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Final

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School
University of Guelph
Department
Statistics
Course
STAT 2040
Professor
Gary Umphrey
Semester
Fall

Description
P P P 2 (xi▯ x▯)2 x ▯ ( i ) Sample Variance: s = . Equivalent alternative formula: s = i n n ▯ 1 n ▯ 1 x ▯ ▯ Sample z-score: z = s ▯ If we transform the data using the linear transformation x = a + bx: x = a + bx▯, x▯ = jbjx , ▯ = b sx x Probability P(A [ B) = P(A) + P(B) ▯ P(A \ B). P(A \ B) = P(A)P(BjA) = P(B)P(AjB). P(A \ B) The conditional probability of A, given B has occurred is P(AjB) = : P(B) Two events A and B are independent if and only if: P(AjB) = P(A);P(BjA) = P(B);P(A \ B) = P(A)P(B): Expected Value and Variance of a Discrete Random Variable P 2 2 P 2 E(X) = ▯ = xp(x), ▯ = E[(X ▯ ▯) ] = (x ▯ ▯) p(x): Properties of Expectation and Variance 2 2 2 E(a + bX) = a + bE(X), ▯ a+bX = b ▯ X, ▯a+bX = jbjX If X and Y are two random variables then E(X + Y ) = E(X) + E(Y ). 2 2 2 2 2 2 If X and Y are independent: ▯X+Y = ▯X + ▯ Y and ▯X▯Y = ▯X + ▯Y Binomial and Poisson Distributions If X is▯ ▯binomial random variable t▯ ▯: p(x) = n p qn▯x, for x = 0, 1, ..., n.= n! . ▯ = np;▯ = npq: x x x!(n▯x)! ▯▯ x e ▯ 2 Poisson distribution: p(x) = x! ;▯ = ▯ = ▯ . Normal Distribution ▯ If X is normally distributed with a mean of ▯ and standard deviation ▯, and X is the mean of n X ▯ ▯ X ▯ ▯ X▯ X ▯ ▯ independent observations: Z = and Z = = p ▯ ▯X ▯= n University of Guelph STAT*2040 W11: Formula sheets for the final exam Inference Procedures for Means If ▯ is known: ▯ ▯ Confidence interval for ▯: X ▯ z ▯=2▯X▯ where ▯ X pn X ▯ ▯ 0 To test H 0 ▯ = ▯ 0 Z = ▯ X If ▯ is unknown: s Confidence interval for ▯: X ▯ t ▯=2SE(X), where SE(X) =▯ p n ▯ X ▯ ▯ 0 To test H 0 ▯ = ▯ 0 t = ▯ SE(X) Inference for ▯ ▯ ▯ : 1 2 2 2 q 2 (n1▯ 1)s 1 (n ▯21)s 2 ▯ ▯ 1 1 The pooled-variance method: s =p , SE(X ▯1X ) 2 s p n1+ n2 n1+ n 2 2 Confidence interval for ▯1▯ ▯ :2X ▯1X ▯ t2 ▯=2SE(X ▯1X ) ▯2 X 1 X ▯2 To test H 0 ▯1= ▯ ,2t = . The degrees of freedom are n1+ n 2 2. SE(X ▯1X ) ▯2 q ▯ ▯ s1 2 The Welch Method: SE (X W X 1 = 2 n1 + n 2 Confidence interval for ▯1▯ ▯ :2X ▯1X ▯ t2 ▯=2SE WX ▯ 1 ) ▯ 2 X 1 X ▯2 To test H 0 ▯1= ▯ ,2t = SE WX ▯ 1 ) ▯2 2 2 (s1+ s2)2 Approximate df = 21 n2 2 (You won’t have to calculate these degrees of freedom by hand) 1 (s1) + 1 (s2)2 n1▯1 n1 n2▯1 n2 Inference Procedures for Proportions r
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