Author(s): Brenda Gunderson, Ph.D., 2012 Unless otherwise ...
Author(s): Brenda Gunderson, Ph.D., 2012
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Statistics 350 Help Card
Summary Measures
Sample Mean x x1 x2 xn n
? xi
n
Sample Standard Deviation
s
? (xi x)2
n 1
? xi 2 nx 2
n 1
Normal Random Variables
x
z score
observation mean standard deviation
x P V
x Percentile: x zV P
x If X has the N (P,V ) distribution, then the variable
Z
X P V
has the
N (0,1)
distribution.
Probability Rules
x Complement rule P( AC ) 1 P( A)
x Addition rule General: P( A or B) P( A) P(B) P( A and B)
For independent events: P( A or B) P( A) P(B) P( A)P(B)
For mutually exclusive events: P( A or B) P( A) P(B) x Multiplication rule
General: P( A and B) P( A)P(B | A)
For independent events: P( A and B) P( A)P(B)
For mutually exclusive events: P( A and B) 0 x Conditional Probability
P( A and B) General: P( A | B)
P(B)
For independent events: P( A | B) P( A)
For mutually exclusive events: P( A | B) 0
Discrete Random Variables
Mean
? E(X ) P
xi pi x1 p1 x2 p2 xk pk
Standard Deviation
? s.d.(X ) V
(xi P)2 pi
? xi2 pi P 2
Binomial Random Variables
P( X
k)
????
n k
????
p
k
(1
p) nk
where
????
n k
????
n! k!(n k)!
Mean E( X ) P X np
Standard Deviation
s.d.( X ) V X np(1 p)
Normal Approximation to the
Binomial Distribution
If X has the B(n, p) distribution and the sample size n is large enough (namely np t 10 and n(1 p) t 10 ),
then X is approximately N np, np(1 p) .
Sample Proportions
p^ x n
Mean
Ep^ P p^ p
Standard Deviation
s.d.( p^ ) V p^
p(1 p) n
Sampling Distribution of p^
If the sample size n is large enough (namely, np t 10 and n(1 p) t 10 )
then p^ is approximately N???? p,
p(1 n
p)
????
.
Sample Means
Mean E X
P X
P
Standard Deviation
s.d.( X )
V X
V n
Sampling Distribution of X If X has the N (P,V ) distribution, then X is
N
P X
,V
X
N???? P,
V n
???? .
If X follows any distribution with mean P and standard deviation V and n is large,
then
X
is approximately
N???? P,
V n
????
.
This last result is Central Limit Theorem .
Page 1
Population Proportion
Parameter p
Statistic
p^
Standard Error
s.e.( p^ )
p^ (1 p^ ) n
Confidence Interval p^ r z*s.e.( p^ )
Conservative Confidence Interval
p^ r z* 2n
Large-Sample z-Test
z
p^ p0
p0 (1 p0 )
n
Sample Size
n
????
z* 2m
????
2
Two Population Proportions
Parameter Statistic
p1 p2 p^1 p^ 2
Standard Error
s.e.( p^1 p^ 2 )
p^1(1 p^1) p^ 2 (1 p^ 2 )
n1
n2
Confidence Interval
p^1 p^ 2 r z *s.e. p^1 p^ 2
Large-Sample z-Test
z
p^1 p^2
p^ (1
p^ )????
1 n1
1 n2
????
where p^
n1 p^1 n2 p^2 n1 n2
Population Mean
Parameter
P
Statistic
x
Standard Error
s.e.(x) s n
Confidence Interval x r t *s.e.(x)
df = n ? 1
Paired Confidence Interval d r t *s.e.(d )
One-Sample t-Test
t x P0 x P0 s.e.(x) s n
df = n ? 1 df = n ? 1
Paired t-Test t d 0 s.e.(d )
d sd n
df = n ? 1
Two Population Means
General
Parameter
P1 P2
Statistic
x1 x2
Standard Error
Pooled
Parameter
P1 P2
Statistic
x1 x2
Standard Error
s.e.x1 x2
s12 s22 n1 n2
pooled s.e.x1 x2
sp
1 1 n1 n2
where s p
(n1
1) s12
(n2
1)
s
2 2
n1 n2 2
Confidence Interval
x1 x2 r t*s.e.(x1 x2 )
Two-Sample t-Test
t
x1 x2 0 s.e.(x1 x2 )
x1 x2 s12 s22 n1 n2
df = min(n1 1, n2 1) df = min(n1 1, n2 1)
Confidence Interval
x1 x2 r t* pooled s.e.(x1 x2 )
Pooled Two-Sample t-Test
t
x1 x2 0 pooled s.e.(x1 x2 )
x1 x2
sp
11 n1 n2
df = n1 n2 2 df = n1 n2 2
One-Way ANOVA
? SS Groups = SSG = ni (xi x) 2 groups
MS
Groups
=
MSG
=
SSG k 1
ANOVA Table
? SS Error = SSE = (ni 1) si 2 groups
? SS Total = SSTO = xij x 2
MS
Error
=
MSE
=
s
2 p
MS Groups F
SSE N k
Source SS
DF MS
F
Groups SS Groups k ? 1 MS Groups F Error SS Error N ? k MS Error
Total
SSTO
N ? 1
values
MS Error
Confidence Interval
xi r t* s p ni
df = N ? k
Under H0, the F statistic follows an F(k ? 1, N ? k) distribution.
Page 2
Linear Regression Model
Population Version:
Mean:
PY x E(Y ) E 0 E1x
Individual: yi E 0 E1 xi H i
where H i is N (0,V )
Sample Version:
Mean:
y^ b0 b1x
Individual: yi b0 b1xi ei
Parameter Estimators
b1
S XY S XX
? x xy y ? x x2
?x xy ? x x2
b0 y b1 x
Residuals e y y^ = observed y ? predicted y
Correlation and its square
r
S XY
S XX SYY
r 2 SSTO SSE SSREG
SSTO
SSTO
? where SSTO SYY
y y2
Regression
Standard Error of the Sample Slope
s.e.(b1 )
s
S XX
s
? x x2
Confidence Interval for E1 b1 r t*s.e.(b1)
df = n ? 2
t-Test for E1 To test H 0 : E1 0 t b1 0 s.e.(b1 )
df = n ? 2
MSREG or F
MSE
df = 1, n ? 2
Confidence Interval for the Mean Response
y^ r t *s.e.(fit)
df = n ? 2
where s.e.(fit) s 1 (x x) 2 n S XX
Prediction Interval for an Individual Response
y^ r t *s.e.(pred)
df = n ? 2
where s.e.(pred) s 2 s.e.(fit)2
Standard Error of the Sample Intercept
s.e.(b0 )
s 1 x2 n SXX
Confidence Interval for E 0 b0 r t*s.e.(b0 )
df = n ? 2
Estimate of V s MSE
? ? SSE
n2
where SSE
y y^ 2
e2
t-Test for E 0 To test H 0 : E 0 0 t b0 0 s.e.(b0 )
df = n ? 2
Chi-Square Tests
Test of Independence & Test of Homogeneity Test for Goodness of Fit
Expected Count E expected row totalu column total total n
Expected Count Ei expected npi0
Test Statistic
? & 2
O E2
E
df = (r ? 1)(c ? 1)
? (observed expected) 2 expected
Test Statistic
? & 2
O E2
E
df = k ? 1
? (observed expected) 2 expected
If Y follows a F 2 df distribution, then E(Y) = df and Var(Y) = 2(df).
Page 3
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used
with permission.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used
with permission.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition, 2012. Used
with permission.
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