# Chi squared test of homogeneity

Suppose you want to test the null hypothesis:

H0: p1 = 0.65, p2 = 0.27, p3 = 0.01, p4 = 0.05, p5 = 0.02

to determine whether a population's distribution matches the proposed proportions. You acquire a random sample of 100 individuals. If the null hypothesis were true, what is the expected value of the test statistic?

I'm not sure I conceptually/intuitively understand how the chi-squared works. Can someone elaborate on why the answer is 4 here? If the null hypothesis is true, then the population does have proportions equal to 0.65, 0.27, 0.01, etc. So what does this tell us of the test statistic?

• If this is related to some subject, could you add the self-study tag please? Apr 22, 2013 at 1:12

Intuitively, if the sample proportions exactly matched the proposed proportions, $\chi^2$ would be 0. But there will be random fluctuation.
On repeated draws, the expected value of $\frac{(O-E)^2}{E}$ will be the degrees of freedom, which in this case is 4.