Rejection region of hypothesis testing

[songoku]

TL;DR

Let say I want to do one tail hypothesis testing using z - test with significance level of 10%. The null hypothesis is μ = 100 and alternative hypothesis is μ < 100. What will be the rejection region ?

From z-table, I get the critical value of z is -1.282

Will the rejection region be z < -1.282 or z ≤ -1.282? If I calculate the test statistics and it has value of -1.282, would I reject or accept null hypothesis?

Thanks

 

[FactChecker]

The odds of that are so small that you can really go either way. You should consider that the evidence of the alternative hypothesis is as weak as a 10% can get.

 

[Mark44]

Summary:: Let say I want to do one tail hypothesis testing using z - test with significance level of 10%. The null hypothesis is μ = 100 and alternative hypothesis is μ < 100. What will be the rejection region ?

Will the rejection region be z < -1.282

This one, since your null hypothesis is μ = 100. Really, though, the null hypothesis should be $\mu \ge 100$, to accommodate samples with a mean larger than 100.

As already noted, it makes no difference practically to have z < -1.282 or $z \le -1.282$ to determine whether to reject the null hypothesis.

 

[FactChecker]

There are several levels at which to set the test: 10%, 5%, 2.5%, all the way down to $5*\sigma = 3*10^{-7}$ for discovering a new particle. The reason that the acceptance levels are set at those levels is to minimize the risk of a type I error (accepting the alternative hypothesis when it is false) or to convince a skeptical audience. You can do whatever you want with a result of $z=-1.282$, but you should keep in mind the consequence of an error or how a skeptical audience would react.

 

[Dale]

Summary:: Let say I want to do one tail hypothesis testing using z - test with significance level of 10%. The null hypothesis is μ = 100 and alternative hypothesis is μ < 100. What will be the rejection region ?

Will the rejection region be z < -1.282 or z ≤ -1.282?

It doesn’t matter. The difference in the probabilities between those two possibilities is 0.

 

[songoku]

Ok so it means that if the test statistics is the same as critical value, then rejecting or accepting null hypothesis should include other factor such as type I error or maybe some other logical reasoning or the experiment needs to be repeated to have more certain data

Thank you very much FactChecker, Mark44, Dale

 

[BWV]

It doesn’t matter. The difference in the probabilities between those two possibilities is 0.

While this is absolute correct mathematically, practically there is some cutoff - if you are using Excel floating point the 90% cutoff is

1.2815515655446​

So at that precision it practically does not matter, however if you are just going to 3 decimal places, 1.282 is slighly more than the 90% percentile while 1.281 is closer