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selection of tail test

selection of tail test

Super Learner
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Hi,
How to know whether we need to select one-sided tail test or two-sided tail test? If it is one-sided test how to choose whether it is left or right?

1 Answer

365 Team
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Hi Tess,
It actually depends on your problem:
I’ll use a coca-cola bottle example. Let’s take a coca-cola can, which in the US is 12oz (~355 ml). 

We are interested if the machine filling it up with liquid is broken. A similar machine (although not the one employed by the Coca-cola company) looks like this:

(a) A one-sided test would be one that tries to determine if the machine is pouring MORE than 12oz (~355ml). 
H0: Machine is pouring less or equal to 12oz (~355ml) -> status quo. The test is essentially trying to reject this statement.
H1: Machine is pouring more than 12oz (~355ml) -> what we are exploring as a problem. If we reject the null, then we are basically confirming our initial doubts. 
If we, in fact, reject the null, the can would be overfilled, and Coca-Cola would lose money, as they will be losing liquid for each fill. 
(b) A different one-sided test would be one that tries to determine if the machine is pouring LESS than 12oz (~355ml). If that is the case, Coca-Cola would be lying to their customers by selling less liquid than written on the bottle. 
H0: Machine is pouring more or equal to 12oz (~355ml) -> status quo. The test is essentially trying to reject this statement.
H1: Machine is pouring less than 12oz (~355ml) -> what we are exploring as a problem. If we reject the null, then we are basically confirming our initial doubts. 
Many companies put a piece of text on their products saying that the liquid inside may be +/- 1% from what is written. Usually, smaller companies with machines that are not as sophisticated as the Coca-Cola ones. They would “insure” themselves in that way against customers claiming the amount of liquid in the bottle does not match the label. 
(c) A two-sided test would be one that tries to determine if the machine is pouring EXACTLY 12 oz (~355ml). Going back to the previous case (b), if we cannot reject the null, then at some significance (e.g. 5%), we are sure that the machine is pouring more or equal to 12oz. But we don’t know if it is more than 12oz, or equal to 12oz. We just know it is NOT less.
One way to solve that is by performing both tests (a) and (b). If it doesn’t pour more and doesn’t pour less at the same time, then it is pouring exactly 12oz.
 
A two-sided test would give us this additional piece of information. It will tick-off both problems with a single test. 
There are two tests we can conduct here (in this part I’ll explore the difference in stating the hypotheses):
(1) 
H0: Machine is pouring exactly 12oz -> Logically, the status quo is that the machine is working fine, and is pouring exactly 12oz. The test is essentially trying to reject this statement. 
H1: Machine is NOT pouring exactly 12 oz -> what we are exploring as a problem. If we reject the null, then we are confirming the initial doubts that the machine is broken.  
If we reject the null hypothesis, we will be sure that there is a problem. So we have kind of conducted both one-sided tests: (a) and (b). Notice, however, that we don’t know if it is pouring more or less. We just know it is NOT exactly 12oz. 
Another situation is again two-sided but with reversed hypotheses:
(2)
H0: Machine is NOT pouring exactly 12oz -> We assume the machine is working fine, so the status quo is it is that it is NOT working fine, and is NOT pouring exactly 12oz. The test is essentially trying to reject this statement. 
H1: Machine is pouring exactly 12 oz -> what we are exploring as a problem. If we reject the null, then we are confirming the idea that the machine is working fine.
As you can see, in this case, we can form the hypotheses both ways. They will give us different errors (Type I error or false positive, and type II error or false negative). It often makes sense to compare the two errors and see which one we are trying to avoid as the worse problem (please refer to the lecture on false positive / false negative).
When deciding between (1) and (2), generally we would choose (1). The status quo would be that the machine is working fine (H0) and you are trying to determine if it is broken (H1). That’s the logical way to approach the problem. You would use (2) if you are trying to prove the point it is pouring exactly 12oz. However, this does not make much sense to prove as the machine was designed to do so. There is no scientific value in this test.   
The scientific way to perform hypothesis testing is to take the status quo as the null, and try to bring the change/innovation through the alternative. So keep that in mind when testing.
Hope this helps!
Best,
The 365 Team