Engineering Math

Hypothesis Test

I think Hypothesis Test has been one of the most confusing concept in Statistics. Probably I am still getting confused a little :). However, I decided to write down my current understandings as it is and keep updating it until this will be very clear to me and hopefully to you as well.

This is just beginning of the writing and will keep updating for pretty long time.

< What is Hypothesis ? >

Before you try anything for testing, you first have clear understanding on what you will test. Hypothesis Test is a method of tesing a Hypothesis. So first thing you need to do is to understand exactly what it mean by "Hypothesis".

I just copied two defintions by Googling as below.

1) A supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation.

2) In science, a hypothesis is an idea or explanation that you then test through study and experimentation. Outside science, a theory or guess can also be called a hypothesis

Do you agree with these definitions ? Is this same as what you thought ?

My feeling is "May be.. I agree.. but not exactly sure" because I've never thought of this definition myself.. I just thought I know the meaning of "Hypothesis".

Is this definition the one used in "Hypothesis Testing" ?  Maybe yes in broad sense. But let's make it in simpler/clearer form that is being used in "Hypothesis Testing" . It can be described as follows :

A Statement (a short sentense) that "you think" is true. i.e,a statement that you claim to be true.

As you see, an Hypothesis just a statement that you think is true, but may not be true in reality because it is not proved yet.

If you toss a coin 100 times, how many heads you would get ?

I THINK it is 50 times. Are you sure ?  Can you prove this ? or you just think like that because you have learned as such at school ?

It doesn't matter... If you think (claim) as such, it can be a Hypothesis.

In our class at school, we have 20 girls and 20 boys. What do you expect if they do math test ?

I THINK (Claim) boys would be better ? Are you sure ? Isn't it a kind of biased opinion (prejudice) ?

It doesn't matter... If you think (claim) as such, it can be a Hypothesis.

< Is Any Hypothesis be testable ? >

This is the most important thing and at the same time the most confusing thing to me.  Is Any Hypothesis that pops  up in my mind can be testable with "Hypothesis Test" method ?

The Answer is NO.

Hypothesis Test requires you to set Hypothesis in a specific way (format). What is the format ?

It requires to convert your claim into two Hypothesis in such a way that if one hypothesis become true, the other hypothesis automatically become false. The Hypothesis that you want to prove it to be true is called 'Ho' and the other one is called H1. It can be described as below.

Ho : A statement you claim to be true

H1 : A statement you calim to be false.

Ho Hypothesis is called "Null Hypothesis" and H1 Hypothesis is called "Alternative Hypothesis".

Let's do some practice. How about trying with "I think Boys' average math score is better than girls' in my class". How do we convert this into proper format for testing.

Case 1 :

Ho : Boys' average math score is better than girls'

H1 : Boys' average math score is Not better than girls' .

Case 2 :

Ho : Boys' average math score is same as girls'

H1 : Boys' average math score is Not same as girls' .

Which one is better form of Hypothesis ? In principle, both one are allowed but I think 'Case 2' is better.

I know you would ask "Why Case 2 is better ?". You will see later. This is very important to clearly understand why Case 2 is better.

Before we don't even start trying "Test", just try to make various kind of Hypothesis on your own. You may find a lot of examples from internet.

< Common Strategy in Hypothesis Testing >

Once you become familiar with setting Hyptotheis and now you have a set of Hypothesis (Ho, H1), the next step is to test it. I think there can be roughly two types of strategies that you case think of.

Case 1 : Try to prove Ho is true. If you are successful here, it would prove Ho is true and automatically H1 become false.

Case 2 : Try to prove Ho is false. If you are successful here, it would prove Ho come false and automatically H1 become true.

You can take whatever approaches depending on situation, but what you need to notice is "We always try with Ho (Null Hypothesis)".

< Steps to Hypothesis Testing >

There are roughly 4 steps to Hypothesis Testing as decribed below.

Step 1 : Make the Hypothesises

Step 2 : Deterime the criterial for decision. that is, set the criteria to prove or disprove Ho (Null) Hypothesis.

Step 3 : Compute the test statistics

Step 4 : Make a decision

Step 1 is explained in previous section and all other steps needs long explanation which will come later.