Question

A friend from your statistics study group, Jack, gave this observation during a study session:

“Assuming the null hypothesis is true, the expected value of every test statistic we have seen this semester, whether it is a z, t, or chi-square test statistic, is 0.”

Form an appropriate response for your study group regarding Jack's statement.
In your response, be sure to comment on the following:

1) For z tests about population proportions: what is the distribution and the corresponding expected value of a z test statistic under the null.
2) For t tests about population means: what is the distribution and the corresponding expected value of a t test statistic under the null.
3) For chi-square tests about categorical models: what is the distribution and the corresponding expected value of the chi-square test statistic under the null.

Answer 1

ANSWER::

1)

TS = (p^ - p)/sqrt(pq/n) follow standard normal distribution
E(TS | null hypothesis is true) = 0

2)
t= (Xbar - mu)/(s/sqrt(n))
assuming the null hypothesis
E(t) = 0

3)

X^2 = sum (Oi- Ei)^2/Ei

E(X^2) = 0 assuming null hypothesis is true

however this is not always case,
in one-way ANOVA
E(F) = 1 when null hypothesis is true

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