Formula sheet
Inference about  when  2 known:
Test statistic:
z=
X −
 n
CI estimator:
z  
n =   /2 
  
Sample size to estimate  ±  :
X  z / 2

n
X  t / 2
s
n
2
Inference about  when  2 unknown:
t=
Test statistic:
 = n −1
X −
s n
CI estimator:
Inference about p (population proportion):
Test statistic:
z=
pˆ − p
p(1 − p) / n
Sample size to estimate p ±  :
Sample Size for Beta error:
CI estimator:
pˆ  z / 2 pˆ (1 − pˆ ) / n
z
pˆ (1 − pˆ ) 

n =   / 2



2
(
z + z B )  2
n=
( 0 −  a ) 2
Inference about (μ 1 – μ 2) when  12 =  22:
2
Test statistic:
t=
( X 1 − X 2 ) − ( 1 −  2 )
sP2 sP2
+
n1 n2
where
(n1 − 1) s12 + (n2 − 1) s22
s =
n1 + n2 − 2
2
P
CI estimator:
sP2 sP2
+
n1 n2
( X 1 − X 2 )  t / 2
DF = n1 + n2 − 2
Inference about (μ 1 – μ 2) when  12 ≠  22:
Test statistic:
t=
( X 1 − X 2 ) − ( 1 −  2 )
s12
n1
+
s12 s22
+
n1 n2
( X 1 − X 2 )  t / 2
s22
n2
CI estimator:
(s n + s n )
DF =
(s n ) + (s n )
2
1
2
1
1
2
1
n1 − 1
2
2
2
2
2
2
2
2
n2 − 1
Inference about p1 – p2:
Test statistic:
[H0: (p1 – p2) = 0]
CI estimator:
z=
( pˆ1 − pˆ 2 ) − ( p1 − p2 )
1 1
pˆ (1 − pˆ ) + 
 n1 n2 
( pˆ1 − pˆ 2 )  z / 2
pˆ1 (1 − pˆ1 ) pˆ 2 (1 − pˆ 2 )
+
n1
n2
Inference about 𝝈𝟐 :
Test statistics:
[H0: 𝝈𝟐 = 𝝈𝟐𝟎 ]
𝝌𝟐𝒏−𝟏 =
(𝒏−𝟏)𝒔𝟐
𝝈𝟐
Page 2 of 4
CI estimator:
(𝒏−𝟏)𝑺𝟐
(𝒏−𝟏)𝑺𝟐
< 𝝈𝟐 < 𝝌𝟐
𝝌𝟐𝜶/𝟐
𝜶
(𝟏− )
𝟐
𝝈𝟐
Inference about 𝟏⁄ 𝟐 :
𝝈𝟐
Test statistics:
[H0: 𝝈𝟐𝟏 = 𝝈𝟐𝟐 ]
𝑺𝟐
𝑭𝒏𝟏 −𝟏,𝒏𝟐−𝟏 = 𝑺𝟏𝟐
𝟐
Simple Linear Regression
Shortcuts:
 (x − X )
n
s =
2
2
i
i =1
n −1
1 n 2
2
=
x
−
n
X

i

n − 1  i =1

1 n

s XY =
x
y
−
n
X
Y
ii

n − 1  i =1

yi =  + xi +  i
SIMPLE REGRESSION:
yˆ = a + bx
Least squares line / linear regression line:
Standard error of estimate:
estimate:
s=
SSE
n−2
s2 =
b=
S xy
a = y − bx
s x2
Standard error of least squares slope
SSE
n−2
sb =
s
(n − 1) s x2
Test statistic & confidence interval estimator for β:
t=
b−
sb
b  t / 2 sb
DF = n − 2
Prediction Interval for y for a given value of x (xg):
value of y given x (xg):
2
1 ( xg − X )
yˆ  t / 2 s 1 + +
n (n − 1) s x2
Confidence Interval for expected
2
1 ( xg − X )
yˆ  t / 2 s
+
n (n − 1) s x2
Simple & Multiple Regression
n
SST =  ( yi − y )
2
i =1
SST = SSR + SSE
n
SSR =  ( yˆ i − y )
i =1
2
n
n
SSE =  e =  ( y − yˆ i i ) 2
i =1
2
i
i =1
Page 3 of 4
Coefficient of Determination:
R2 = 1 − n
R2 =
SSR
SST
R2 = 1 −
SSE
SST
SSE
 ( yi − y ) 2
i =1
MULTIPLE REGRESSION:
yi =  + 1 x1i +  2 x2i +  +  k xki +  i
Standard error of estimate:
n
 (ei − 0) 2
SSE
s 2 = i =1
=
n − k −1
n − k −1
SSE (n − k − 1)
Adj. R 2 = 1 −
2
 ( yi − y ) (n − 1)
SSE
n − k −1
s=
Test statistic & confidence interval estimator for βj:
t=
bj −  j
sb
b j  t / 2 s b
j
v = n − k −1
j
Test of overall statistical significance of linear regression model:
2
F=
( SST − SSE ) / k
R /k
F
=
SSE /( n − k − 1)
(1 − R 2 ) /( n − k − 1)
SSR / k
F=
SSE /( n − k − 1)
Numerator degrees of freedom:
k
Denominator degrees of freedom:
n − k −1
Page 4 of 4