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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