Title 2nd stats ch 05.pdf ✅

Chapter 5 Theoretical Distribution Question 4. Write down the p.m.f of Hyperegeometric distribution and with its two features and examples. Answer: Hypergeometric distribution tends to Binomial distribution when: 1. a is large ie.a → ∞ 2. b is large ie. b → ∞ and 3. aa+b = p is fixed. `Ex:- Number of vegetarians in an office of 18 vegetarians and 12 non- vegetarians; number of apples selected from a bag containing 20 oranges and 10 apples. Question 5. Write down the p.m.f of Normal distribution and with its two features and examples. Answer: The Normal distribution is:- Where -∞ < x < ∞, (Range) σ > 0 Ht./Wt.of students of a class; I.Q.of a large group of children; Wages/Income of employees Properties:- 1. Curve is bell shaped, Mean = Median = Mode ie., The curve is Symmetric ie., non-skew (β1 = 0) 2. Mean = μ, Variance = σ2 ; S.D = σ, Q.D = 2/3 σ, M.D = 4/5 σ 3. The curve is Mesokurtic β2 = 3 4. The area under the Normal curve is 1
Chapter 5 Theoretical Distribution 5. The curve is asymptotic to the x-axis, ie. It touches the x-axis at -∞ and ∞, 6. Q1 & Q3 are equidistant from: X̄ / M / Z = Q3+Q12 Question 6. Write down the p.m.f of Chi-square distribution and students t – distribution with its two features and examples. Answer: Chi-square distribution:- Definition-Let Z1, Z2, Z3, Zn are n S.N.V’s; Then X2= Z1 2+ Z2 2 + Z3 2 .......+ Zn 2~ X2 (n) Features/Properties:- parameter ‘n’; Mean = n, Variance = 2n. Mode = (n – 2) for n > 2, Range – (0, ∞ ); The curve is + vely Skewed. Application/uses:- 1. Test for population variance; 2. Test for Goodness of fit 3. Test for Independence of Attribute. Properties : – n – parameter, Mean = 0, (X̄ = M = Z = 0), Variance = nn−2 for n>2, -∞ < t < ∞; Application:- 1. To test for Population mean 2. Test for equality of means, 3. Test for equality of two population’s means when observations are paired (paired t-test).