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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 $x$

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  • $\begingroup$ Something is missing. What are the null and alternative hypotheses? When you write $x$, do you mean the sample mean $\bar x?$ $\endgroup$
    – Dave
    Aug 3, 2021 at 18:38
  • $\begingroup$ $x$ means any number. The problem doesn't states any specifically about the null and alternative H, it's only true or false. $\endgroup$
    – Chris
    Aug 3, 2021 at 19:02
  • $\begingroup$ (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? $\endgroup$
    – whuber
    Aug 3, 2021 at 19:15
  • $\begingroup$ 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 $\endgroup$ Aug 3, 2021 at 19:42

1 Answer 1

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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.

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