# critical Z-statistic vs sample Z-statistic

A hypothesis that the population mean is less or equal to $$x$$ should be rejected when the critical Z-statistic is greater than the sample Z-statistic. $$x$$ is any number.

1. If $$x=5$$, the statement is true or false? The answer is false, but why?
2. For what values of $$x$$ the statement is true or false?

I know this is a one-tail hypothesis, but how do I know which tail are we talking about? Or is there some general rule that can answer 2)?

UPDATE: I will post the question completely:

Which of the following statements about hypothesis testing is most accurate?

A) A Type I error is rejecting the null hypothesis when it is true, and a Type II error is rejecting the alternative hypothesis when it is true.

B) A hypothesis that the population mean is less than or equal to 5 should be rejected when the critical Z-statistic is greater than the sample Z-statistic.

C) A hypothesized mean of 3, a sample mean of 6, and a standard error of the sampling means of 2 give a sample Z-statistic of 1.5.

The answer is C. I'm trying to understand when B would be true (first understanding when B is true for $$x=5$$ as in my original question, then understanding when it's true for any value of $$x$$

• Something is missing. What are the null and alternative hypotheses? When you write $x$, do you mean the sample mean $\bar x?$
– Dave
Aug 3, 2021 at 18:38
• $x$ means any number. The problem doesn't states any specifically about the null and alternative H, it's only true or false. Aug 3, 2021 at 19:02
• (1) and (2) make no sense: could you rephrase them? What "statement" are they trying to refer to? What is the meaning of "critical Z-statistic"? Is it some kind of statistic or is it trying to refer to a critical value for the test?
– whuber
Aug 3, 2021 at 19:15
• If we don't know the sample mean and variance, knowing $x$ will not help to answer the first question. It can be true and it can be false Aug 3, 2021 at 19:42

To make it clear we can give it some context. Let $$X_1,...,X_n\sim N(\mu,\sigma^2)$$ with $$\sigma^2$$ known. To test $$H_0: \mu \le 5$$ at level $$\alpha$$ we would compare the observed sample test statistic
$$z=\frac{\bar{x}-5}{\sigma/\sqrt{n}}$$
to the critical value $$z_{1-\alpha}$$. If $$z>z_{1-\alpha}$$ then $$H_0$$ is rejected. Otherwise, $$H_0$$ is not rejected.
To generalize we would write $$H_0: \mu \le \mu_0$$ and examine $$z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$. Nevertheless, if $$z>z_{1-\alpha}$$ then $$H_0$$ is rejected.
Option B) above is incorrect because it states that $$H_0$$ is rejected if $$z_{1-\alpha}>z$$. Let me know if I have understood your question correctly and if I have made any mistakes.