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 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  1s   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

p1  p n

where

x p n

p1  p 2

ˆ  

z

ˆ

ˆ

p1  p p 1  p  n1 n2 ˆ

ˆ

ˆ

ˆ

, where p  ˆ

x1  x 2 n1  n 2

2

χ 

*Clopper-Pearson (nN)

* 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  E2 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  nY   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  E2 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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