Formulas in Inferential Statistics
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FORMULAS IN INFERENTIAL STATISTICS
Number of Groups Being Compared
single
two (independent)
More than two (independent)
Two (dependent)
z-test for single mean
z-test for two means
Analysis of Variance (ANOVA): Single-Factor
t-test
x z
x 1
Parameter
z
σ
x
2
1
n
n 1
2
σ
F
2
2
mean squares columns (between) mean squares error (within)
n 2
MSC MSE
t
d df =
sd n
Three or more (dependent)
Analysis of Variance for Repeated Measures (ANOVARM)
n-1
t-test for single mean x t , df = n-1 s
( 1,
x2
x1
n1 1 n 2 1 n1 n 2 2 2 s1
n
Mean(s) 2, 3,…, k)
t
2 s2
1 1 n n 1 2
**Post hoc Analyses: Tukey’s HSD, Duncan Multiple Range Test (DMRT) Fisher’s LSD, Dunnett’s Test (if with control group), Bonferonni, Scheffe, Student-Newman Keuls (S-N-K)
, df = n1 + n2 – 2
t-test (with unequal variances) for two means
x2
x1
t
2
n1
2
2 2
1
, df =
2
2
2
2
1
2
s1
2
s s n n s s n n n 1 n 1 1
s2
2
2
1
n2
*Friedman Test
Analysis of Covariance (ANCOVA)
t-test (with equal variances) for two means
2
1
*Kruskal-Wallis Test
2
**Post hoc Analysis: Mann-Whitney (U)
(
Variance(s) 2 2 2 1, 2, 3,…, 2 k)
F test for two variances
Chi-square test for single variance 2
n 1s 2
2
F
, df = n-1
s12 s22
,
Bartlett’s Test
Chi-square test for single variance Within ( )
Chi-square test of homogeneity
*Mc Nemar’s Test
where s1 s2
*Levene’s Test
Proportion(s) (p1, p 2, p 3,…, p k)
z-test
z-test (n 30) z
p p ˆ
p1 p n
where
x p n
p1 p 2
ˆ
z
ˆ
ˆ
p1 p p 1 p n1 n2 ˆ
ˆ
ˆ
ˆ
, where p ˆ
x1 x 2 n1 n 2
2
χ
*Clopper-Pearson (nN)
* Non-parametric Test (does not assume normality/randomization and constancy of variance(s)) ** Post hoc analyses are applied if data provides sufficient evidence that the means or proportions across two or more groups have significant difference
O E2 E
, df = (rows – 1)(columns – 1) **Post hoc analysis: Marascuilo Test
Correlation Techniques (Tests of Dependence) Levels of Measurement
Interval/Ratio Pearson Correlation
Interval/Ratio
Ordinal
Nominal (non-dichotomous)
***Point-Biserial Correlation
*Spearman Correlation
*Spearman Correlation
Nominal (dichotomous)
Rank-Biserial Correlation Ordinal
*Spearman Correlation
*Spearman Correlation
Nominal (dichotomous)
n/a
n/a
***Point-Biserial Correlation
Nominal (dichotomous)
Chi-square test of Independence
Chi-square Test of Independence Chi-square test of Independence Chi-square Test of Independence
Chi-square test of Independence
Chi-square test of Independence
Pearson Correlation Phi coefficient ( )
* Non-parametric *** Correlation Technique derived from Pearson Correlation
XY X Y , n X X nY Y n
Pearson Correlation coefficient may be computed as r
2
2
2
2
where its test of significance may be computed using
z r n 1 for n 30 or t r n 2 ; df 1 r 2 Spearman Correlation coefficient may be computed as 1
2 for n < 30
n–
d
2
6
1 )
2
n(n
where its test of significance may be computed using t
r n 2
Chi-square test of Independence test statistic may be computed as
1 r 2
2
χ
,
; df
O E2 df = (rows – 1)(columns – 1) if all E 5 , where E Row i total Column j total . , ij ij E
Otherwise, collapse or remove rows/columns. If df = 1, Fisher’s exact test , where p
A B !C D ! A C ! B D ! A! B!C ! D! N !
Simple Regression Analysis Y = a + bX +
or
Y = β0 + β1X +
0 y 1 x , and 1 ˆ
ˆ
ˆ
n
; where
xy x y n x x 2
2
- Test of significance of 1 may be performed to determine if β1 = 0 ˆ
1 0 ˆ
t
, with df = n – 2
2
s y x
x x
2
2
n–
GrandTotal
CRITICAL VALUES
FOR CHI-SQUARE ( 2)TESTS:
FOR Z-TESTS:
Two-tailed test .01 .05 .10
FOR t-TESTS:
2.575
2.33 or –2.33
1.96
1.645 or – 1.645
1.645
1.28 or – 1.28
One-tailed test
FOR F TESTS:
= 0.05
= 0.01
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