**Statistics (Elective) **

**OUTLINES OF TESTS**

Paper-A | 75 marks |

Paper-B |
75 marks |

Paper-C (Practical) |
50 marks |

Total |
200 marks |

#### Syllabi and Courses of Reading

__ ____Paper- A__

*Candidates are required to attempt at least two questions from each Section.*

__Section- I.__

** Descriptive Statistics:** (Weight 2/10)

Descriptive and inferential Statistics. Population and sample. Variables. Measurement scales. Sources of Statistical data in Pakistan. Description of data by frequency tables and graphs. Stem and leaf plots and Box and whisker plots. Measures of Central Tendency. Measures of location: A.M., H.M., G.M., Mode, Median, Quartiles. Properties of Mean with proofs. Weighted Arithmetic mean. Empirical Relation between Mean, Median and Mode. Relative Merits and Demerits of various averages. Measures of dispersion: Absolute and Relative Measures, Range, Semi-Inter Quartile Range, Mean Deviation, Variance, Standard Deviation. Coefficient of Variation. Coefficient of Mean Deviation. Coefficient of Quartile Deviation. Properties of Variance and Standard Deviation with proofs. Standardized Variables. Description both for population and sample and their properties. Chebychev’s Theorem and its application. Moments, Moments Ratios, Sheppard’s Correction, Skewness and Kurtosis.

**(Weight 1/10)**

__Method of Least Squares:__Scatter diagram. Principle of least squares. Deduction and solution of normal equations of general linear model. Curve fitting. Equations of approximating curves by the method of least squares up to third degree polynomials. Fitting of exponential of the type (1) y= ae

^{bx}(2) y= ab

^{x}(3) y= ax

^{b. }Graphic representation of the curves. Interpolation and Extrapolation graphically. Criteria for fitting a suitable curve.

**(Weight 1/10)**

__Time Series:__Time series. Decomposition of Time Series. Measurement of Trend, Seasonal (Additive and multiplicative models), and cyclical variations. Seasonal indices. Deseasonalisation of data.

**(Weight 1/10)**

__Index Numbers:__Index number. Simple and composite indices. Problems in construction of index numbers. Laspayre, Paasche, Marshall-Edgworth, Fisher ideal, Walsh and Palgraves indices. Shifting of base. Quantity index numbers. Theoretical tests for index numbers. Consumer price index. Construction and uses of index numbers in Pakistan. Sensitive Price Indicator.

__Section – II.__

**(Weight 1/10)**

__Concepts of Probability:__Operation in sets. Cartesian product set. Random experiment. Sample space and event. Rules of counting introduction to probability and axioms of probability, emphasizing to concepts, facts, interpretation and illustrating examples. Basic laws of probability, Conditional and marginal probabilities. Independence of events. Baye’s theorem and its application. (Proof not required).

__Random Variable: (Discrete)__**(Weight:**

*P*

*lease see Note No. 1 below*)

Random Variable, Discrete random variable. Probability function, probability distribution function. Mathematical expectation and its properties. Joint distribution of two discrete random variables. Marginal and conditional distributions. Mean, Variance, moments, covariance and correlation of two discrete random variables. Moment generating function and its properties.

**(Weight:**

__Discrete Probability Distributions:__*Please see Note No. 1 below*)

Uniform, Bernoulli, Multinomial, Hypergeometric, poisson. Negative Binomial and Geometric distributions with their derivations, properties, applications and their fitting to statistical data.

__(Note No. 1__

__For Random variable (Discrete) and Discrete probability Distributions common weight is 2/10)__

__Random Variable: (Continuous)__**(Weight 1/10)**

Continuous random variable. Probability distribution of a continuous random variable. Probability density function and probability distribution function. Joint distribution of two continuous random variables. Marginal and conditional distributions. Mathematical expectation and its properties. Moment generating function. Covariance and correlation of two random variables. Mean, Median, Mode, Geometric mean, Harmonic mean, Mean deviation. Variance and moments of simple continuous functions.

**(Weight 1/10)**

__Continuous Probability Distributions:__Uniform, Exponential and Normal distributions with derivations. Their properties, applications and fitting of statistical data. Normal approximation to the Binomial and Poisson distributions (just applications)

#### Paper- B

* **Candidates are required to attempt at least two questions from each section.*

__Section -I__

__ __** Sampling and Sampling Distributions:** (Weight 2/10)

Basic concepts. Advantages of Sampling. Probability and non-probability sampling, Sampling and non-sampling errors. Sampling designs of simple random. Stratified, systematic, and cluster sampling. Judgement and quota sampling. Random numbers and their use in sampling calculation of sample mean, proportion and variance of simple random samples and stratified random samples. Sampling distribution of a statistic and its standard error. Distributions of sample mean and difference between two sample means with their properties. Distributions of sample proportion and difference between two sample proportions with properties. Central limit theorem with illustrations. Sampling distribution of sample variance and distribution of ratio of two sample variances. Concept of t,

**c**

**– and F- distributions.**

^{2}**(Weight 2/10)**

__Statistical Inference:__Concept of statistical inference. Estimates and estimators. Point estimation by the methods of moments and maximum likelihood. Properties of point estimators; unbiasedness. Consistency and efficiency. Interval estimation.

Null and alternative hypothesis, simple and composite hypothesis. Two types of errors. Level of significance, p-value and power of the test. Acceptance and rejection regions, one-sided and two-sided tests. Testing of hypothesis for mean, difference between two means, proportion and difference between two proportions; based on large samples. Interval estimation. Confidence interval and its interpretation. Interval estimation of the mean, difference between two means, the proportions and the difference between two proportions of populations with known and unknown variances; based on large samples. Determination of sample size. Testing of hypothesis (based on small samples and unknown population variance) for the mean, difference between two means for paired and unpaired observations. Testing of hypothesis of single population variance (Large sample) and equality of two variances (Large and small samples)

__Statistical inference concerning__

__c__

^{2_}**(Weight 1/10)**

__Distribution:__Testing of hypothesis about the variance. Testing of hypothesis about the equality of more than two variances. Pearson’s test for goodness of fit. Contingency tables and tests for independence and homogeneity. Co- efficient of mean square contingency and its maximum value. Yates correction for continuity. Chi-Square test for the Multinomial probabilities.

#### Section -II

** Analysis of Variance:** (Weight:

*Please see Note No. 2 below*)

Definition, importance and assumptions of Analysis of Variance. Partitioning of sum of squares and degrees of freedom in one-way classification. Testing the equality of means for one-way classification. Partitioning of sum of squares and degrees of freedom in two-way classification. Testing the equality of means for two-way classification. Multiple comparison tests: Least significant difference test, Duncan’s and Newman- Keuls Multiple range test.

** Basic Experimental Designs:** (Weight:

*Please see Note No. 2 below*)

Basic principle of experimental design. Completely randomised, complete block and Latin square designs. Description, layout, statistical analysis, advantages and disadvantages of these designs. Relative efficiency of three basic designs. Applications of these designs.

__(Note No. 2__** ***For Analysis of variance and Basic Experimental Designs common weight is 2/10)*

** Regression and Correlation Analysis:** (Weight 1/10)

Logic of regression and correlation. Scatter diagram. Regression models. Simple linear regression, least square estimates and their properties. Properties of Least Square regression line, standard error of estimate, co-efficient of determination. Multiple linear regression with two regressors, co-efficient of multiple determination Partial and multiple correlation up to three variables. Linear correlation. Correlation co-efficient and its properties. Correlation of bivariate frequency distribution. Partial and multiple correlation for three variables. Rank correlation, Tied ranks. Testing of hypothesis about regression and simple correlation partial and multiple correlation. Interval estimates and tests of hypothesis about regression paramenters, mean prediction and individual prediction. Interval estimation of regression parameters.

**(Weight 1/10)**

__Non-Parametric Tests:__Sign test, Run test. Mann-Whitney U-test, Wilcoxon Signed rank test, Wilcoxon rank sum test.

**(Weight 1/10)**

__Vital Statistics:__Definition of vital events and vital statistic. Uses and shortcomings of vital statistics. Sources of demographic data. Vital rates and ratios: Sex and child woman ratio. Vital Index, Crude, specific and standardized death rates. Crude, specific and standardized birth rates, general and specific fertility rate. Reproduction rates: Gross and Net reproduction rates. Census, registration system of deaths and births in Pakistan.

__Paper- C (Practical)__

*Candidates are required to attempt one question from each section.*#### Section-I

One question from each section of __Paper A__ should be set.

#### Section-II

One question from each section of __Paper B__ should be set.

Each question of 18 marks: .i.e. 18-18 | 36 marks |

Practical Note Book | 05 marks |

Viva Voce: | 09 marks |

Total: | 50 marks |

__Books Recommended:__

- Beg,M.A. and Mirza, M.D.(1997). Statistics, Theory and Methods. Vol.I&II. Carvan Book House, Kachehry Road, Lahore.
- Chaudhrey, S.M. and Kamal, S.(2002). Introduction to Statistical Theory Part-I &II. Illmi Kitab Khana Urdu Bazar, Lahore.
- Chaudhry, R.M.(1998) Polymer Modern Statistics, Polymer’s Urdu Bazar, Lahore.
- Freedman, D; Pisani, R;Parues, R and Adhikari, A (1997). Statistics 3
^{rd}Norton, New York. - Freund, J.E(1990).Modern Elementary Statistics. Prentice Hall, Inc.New Jersy.
- Graybill,I and Burdick (1998).Applied Statistics: A first course in inference. Prentice Hall, New Jersy.
- Haq, Masood-ul (1983) Foundation of Probability and Statistics, Tahir Sons, Urdu Bazar, Karachi.
- Lipschutz, S and Schiller,J (1998). Introduction to Probability and McGraw Hill, New York.
- Polland, A.H; Yousaf, F and Pollard, G,N (1981). Demographic Techniques. Second Edition. Pergaman Press, Oxford.
- Speigal, M.R and Stephens, L.J (1999). Statistics 3
^{rd}McGraw Hill, New York. - Speigal, M.R; Schiller, J.L; Srinivban, R.L (2000). Probability and Statistics 2
^{nd}Schaums out line Series. McGraw Hill, New York. - Walpole, R.E (2001 R). Introduction to Statistics. Macmillan publishing Company.New York/London.
- Wonnacott, T.H. and Wonnacott, R.J (1990). Introductory Statistics. John Wily & Sons. New York.

**Statistics (Optional)**

### Outlines of Tests

Marks: 100

**Paper: (Written) Time – 3 hours**

** **Syllabi and Courses of Reading

*Paper***: (Written) **

**Introduction**

Definition, Characteristics and limitations of statistics, Collection, classification and tabulation of data.

**Graph****and Diagrams**

Bar and pie diagrams. Graphs of frequency distribution viz. Histogram, frequency polygon, frequency curve, cumulative frequency curve, Graphic interpolation.

**Averages**

Elementary knowledge and numerical illustrations of arithmetic mean, median mode, and weighted average. Smoothing of fluctuations by moving average method.

**Dispersion****:**

Elementary knowledge and numerical illustrations of range, fractiles, quartile deviation, standard deviation, co-efficient of skewness and co-efficient of variantion.

**Attributes****and Chi-Square:**

** **Concept of attribute, idea of independence and association, dichotomy, co-efficient of association, contingency table. Chi-square.

**Correlation:**

** **Concept of regression, simple correlation and rank correlation with numerical illustrations.

**Sampling****:**

** **Concept of sampling. Definition of population and sampling unit. Purposive and random sampling. Drawing of a random sample without replacement from finite population

*Books Recommended:*

- Zia-ud-Din, M., Practical Statistics with Fundamentals of Theory, 8
^{th}edition, The Punjab Educational Press, Lahore. - Chambers, E.G., Statistical Calculations for Beginners. Cambridge University Press, London.