A discrete test and its actual significance

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Before you start

A one-sided binomial critical region. Read the relevant method, then decide which parts you can solve without hints.

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Read an original example question

Under $H_{0}$ , X ∼ Bin(4, $\frac{1}{2}$ ). Test $H_{1}$ : p > $\frac{1}{2}$ at significance at most 10%.

Question data
x	0	1	2	3	4
P(X=x)	1/16	1/4	3/8	1/4	1/16

Part a

Find the smallest upper-tail rejection threshold c.

Part b

State the actual significance probability.

Part c

For an observation X = 4, state the conclusion in context.

Model solution and review criteria

Part a

Sum upper-tail masses. P(X≥4)= $\frac{1}{16}$ ≤ $\frac{1}{10}$ ; lowering c would exceed $\frac{1}{10}$ .

Select the upper tail for p> $\frac{1}{2}$ .
Check the adjacent threshold.

Part b

The actual null rejection probability is $\frac{1}{16}$ , not automatically $\frac{1}{10}$ .

Use the discrete tail at the chosen threshold.

Part c

Reject $H_{0}$ at this rule’s significance level. There is evidence of p> $\frac{1}{2}$ ; rejection does not prove the alternative or assign a probability that $H_{0}$ is true.

Use the rejection boundary including equality.
State evidence rather than certainty.

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