# Non-Parametric Hypothesis Testing Prof. Richard B. Goldstein

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```					                           Non-Parametric Hypothesis Testing

Prof. Richard B. Goldstein
single mean         r − 0.5            #of plus signs
z=         where r =
Sign test               or               0.25                   n                  0.63
two paired means          n
n1 = smaller sample
n2 = larger sample
n1 ( n1 + n2 + 1)
µR =                                      0.95
Mann-Whitney test                                             2
or             two independent                n1n2 ( n1 + n2 + 1)          n1, n2 at least 8
Wilcoxin               means              σR =
12
two-sample test                                                                  Otherwise, use
R− µR
z=                                      tables
σR
R = sum of the ranks
of the smaller sample
12       k
ri2
H=             ∑ − 3( n + 1)
n( n + 1) i = 1 ni
is approx. χ 2 with k − 1 d. f .
Kruskal-Wallis test     k ≥ 2 means                                               All ni at least 5
ri = sum of ranks of i th sample
ni = i th sample size
n = n1 + n2 + L + n k
V = number of runs of odds/evens
or any two similar groups

2n1n2
µV =             +1
n1 + n2
Runs test          randomness
( 2n1n2 )( 2n1n2 − n1 − n2 )
σV =
(n1      + n2 ) ( n1 + n2 + 1)
2

V− µV
z=
σV
n                                                     0.91
6∑ (d i )
2

rS = 1 −       i=1                                              If H0: no correlation,
Spearman Rank                                                          n( n2 − 1)                                              then for large n
correlation
Correlation                                               where
rs − 0
d i = difference of rankings                                    z=          = rs n − 1
1
of n pairs of data
n− 1
+          i               −                            i − 1
D n = max  − F( x ( i ) , θ), D n = max F( x ( i ) , θ) −      
1≤i ≤ n n                                                n 
                       1≤i ≤ n

Kolmogorov-Smirnov test goodness of fit               (   + −
D n = max D n , D n      )
{              }                                             (                   )
∞
P n D n > t → K ( t ) = 2∑ (−1) k −1 e − 2 k
2 2           2        2
t
= 2 e − 2 t − e −8 t + L
k =1

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