Fundamentals of Hypothesis Testing
Let’s take an example to understand the concept of Hypothesis Testing. A person is on trial for a criminal offense and the judge needs to provide a verdict on his case. Now, there are four possible combinations in such a case:
- First Case: The person is innocent and the judge identifies the person as innocent
- Second Case: The person is innocent and the judge identifies the person as guilty
- Third Case: The person is guilty and the judge identifies the person as innocent
- Fourth Case: The person is guilty and the judge identifies the person as guilty
As you can clearly see, there can be two types of error in the judgment – Type 1 error, when the verdict is against the person while he was innocent and Type 2 error, when the verdict is in favor of Person while he was guilty
According to the Presumption of Innocence, the person is considered innocent until proven guilty. That means the judge must find the evidence which convinces him “beyond a reasonable doubt”. This phenomenon of “Beyond a reasonable doubt” can be understood as Probability (Judge Decided Guilty | Person is Innocent) should be small.
The basic concepts of Hypothesis Testing are actually quite analogous to this situation.
We consider the Null Hypothesis to be true until we find strong evidence against it. Then. we accept the Alternate Hypothesis . We also determine the Significance Level (⍺ ) which can be understood as the Probability of (Judge Decided Guilty | Person is Innocent) in the previous example. Thus, if ⍺ is smaller, it will require more evidence to reject the Null Hypothesis. Don’t worry, we’ll cover all of this using a case study later.
Steps to Perform Hypothesis testing
There are four steps to perform Hypothesis Testing:
- Set the Hypothesis
- Set the Significance Level, Criteria for a decision
- Compute the test statistics
- Make a decision
Steps 1 to 3 are quite self-explanatory but on what basis can we make a decision in step 4? What does this p-value indicate?
We can understand this p-value as the measurement of the Defense Attorney’s argument. If the p-value is less than ⍺ , we reject the Null Hypothesis or if the p-value is greater than ⍺, we fail to reject the Null Hypothesis.
Critical Value, p-value
Let’s understand the logic of Hypothesis Testing with the graphical representation for Normal Distribution.
Typically, we set the Significance level at 10%, 5%, or 1%. If our test score lies in the Acceptance Zone we fail to reject the Null Hypothesis. If our test score lies in the critical zone, we reject the Null Hypothesis and accept the Alternate Hypothesis.
Critical Value is the cut off value between Acceptance Zone and Rejection Zone. We compare our test score to the critical value and if the test score is greater than the critical value, that means our test score lies in the Rejection Zone and we reject the Null Hypothesis. On the opposite side, if the test score is less than the Critical Value, that means the test score lies in the Acceptance Zone and we fail to reject the null Hypothesis.
But why do we need p-value when we can reject/accept hypotheses based on test scores and critical value?
p-value has the benefit that we only need one value to make a decision about the hypothesis. We don’t need to compute two different values like critical value and test scores. Another benefit of using p-value is that we can test at any desired level of significance by comparing this directly with the significance level.
This way we don’t need to compute test scores and critical value for each significance level. We can get the p-value and directly compare it with the significance level.
Directional Hypothesis
In the Directional Hypothesis, the null hypothesis is rejected if the test score is too large (for right-tailed and too small for left tailed). Thus, the rejection region for such a test consists of one part, which is right from the center.
Non-Directional Hypothesis
In a Non-Directional Hypothesis test, the Null Hypothesis is rejected if the test score is either too small or too large. Thus, the rejection region for such a test consists of two parts: one on the left and one on the right.
