Definition of chisquare r.v. as a sum of squares of i.i.d. standard normal variables. Derivation of the p.d.f. of Chi-square variable with n degrees of freedom (d.f.) using M.G.F.
Notation: X-Mean, variance, M.G.F., C.G.F., central moments skewness, kurtosis, mode, additive property. Use of chi-square tables for calculations of probabilities.
(-n)
Normal approximation:
(statement only) Distribution of X
√2n
and
i = 1
(x-x)2 for a random sample from a normal
distribution using orthogonal transformation, independence of X and S2.
Definition of t r.v. with n d.f. in the form t =
where
√v/n'
U ~ N(0, 1) and V with n d.f. and U and V are independent random variables.
Notation: t-to
Derivation of the p.d.f of t distribution, nature of probability curve, mean, variance, moments, mode. Use of t-tables for calculations of probabilities, statement of normal approximation.
X1/n1
Definition of F r.v. with n1 and n2 d.f. as Fn1, n2 = X2/n2 Where X1
and X2 are independent chi-square r.v.s with n, and n2 d.f. Notation: F - Fn1, n2 Derivation of p.d.f., nature of probability curve, mean, variance, moments, mode.
Distribution of (Fn, n2), use of tables of F-distribution of calculation of probabilities.
Interrelations between Chi-square, t and F distributions.
Tests based on chi-square distribution:
(a) Test for independence of two attributes arranged in 2 x 2 contingency table. (With Yate's correction) (to be covered in practical only).
(b) Test for independence of two attributes arranged in rxs contingency table, McNemar's test. (to be covered in practical only).
(c) Test for 'Goodness of Fit'. (to be covered in practical only).
(d) Test for (Ho: o2 = 6) against one-sided and two-sided
alternatives when (i) for known mean, (ii) for unknown
mean.
Tests based on t-distribution:
(a) Tests for population means: (i) Single sample with unknown variance and two sample for unknown equal variances tests for one-sided and two-sided alternatives, (ii) 100 (1a) % two sided confidence interval for population mean and difference of means of two independent normal population.
(b) Paired t-test for one-sided and two-sided alternatives.
Tests based on F-distribution:
(a) Tests for Ho: 01 02 against one-sided and two-sided alternatives when (i) means are known, and (ii) means are unknown. Take F = S/S
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