Hypotheses, critical region, and p-value

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Problem

For a normal population with known standard deviation $5$ , a sample of size $25$ has mean $2$ . Test $H_0:\mu=0$ against $H_1:\mu>0$ at $\alpha=0.05$ . Give the $z$ statistic, p-value, critical region in terms of $\bar X$ , and decision. Use $z_{0.95}=1.645$ .

Reveal answer or reference solution

The standard error is $5/\sqrt{25}=1$ , so $z=(2-0)/1=2$ . The one-sided p-value is

1-\Phi(2)\approx0.0228

The level- $0.05$ critical region is $z>1.645$ , equivalently $\bar X>1.645$ . Since $2>1.645$ (and $0.0228<0.05$ ), reject $H_0$ .

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Hypotheses, critical region, and p-value

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