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STAT 2040
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Gary Umphrey
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Final

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University of Guelph

Statistics

STAT 2040

Gary Umphrey

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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