FORMULAS FOR STATISTICS 1 Sample statistics X = 1 n n X i=1 X i or x = 1 n n X i=1 x i (sample mean) S 2 = 1 n - 1 n X i=1 (X i - X ) 2 = 1 n - 1 n X i=1 X 2 i - n n - 1 X 2 or s 2 = 1 n - 1 n X i=1 (x i - x) 2 = 1 n - 1 n X i=1 x 2 i - n n - 1 x 2 (sample variance) E( X )= μ , var( X )= σ 2 n , E(S 2 )= σ 2 Sample statistics for normal distribution Z = X - μ σ/ √ n ∼ N(0, 1) T = X - μ S/ √ n ∼ t with n - 1 degrees of freedom V = (n - 1)S 2 σ 2 = n X i=1 (X i - X ) 2 σ 2 ∼ χ 2 with n - 1 degrees of freedom Z = ( X 1 - X 2 ) - (μ 1 - μ 2 ) p σ 2 1 /n 1 + σ 2 2 /n 2 ∼ N(0, 1) T = ( X 1 - X 2 ) - (μ 1 - μ 2 ) S p p 1/n 1 +1/n 2 ∼ t with n 1 + n 2 - 2 degrees of freedom, where S 2 p = (n 1 - 1)S 2 1 +(n 2 - 1)S 2 2 n 1 + n 2 - 2 assuming σ 1 = σ 2 W = ( X 1 - X 2 ) - (μ 1 - μ 2 ) p S 2 1 /n 1 + S 2 2 /n 2 ≈ t with (a 1 + a 2 ) 2 a 2 1 /(n 1 - 1) + a 2 2 /(n 2 - 1) degrees of freedom, where a 1 = s 2 1 n 1 and a 2 = s 2 2 n 2 (Welch–Satterthwaite approximation) F = S 2 1 /σ 2 1 S 2 2 /σ 2 2 ∼ F with n 1 - 1 and n 2 - 1 degrees of freedom i
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FORMULAS FOR STATISTICS 1 - TUTmath.tut.fi/~ruohonen/S1L/Formulas.pdf · 2018-11-16 · SST = Xn i=1 (y i 2y) ; SSR = n i=1 (^y i y)2 SST = SSE + SSR MST = SST n 1; MSR = SSR k ANOVA-table:
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FORMULAS FOR STATISTICS 1
Sample statistics
X =1
n
n∑i=1
Xi or x =1
n
n∑i=1
xi (sample mean)
S2 =1
n− 1
n∑i=1
(Xi −X)2=1
n− 1
n∑i=1
X2i −
n
n− 1X
2or
s2 =1
n− 1
n∑i=1
(xi − x)2 =1
n− 1
n∑i=1
x2i −n
n− 1x 2 (sample variance)
E(X) = µ , var(X) =σ2
n, E(S2) = σ2
Sample statistics for normal distribution
Z =X − µσ/√n∼ N(0, 1)
T =X − µS/√n∼ t with n− 1 degrees of freedom
V =(n− 1)S2
σ2=
n∑i=1
(Xi −X)2
σ2∼ χ2 with n− 1 degrees of freedom
Z =(X1 −X2)− (µ1 − µ2)√
σ21/n1 + σ2
2/n2
∼ N(0, 1)
T =(X1 −X2)− (µ1 − µ2)
Sp
√1/n1 + 1/n2
∼ t with n1 + n2 − 2 degrees of freedom, where
S2p =
(n1 − 1)S21 + (n2 − 1)S2
2
n1 + n2 − 2assuming σ1 = σ2
W =(X1 −X2)− (µ1 − µ2)√
S21/n1 + S2
2/n2
≈ t with(a1 + a2)
2
a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where
a1 =s21n1
and a2 =s22n2
(Welch–Satterthwaite approximation)
F =S21/σ
21
S22/σ
22
∼ F with n1 − 1 and n2 − 1 degrees of freedom
i
Point estimation
Parameter θ Estimate θ Estimator Θ
µ µ = x X
σ2 σ2 = s2 S2
m m = q(0.5) Q(0.5)
Interval estimation of expectation
Z =X − µσ/√n
: x± zα/2σ√n
T =X − µS/√n
: x± tα/2s√n
(with n− 1 degrees of freedom)
Z =(X1 −X2)− (µ1 − µ2)√
σ21/n1 + σ2
2/n2
: (x1 − x2)± zα/2
√σ21
n1
+σ22
n2
T =(X1 −X2)− (µ1 − µ2)
Sp
√1/n1 + 1/n2
: (x1 − x2)± tα/2sp√
1
n1
+1
n2
(with n1 + n2 − 2 degrees of freedom)
W =(X1 −X2)− (µ1 − µ2)√
S21/n1 + S2
2/n2
: (x1 − x2)± tα/2
√s21n1
+s22n2
(Welch–Satterthwaite)
(with ∼=(a1 + a2)
2
a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where a1 =
s21n1
and a2 =s22n2
)
Estimation of proportion for the binomial distribution
P(X = x) =
(n
x
)px(1− p)n−x
E(X) = np , var(X) = np(1− p)
p =x
n
n∑i=x
(n
i
)piL(1− pL)n−i =
α
2,
x∑i=0
(n
i
)piU(1− pU)n−i =
α
2
Interval estimation of variance
V =(n− 1)S2
σ2:
(n− 1)s2
h2,α/2and
(n− 1)s2
h1,α/2(with n− 1 degrees of freedom)
F =S21/σ
21
S22/σ
22
:s21s22
1
f2,α/2and
s21s22
1
f1,α/2(with n1 − 1 and n2 − 1 degrees of freedom)
ii
Testing expectations
z =x− µ0
σ/√n
and H0 : µ = µ0 :
H1 Critical region P-probability
µ > µ0 z ≥ zα 1− Φ(z)µ < µ0 z ≤ −zα Φ(z)µ 6= µ0 |z| ≥ zα/2 2 min
(Φ(z), 1− Φ(z)
)Φ is the standard normal cumulative distribution function.
t =x− µ0
s/√n
and H0 : µ = µ0 :
H1 Critical region P-probability
µ > µ0 t ≥ tα 1− F (t)µ < µ0 t ≤ −tα F (t)µ 6= µ0 |t| ≥ tα/2 2 min
(F (t), 1− F (t)
)F is the cumulative t-distribution function with n− 1 degrees of freedom.
z =x1 − x2 − d0√σ21/n1 + σ2
2/n2
and H0 : µ1 − µ2 = d0 :
H1 Critical region P-probability
µ1 − µ2 > d0 z ≥ zα 1− Φ(z)µ1 − µ2 < d0 z ≤ −zα Φ(z)µ1 − µ2 6= d0 |z| ≥ zα/2 2 min
(Φ(z), 1− Φ(z)
)Φ is the standard normal cumulative distribution function.
t =x1 − x2 − d0
sp√
1/n1 + 1/n2
, where s2p =(n1 − 1)s21 + (n2 − 1)s22
n1 + n2 − 2, and H0 : µ1−µ2 = d0 :
H1 Critical region P-probability
µ1 − µ2 > d0 t ≥ tα 1− F (t)µ1 − µ2 < d0 t ≤ −tα F (t)µ1 − µ2 6= d0 |t| ≥ tα/2 2 min
(F (t), 1− F (t)
)F is the cumulative t-distribution function with n1 + n2 − 2 degrees of freedom.
t =x1 − x2 − d0√s21/n1 + s22/n2
and H0 : µ1 − µ2 = d0 (Welch–Satterthwaite) :
H1 Critical region P-probability
µ1 − µ2 > d0 t ≥ tα 1− F (t)µ1 − µ2 < d0 t ≤ −tα F (t)µ1 − µ2 6= d0 |t| ≥ tα/2 2 min
(F (t), 1− F (t)
)F is approximatively the cumulative t-distribution function with
(a1 + a2)2
a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where a1 =
s21n1
and a2 =s22n2
.
iii
Testing variances
v =(n− 1)s2
σ20
and H0 : σ2 = σ20 :
H1 Critical region P-probability
σ2 > σ20 v ≥ h2,α 1− F (v)
σ2 < σ20 v ≤ h1,α F (v)
σ2 6= σ20 v ≤ h1,α/2 or v ≥ h2,α/2 2 min
(F (v), 1− F (v)
)F is the cumulative χ2-distribution function with n− 1 degrees of freedom.
f =1
k
s21s22
and H0 : σ21 = kσ2
2 :
H1 Critical region P-probability
σ21 > kσ2
2 f ≥ f2,α 1−G(f)σ21 < kσ2
2 f ≤ f1,α G(f)σ21 6= kσ2
2 f ≤ f1,α/2 tai f ≥ f2,α/2 2 min(G(f), 1−G(f)
)G is the cumulative F-distribution function with n1 − 1 and n2 − 1 degrees of freedom.
Right tail quantiles f2, for F-distribution for = 0.05, 0.025, 0.01 and degrees of freedom v1 and v2. The inverses are the left tail quantiles f1, for the same :s, and degrees of freedom v2 and v1.