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GIVEN INFORMATION
1- Proportion Z-Test
x
pˆ 
n
2- Proportion Z-Test
x
pˆ 1  1
n1
pˆ 2 
x2
n2
GIVEN INFORMATION
 2 Goodness of Fit Test
Observed counts
 2 Test of Independence
2-way table
CONDITIONS
 assume SRS
 10n < population
 np 0  10
 n(1  p0 )  10
 assume SRS
 independent samples
 10n1 < population
 10n2 < population
 n1 p c  5 n1 (1  pc )  5
n2 pc  5 n2 (1  pc )  5
pc 
H0
TEST STATISTIC
z
H0: p  p 0
H0: p1  p2
pˆ  p 0
Use p̂ instead of p 0 for conditions
p 0 (1  p 0 )
n
( pˆ 1  pˆ 2 )  0
z
p c (1  p c )
1
1

n1 n 2
x1  x 2
n1  n 2
CONDITIONS
 assume SRS
 all expected counts >1
 no more than 20% of expected
counts < 5
SHOW EXPECTED
COUNTS
 same conditions as  2 GOF
Expected Counts =
(row _ total)(column _ total)
grand _ total
CONFIDENCE INTERVAL
pˆ  z *
pˆ (1  pˆ )
n
Use p̂1 , p̂ 2 instead of pc for conditions
( pˆ 1  pˆ 2 )  z *
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
H0
TEST STATISTIC
H0: The sample distribution is the same as
the population distribution
(O  E ) 2
E
df = (# proportions – 1)
H0: The two variables are independent
or
H0: There is no association between the two
variables
2 
(O  E ) 2
E
df = (# rows – 1)(# columns – 1)
2 
 independent samples
 same conditions as  2 GOF
 2 Test of Homogeneity
2-way table
Expected Counts =
(row _ total)(column _ total)
grand _ total
H0: The distributions are the same for all
populations (homogeneity of populations)
(O  E ) 2
E
df = (# rows – 1)(# columns – 1)
2 
GIVEN INFORMATION
1- Sample T-Test
x , s, n
Matched Pairs T-Test
xD , s D , n
(calculate paired
differences 1st)
2- Sample T-Test
x1 , s1 , n1
x 2 , s 2 , n2
Linear Regression T-Test
yˆ  a  bx
* Computer Printout
SEb is given
CONDITIONS
 assume SRS
 10n < population
 for n  30 t-distribution valid
 for n  30 check data for
skewness or outliers
- make a normal
probability plot, if linear
then data is ≈ normal
or
- make a boxplot & assess
 independent samples
 same conditions as 1-sample ttest for BOTH SAMPLES
 SRS
 form of relationship is linear
- scatterplot linear
 errors independent
- residual plot
 variability of errors is constant
- residual plot
 errors follow normal model
- histogram of residuals
or
- normal probability plot
of residuals
H0
H0:    0
TEST STATISTIC
t
x  0
df = n - 1
s
n
H0:  D  0
t
xD  0
df = n - 1
sD
n
t
H0: 1   2
H0:   0
Standard Error (SE): the estimate of the standard deviation
Margin of Error: the ± portion of a confidence interval, or half of the
interval’s width
±(critical value)(SE)
CONFIDENCE INTERVAL
 s 
 df = n - 1
x  t * 
 n
s 
x D  t *  D  df = n - 1
 n
( x1  x 2 )  0
2
2
s1
s
 2
n1
n2
df = from calculator
b0
SE b
df = n – 2
t
2
2
s1
s
 2
n1
n2
df = from calculator
( x1  x 2 )  t *
b  t * ( SE b )
df = n – 2
Type I Error: rejecting a true null hypothesis
α = probability of Type I Error
Type II Error: not rejecting a false null hypothesis
Power: the probability of correctly rejecting a false null hypothesis
- increase α, increase n (& decrease variation, extreme obs.)