Tamil Nadu HSE Second Year Statistics study plan

Distinguish discrete and continuous random variables, and derive mean and variance for the binomial, Poisson and normal distributions from first principles.

Compute probabilities for binomial and Poisson situations, including cumulative cases, without referring to a table.

State the null and alternative hypotheses for a given research claim and classify errors of Type I and Type II with their probabilities.

Set up one-tailed and two-tailed tests at 5% and 1% levels and identify the correct critical region for each.

Carry out large-sample tests for a single mean and for the difference of two means, showing every step of the test statistic.

Apply the chi-square goodness-of-fit and independence tests, computing expected frequencies correctly.

Explain the difference between a population, a sample, a sampling frame and a statistic, using a concrete Tamil Nadu survey example.

Derive the standard error of the sample mean for simple random sampling with and without replacement.

Distinguish point and interval estimation, and state the properties of a good estimator: unbiasedness, consistency, efficiency and sufficiency.

Construct confidence intervals for a population mean with known variance and for a proportion, at 95% and 99% confidence.

Compute the standard error of a proportion and of a difference of proportions, and interpret each numerically.

Describe the central limit theorem and use it to justify normal approximations in sampling problems.

Compute Karl Pearson's coefficient of correlation from raw and grouped data and interpret its sign and magnitude.

Derive both regression lines, identify which line predicts which variable, and compute the coefficient of determination.

Use the relation between correlation and regression coefficients, including the product of regression coefficients, in numerical problems.

Solve a mixed set of distribution and correlation problems under timed conditions to build recall speed.

Compare simple random, stratified, systematic and cluster sampling, and justify a choice for a stated practical situation.

Solve problems on sample size determination for a desired margin of error for means and proportions.

Explain the t-distribution and use it for small-sample confidence intervals and tests for a single mean.

Work through past-style numerical questions on standard error and sampling distribution of the mean and of the proportion.

Find the inverse of a matrix using the adjoint method and by elementary row operations, and verify the result.

Solve systems of linear equations by matrix inversion and by Cramer's rule, including consistency checks via rank.

Perform addition, multiplication, division and conjugation of complex numbers in algebraic form, and convert between algebraic, polar and exponential forms.

Apply De Moivre's theorem to find powers and roots of complex numbers and solve simple polynomial equations over the complex numbers.

Expand binomial expressions with positive, negative and fractional indices and identify the general term.

Find a specified term, coefficient or constant term in an expansion, and compute approximations using the binomial theorem.

Recognise arithmetic, geometric and special series, and compute sums to n terms and sums to infinity where they exist.

Apply permutation and combination formulas to arrangement and selection problems, including circular permutations and repetition cases.

Use the relationship between roots and coefficients of polynomial equations to find sums and products of roots and to form equations with given roots.

Apply Descartes' rule of signs and the nature-of-roots tests to locate and bound real roots.

Evaluate inverse trigonometric expressions and simplify composite forms using principal value ranges.

Prove standard inverse trigonometric identities and solve simple inverse trigonometric equations.