Chi-square Distribution {?2 -Distribution}

Chi-square Distribution {?2 -Distribution}
l Chi-square: is defined as the sum 0f squares of independent
normally distributed variables with zero mean and unit variance.
l The mean and variance of chi-square distribution are
l .
l Goodness-Of-Fit Test:
l This test refers to the comparison between some observed sample
distribution and theoretical frequency distribution. This test is based
on the quantity
2
v
22
2
1
v
i 1
2
i
v i
i 1
2
i
2 Z Z ... Z
?
x ?
Z ? + + + = ? ÷ ÷
?
?
ç çè
? -
= ? =
= =
m = v and s 2 = 2v , v = df
( )
o observed e ected values
e
o e
i i
i
i i , where and exp
2
2 ? = =
-
c =
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Example: Calculation of ?2 goodness-of-fit of data consisting of 100
hair colors to a hypothesized color ratio of 3:1. Let a =0.05.
l Hair color
l Black Brown Total
l fi= Oi 84 16 100
l Fi= eI 75 25 100
l
l e1= 100*3/4=75 , e2 = 100*1/4 =25
l H0: The sample data came from a population having a 3:1 ratio of
black to brown hair.
l H1: The sample data came from a population not having a 3:1 ratio
of black to brown hair.
l Statistical decision: Reject H0, since Clinical decision: Conclude that,
on the bases of these data, the sample not having a ratio 3:1 of
black to brown hair. (P<0.05).
( ) ( )
4.320
25
16 25
75
84 75 2 2
2 =
-
+
-
c =
( ) ( ) 3.841
Fromprecentiles of the chi - sequare distribution table,
2
0.95 ,1
2
1 , = = - c c a v
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