Download MAT 222 FORMULA CARD

MAT 222 FORMULA CARD
Standard score: z =
Two-sample t statistic for H0 : ¹1 = ¹2
(¾1 ; ¾2 unknown):
x¡¹
¾
Standard score for sample mean x
¹: z =
x
¹¡¹
p
¾= n
t=
df = minimum of n1 ¡ 1, n2 ¡ 1
Chapter 6: Introduction to Inference
Two-Sample Con¯dence interval for ¹1 ¡ ¹2 :
s
s2
s21
¤
(¹
x1 ¡ x
¹2 ) § t
+ 2;
n1
n2
Con¯dence interval for mean ¹ (¾ known):
¾
x
¹ § z¤ p ;
n
z¤
C
1.645
90%
1.960
95%
2.576
99%
df = minimum of n1 ¡ 1, n2 ¡ 1
Sample size for con¯dence interval for ¹ with
margin of error m:
n=
(¹
x1 ¡ x
¹2 ) ¡ (¹1 ¡ ¹2 )
r 2
;
s22
s1
+
n1
n2
Pooled two-sample estimator of ¾ 2 :
h z ¤ ¾ i2
s2p =
m
(n1 ¡ 1)s21 + (n2 ¡ 1)s22
n1 + n2 ¡ 2
z Statistic for H0 : ¹ = ¹0 (¾ known):
z=
Pooled two-sample t statistic for H0 : ¹1 = ¹2 when
¾1 = ¾2 :
x
¹ ¡ ¹0
p
¾= n
t=
s
Standard error of x
¹: SEx¹ = p
n
Con¯dence interval for mean ¹ (¾ unknown):
Two-sample F statistic for ¾1 = ¾2 :
df= n ¡ 1
F =
x
¹ ¡ ¹0
p ;
s= n
s21
;
s22
df = n1 ¡ 1 (numerator) and n2 ¡ 1 (denominator).
One-sample t Statistic for H0 : ¹ = ¹0 (¾ unknown):
t=
df = n1 + n2 ¡ 2
Pooled two-sample con¯dence interval for ¹1 ¡ ¹2
when ¾1 = ¾2 :
r
1
1
¹2 ) § t¤ sp
+
;
df = n1 + n2 ¡ 2
(¹
x1 ¡ x
n1
n2
Chapter 7: Inference for Distributions
s
x
¹ § t¤ p ;
n
x
¹1 ¡ x
¹2
q
;
1
sp n1 + n12
df = n ¡ 1
Chapter 8: Inference for Proportions
Two-sample z statistic for H0 : ¹1 = ¹2
(¾1 ; ¾2 known):
z=
Sample proportion: p^ = X=n, X = number of \successes"
(¹
x1 ¡ x
¹2 ) ¡ (¹1 ¡ ¹2 )
r 2
¾2
¾1
+ 2
n1
n2
Standard error of p^: SEp^ =
1
r
p^(1 ¡ p^)
n
Chi-square test statistic:
Con¯dence interval for p:
X2 =
¤
p^ § z SEp^
X (observed ¡ expected)2
expected
df = (# of rows { 1)(# of columns { 1)
z statistic for H0 : p = p0
z= r
p^ ¡ p0
Chapter 10: Inference for Regression
p0 (1 ¡ p0 )
n
1 P
(xi ¡ x
¹) 2
n¡1
1 P
Sample variance of y's: s2y =
(yi ¡ y¹)2
n¡1
Sample variance of x's: s2x =
Sample size for desired margin of error m:
n=(
or
z¤ 2 ¤
) p (1 ¡ p¤ )
m
z¤ 2
)
n=(
2m
(p¤ = guessed value)
Sample correlation:
1 X xi ¡ x
¹ yi ¡ y¹
r=
)(
)=
(
n¡1
sx
sy
1
(conservative approach with p = )
2
¤
Di®erence of sample proportions: D = p^1 ¡ p^2
Regression line: y^ = b0 + b1 x,
Standard error of sample di®erence D:
r
p^2 (1 ¡ p^2 )
p^1 (1 ¡ p^1 )
+
SED =
n1
n2
Slope: b1 = r
Mean squared error:
s2 =
(^
p1 ¡ p^2 ) § z ¤ SED
1 X 2
ei ;
n¡2
Standard error of b0 :
s
SEb0 = s
Pooled estimator of p when p1 = p2 :
where ei = yi ¡ y^i
1
x
¹2
+P
n
¹ )2
(xi ¡ x
Level C con¯dence interval for b0 :
X1 + X2
n1 + n2
b0 § t¤ SEb0 ; df = n ¡ 2
Standard error of D under H0 : p1 = p2 :
r
1
1
SEDp = p^(1 ¡ p^)(
+
)
n1
n2
Standard error of b1 :
s
SEb1 = pP
¹) 2
(xi ¡ x
z statistic for H0 : p1 = p2 :
z=
sy
sx
Intercept: b0 = y¹ ¡ b1 x
¹
Con¯dence interval for p1 ¡ p2 :
p^ =
sP
(^
y ¡ y¹)2
P i
(yi ¡ y¹)2
Level C con¯dence interval for b1 :
b1 § t¤ SEb1 ; df = n ¡ 2
p^1 ¡ p^2
SEDp
Test statistic for H0 : ¯1 = 0 :
Chapter 9: Inference for Two-Way Tables
t=
b1
; df = n ¡ 2
SEb1
Expected cell counts:
Estimate for mean response ¹y when x = x¤ :
row total £ column total
expected cell count =
n
¹
^y = b0 + b1 x¤
2
Standard error of ¹
^y when x = x¤ :
s
(x¤ ¡ x
1
¹) 2
SE¹^ = s
+P
n
¹)2
(xi ¡ x
Test statistic for H0 : ¯j = 0:
t=
bj
; df = n ¡ p ¡ 1:
SEbj
Level C con¯dence interval for ¹y when x = x¤ :
¹
^y § t¤ SE¹^ ;
Sum of squares SS: SST = SSM + SSE
df = n ¡ 2
Estimate for future observation of y when x = x¤ :
Degrees of freedom DF:
¤
y^ = b0 + b1 x
DFT = DFM + DFE,
Standard error of y^ when x = x¤ :
s
1
(x¤ ¡ x
¹)2
SEy^ = s 1 + + P
n
¹ )2
(xi ¡ x
DFT = n ¡ 1, DFM = p, DFE = n ¡ p ¡ 1,
SSM
DFM
SSE
Mean square error: MSE =
DFE
Mean square model: MSM =
Level C prediction interval for y when x = x¤ :
y^ § t¤ SEy^ ;
Sum of Squares
X
SST =
(yi ¡ y¹)2 ;
df = n ¡ 2
Test statistic for H0 : ¯1 = ¯2 = ¢ ¢ ¢ = ¯p = 0:
(Total Sum of Squares)
X
(^
yi ¡ y¹)2 ; (Model Sum of Squares)
X
SSE =
(yi ¡ y^i )2 ; (Error Sum of Squares)
Squared multiple correlation: R2 =
SSM =
SST=SSM+SSE
sum of squares
MS =
degrees pf freedoms
One-way ANOVA model:
xij = ¹i + "ij ;
MSM
SSM/DFM
=
; df = (1; n ¡ 2)
MSE
SSE/DFE
for i = 1; : : : ; I and j = 1; : : : ; ni ; and N = n1 + ¢ ¢ ¢ + nI .
p
r n¡2
Test statistic for H0 : ½ = 0: t = p
1 ¡ r2
df = n ¡ 2.
Pooled-sample variance:
s2p =
(n1 ¡ 1)s21 + ¢ ¢ ¢ (nI ¡ 1)s2I
= MSE
(n1 ¡ 1) + ¢ ¢ ¢ + (nI ¡ 1)
Sum of squares (SS): SST = SSG + SSE
Chapter 11: Multiple Regression
SSG
Multiple Regression Model:
=
X
ni (¹
xi ¡ x
¹)2
X
(ni ¡ 1)s2i
allgroups
yi = ¯0 + ¯1 xi1 + ¯2 xi2 + ¢ ¢ ¢ + ¯p xip + "i
SSE
Least squares estimates of ¯0 ; ¯1 ; : : : ; ¯p :
=
allgroups
Degrees of freedom (DF):
b0 ; b1 ; : : : ; bp
Estimate of ¾: s =
SSM
SST
Chapter 12: One-way ANOVA
The ANOVA F test for H0 : ¯1 = 0
F =
MSM
; df = (p; n ¡ p ¡ 1):
MSE
F =
DF T = DF G + DF E;
p
MSE.
where DFT = N ¡ 1, DFG = I ¡ 1, DFE = N ¡ I.
Level C con¯dence interval for ¯j :
Mean square (MS): MSG =
¤
bj § t SEbj ; df = n ¡ p ¡ 1
3
SSG
SSE
, MSE =
DFG
DFE
DFT = DFA + DFB + DFAB + DFE,
Test statistic for H0 : ¹1 = ¹2 = ¢ ¢ ¢ = ¹I :
F =
DFM = DFA + DFB + DFAB ,
MSG
; df = (I ¡ 1; N ¡ I):
MSE
where
DFT = N ¡ 1,
DFA = I ¡ 1,
DFB = J ¡ 1,
DFAB = (I ¡ 1)(J ¡ 1),
DFE = N-IJ.
SSG
SST
P
P
Population contrast: Ã =
ai ¹i , where
ai = 0
Coe±cient of determination: R2 =
Sample contrast: c =
P
ai x¹i
Mean square (MS): For the factors A and B, for the
interaction AB, and for the error E:
Standard error of c :
SEc = sp
s
X a2i
MS =
ni
Test statistic for H0 : Main e®ect of A is zero:
Test statistic for H0 : Ã = 0:
t=
c
; df = N ¡ I
SEc
F =
Level C con¯dence interval for Ã:
c § t SEc ; df = N ¡ I:
F =
Multiple Comparisons t statistic:
tij
MSA
;
MSE
df = (I ¡ 1; N ¡ IJ):
Test statistic for H0 : Main e®ect of B is zero:
¤
x
¹i ¡ x
¹j
= q
;
1
sp ni + n1j
SS
DF
MSB
;
MSE
df = (J ¡ 1; N ¡ IJ):
Test statistic for H0 : Interaction e®ect of A and
B is zero:
df = N ¡ I
F =
Simultaneous Con¯dence Intervals for Mean Differences:
s
1
1
¤¤
(¹
xi ¡ x
¹ j ) § t sp
+
; df = N ¡ I
ni
nj
Chapter 13: Two-way ANOVA
Factors: Two factors A and B, factor A has I levels, and
factor B has J levels
Two-way ANOVA model:
xijk = ¹ij + "ijk ;
for i = 1; : : : ; I; j = 1; : : : ; J and k = 1; : : : nij :
Pooled-sample variance:
P
(nij ¡ 1)s2ij
= MSE
s2p = P
(nij ¡ 1)
Sum of squares (SS): SST = SSA + SSB + SSAB + SSE
Degrees of freedom (DF):
4
MSAB
;
MSE
df = ((I ¡ 1)(J ¡ 1); N ¡ IJ):