SECTION 6.6 χ^2 and Goodness of Fit 407
Z =
N 1 −np
»
np(1−p)is approximately normally distributed with mean 0 and standard devi-
ation 1. This means that
Z^2 =
(N 1 −np)^2
np(1−p)has approximately theχ^2 distribution with one degree of freedom, com-
pleting the argument in this very special case.
Example 1. Let’s flesh out the above in a very simple hypothesis-
testing context. That is, suppose that someone hands you a coin and
tells you that it is a fair coin. This leads you to test the hypotheses
H 0 :p= 1/2 against the alternative Ha:p 6 = 1/ 2 ,wherepis the probability that the coins lands on heads on any given
toss.
To test this you might then toss the coin 100 times. Under the null
hypothesis, we would have E(nH) = 50,where nH is the number of
heads (the random variable in this situation) observed in 100 tosses.
Assume that as a result of these 100 tosses, we getnH = 60, and so
nT = 40, where, obviously,nT is the number of tails. We plug into the
aboveχ^2 statistic, obtaining
χ^2 =(60−50)^2
50
+
(40−50)^2
50
= 2 + 2 = 4.
So what is theP-value of this result? As usual, this is the probability
P(χ^2 ≥4) which is the area under theχ^2 -density curve forx≥4: