Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

Course Information

Title	Inferential Statistics
Theory Credits	3 (3 hrs/week)
Practical Credits	1 (2 hrs/week)

Course Outcomes

1. Acquaint with estimator, estimates, estimation techniques and their properties.
2. Acquire knowledge of testing the hypothesis of different distributions.
3. Learn about large sample techniques using various tools.
4. Learn about small sample techniques using various tools.
5. Deal with situations where there are no parameters in the distributions.

Theory — Five Units

Unit 1: Theory of Estimation

Estimation of a parameter; criteria of a good estimator — unbiasedness, consistency, efficiency, sufficiency. Estimation of parameters by the method of moments and maximum likelihood (MLE); properties of MLEs. Rao–Cramer inequality, properties. Binomial, Poisson and Normal population parameters estimated by MLE method. Confidence intervals.

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Unit 2: Testing of Hypothesis

Concepts of statistical hypotheses, null and alternative hypothesis, critical region, two types of errors, level of significance, concept of p-value and power of a test. One- and two-tailed tests. Neyman–Pearson's lemma. Examples in case of Binomial, Poisson, Exponential and Normal distributions.

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Unit 3: Large Sample Tests

Large sample test for single mean and difference of two means; confidence intervals for mean(s). Large sample test for single proportion, difference of proportions. Standard deviation(s) and correlation coefficient(s).

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Unit 4: Small Sample Tests

Assumptions and t-test for single mean, difference of means and paired t-test. \(\chi^2\) test for goodness of fit and independence of attributes. \(\chi^2\) test for single variance, F-test for equality of variances.

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Unit 5: Non-parametric Tests

Advantages and disadvantages, comparison with parametric tests. One-sample runs test, sign test and Wilcoxon signed-rank tests (single and paired samples). Two independent sample tests: Median test, Wilcoxon–Mann–Whitney U test, Wald–Wolfowitz runs test.

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Practical — List of Experiments (12)

1. Large sample test for mean and difference of means.
2. Large sample test for proportion and difference of proportions.
3. Large sample test for SD and difference of SDs.
4. Large sample test for correlation coefficient.
5. Small sample test for mean and difference of means.
6. Small sample test for correlation coefficient.
7. Paired t-test (paired samples).
8. Small sample test for single variance (\(\chi^2\) test) and difference of variances (F-test).
9. \(\chi^2\) test for goodness of fit and independence of attributes.
10. Non-parametric tests for single sample (run test, sign test, Wilcoxon signed-rank).
11. Non-parametric tests for related samples (sign test, Wilcoxon signed-rank).
12. Non-parametric tests for two independent samples (Median, Mann–Whitney U, Wald–Wolfowitz).

Note: Conclusions of practical problems must be drawn based on p-value as well as critical values.

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Text Books

1. S. C. Gupta & V. K. Kapoor — Fundamentals of Mathematical Statistics, Sultan Chand & Sons.
2. K. Rohatgi & Ehsanes Saleh — An Introduction to Probability and Statistics, John Wiley & Sons.

References

1. O. P. Gupta — Mathematical Statistics, Kedarnath Ramnath & Co.
2. P. N. Arora & S. Arora — Quantitative Aptitude Statistics — Vol II, S. Chand & Company Ltd.

Suggested Co-Curricular Activities

1. Training of students by related industrial experts.
2. Assignments including technical assignments, if any.
3. Seminars, group discussions, quiz, debates etc. on related topics.
4. Preparation of audio and videos on tools of diagrammatic and graphical representations.
5. Collection of material / figures / photos of related topics.
6. Invited lectures and presentations of stalwarts on those topics.
7. Visits / field trips of firms, research organizations etc.

Exam Blueprint (recommended)

Unit	Topic	Approx. Weightage
1	Theory of Estimation	20 %
2	Testing of Hypothesis	20 %
3	Large Sample Tests	20 %
4	Small Sample Tests	20 %
5	Non-parametric Tests	20 %

Quick Reference — Test Statistics

Hypothesis	Test Statistic	Distribution
μ = μ₀ (large n)	(X̄ − μ₀)/(σ/√n)	N(0, 1)
μ = μ₀ (small n)	(X̄ − μ₀)/(s/√n)	tn−1
μ₁ = μ₂ (small, pooled)	(X̄₁ − X̄₂)/[s_p √(1/n₁+1/n₂)]	tn₁+n₂−2
p = p₀	(p̂ − p₀)/√(p₀q₀/n)	N(0, 1)
σ² = σ₀² (small n)	(n−1)s²/σ₀²	χ²n−1
σ₁² = σ₂²	s₁²/s₂²	Fn₁−1, n₂−1
Goodness of fit	Σ(O − E)²/E	χ²k−1−r
ρ = 0	r √(n−2)/√(1 − r²)	tn−2