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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?

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

1 Answer 1

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Intuitively, if the sample proportions exactly matched the proposed proportions, $\chi^2$ would be 0. But there will be random fluctuation.

e.g.

set.seed(239920)
(testmult <- rmultinom(1, 100, c(0.65, 0.27, 0.01, 0.05, 0.02)))

does not yield exactly 65, 27, 1, 5, 2.

On repeated draws, the expected value of $\frac{(O-E)^2}{E}$ will be the degrees of freedom, which in this case is 4.

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    $\begingroup$ there is a typo in your statistic, the square is on the top, not bottom $\endgroup$
    – colinfang
    Apr 21, 2013 at 21:57

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