P)

PART-II

(8) Drawing of random sample by using simple random sampling stratified random sampling and systematic random sampling. (1P)

(9) Computation of product moment correlation coefficient and regression lines for ungrouped data(problems based on only raw data are expected ; Students should be able to determine explained and unexplained variations , standard error of estimate and estimation of unknown value by using regression equations). (2P)

(10) Construction of bivariate frequency distribution , computation of product moment correlation coefficient and regression lines for grouped data. (2P)

(11) Fitting of non linear curves(students are expected to fit the curves I)y=a+bx+cx2 x

(ii)y=axb , (iii) y=ab (problems based on only raw data are expected). (2P)

(12) Fitting of binomial distribution and model sampling form binomial distribution (for model sampling parameter values should be explicitly specified) (2P)

(13) Applications of binomial and hypergeometric distribution (1P) REMARKS :- P Indicates practical period . One practical is equivalent to 3 clock hours.

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