How Do You Determine Direction in Hypothesis Testing?

[theBEAST]

Homework Statement

For example in my notes in class the prof went over an example with the following null and alternative hypothesis:

I feel like you can switch them around but I am not too sure. I have been trying to figure out when to use greater than or less than for the null hypothesis but to no avail.

Does anyone know how you know when to use greater than or when to use less than for the null/alternative hypothesis?

 

[Mark44]

Homework Statement

For example in my notes in class the prof went over an example with the following null and alternative hypothesis:

I feel like you can switch them around but I am not too sure.

No, you can't switch them around. Usually the problem statement will give you and idea of what the alternate hypothesis (H₁ or H_a) is. In your problem it says to "test the claim that the mean breakdown voltage is less than 9 volts."
So your alternate hypothesis is H₁: $\mu < 9$
This forces the null hypothesis to be H₀: $\mu \ge 9$

I have been trying to figure out when to use greater than or less than for the null hypothesis but to no avail.

Does anyone know how you know when to use greater than or when to use less than for the null/alternative hypothesis?

 

[theBEAST]

No, you can't switch them around. Usually the problem statement will give you and idea of what the alternate hypothesis (H₁ or H_a) is. In your problem it says to "test the claim that the mean breakdown voltage is less than 9 volts."
So your alternate hypothesis is H₁: $\mu < 9$
This forces the null hypothesis to be H₀: $\mu \ge 9$

Oh okay that makes sense.

Also, in the solution we get that t_obs > t_(α=0.05), and it then says fail to reject H0. Does this mean that there is a good chance that the population mean is greater than 9?

If it was the case that t_obs < t_(α=0.05), then it would mean that the claim that the mean breakdown voltage is less than 9 volts is most likely true.

Am I right?

 

[Mark44]

Oh okay that makes sense.

Also, in the solution we get that t_obs > t_(α=0.05), and it then says fail to reject H0. Does this mean that there is a good chance that the population mean is greater than 9?

If the calculated value of t happened to be in the critical region (the region you show as shaded), we would reject the null hypothesis, which is the same as accepting the alternate hypothesis. Since the alternate hypothesis was $\mu < 9$, we would say with 95% confidence that the population mean was less than 9.

If the calculated t value was NOT in the critical region (t ≥ t_α=0.05), we would fail to reject the null hypothesis, which is equivalent to saying that we accept the null hypothesis.

If it was the case that t_obs < t_(α=0.05), then it would mean that the claim that the mean breakdown voltage is less than 9 volts is most likely true.

Am I right?

Yes.