1. One sided test

$H_0: \mu <= \mu_{0}$

$H_1 : \mu > \mu_{0}$

 \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}, \end{align}

If the null hypothesis is true we expect the sample mean to be relatively small.

I don't understand why it would expect it to be small. There's no restriction on $\mu_{0}$ that I can see.

2. Originally Posted by IvanCarrie
$H_0: \mu <= \mu_{0}$

$H_1 : \mu > \mu_{0}$

 \begin{align}%\label{} W(X_1,X_2, \cdots,X_n)=\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}, \end{align}

If the null hypothesis is true we expect the sample mean to be relatively small.

I don't understand why it would expect it to be small. There's no restriction on $\mu_{0}$ that I can see.
I take it the first sentence is what your book says, and the second is your comment?

It would appear that "relatively small" means "relative to $\mu_0$". Is there any context that might confirm whether this is all they meant? How do they use that fact going forward?

3. Originally Posted by Dr.Peterson
I take it the first sentence is what your book says, and the second is your comment?

It would appear that "relatively small" means "relative to $\mu_0$". Is there any context that might confirm whether this is all they meant? How do they use that fact going forward?
Yes, the first is from the text and the second is my comment.

The full context is here

https://www.probabilitycourse.com/ch...g_for_mean.php

Example 8.28

Apart from the fact that X is a normal variable, I guess obvious from the fact it's being normalised, I can't see what other context there is. The fact is used to establish a threshold for a hypothesis test.

Spent quite a few hours now re-reading this section, but can't seem to see what it is I'm missing.

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