Hypothesis Testing - Critical Value Method

# Critical Value Method

## Critical Region Method

Now that we know about Null and Alternate Hypothesis and how to choose our Null Hypothesis, let’s see how we can use the Critical Value Method to to make decisions about our Null Hypothesis.

Before we move on to that, keep the following clear in your mind:

We NEVER accept a Null Hypothesis.
We either Reject it, meaning we found sufficient evidence to say that the Null Hypothesis is not good.
Or,
We FAIL to reject it, meaning we could not find sufficient evidence to reject the Null Hypothesis.

Note: The second part does not mean that we are accepting the Null Hypothesis. It only says that we could not find sufficient evidence to reject it.
Lack of evidence to reject something is not equivalent to accepting it.

So, how do we decide whether to reject or fail to reject the Null Hypothesis?
Let’s take the same example that we took in our discussion on inferential statistics in this earlier post.

We had a plant that was manufacturing tablets. (1 lakh a day) and so in 30 days we had 30 Lakh tablets. Each of these tablets has a chemical called X.
Now, we make a claim that the average amount of chemical X in these 30 Lakh tablets is equal to 9.8 mg.

So, our Null Hypothesis is Ho = 9.8 mg
And Alternate Hypothesis is H1 not = 9.8 mg

Since we can go wrong in both directions (underestimating and over estimating), this would be a two tailed test.
Now, we need to find a cut-off value on both sides. Beyond this cutoff point we would reject the null Hypothesis.
The upper cutoff is known as Upper Critical Value and the lower cutoff is known as the Lower Critical Value.

How to find these Critical Values?

So we take a sample of let’s say 100 tablets and find its mean to be 10.3 and Standard Deviation as 2.1.
Following is the formula that is used for calculating the Critical Values:

UCV = U + Zc * Sigma / Sqrt(n)
LCV = U - Zc * Sigma / Sqrt(n)

Where, U is the Claimed Mean (in our Null Hypothesis)
Zc is the Z score associated with the given level of significance (we will see later how to calculate Z score in this post )
Sigma is the Standard Deviation of the Sample, and
N is the Sample SIze

We know the value for U (9.8) from our Null Hypothesis,
The value for Sigma (2.1) from the Sample we took,
The value of n (100) from from the sample we took.
Only Zc is what we do not know yet.
For now, let’s take it as 2.17 ( In this next post we will discuss on how to calculate this value).

So, putting these values in the formula we have:

UCV = 9.8 + 2.17 * 2.1 / SqrRoot(100)
= 9.8 + 0.45 = 10.25
LCV = 9.8 - 2.17 * 2.1 / Root(100)
= 9.8 - 0.45 = 9.35

So, our acceptable region is from 9.35mg to 10.25 mg (LCV to UCV).

However, our sample mean was 10.3 mg (mean of the 100 samples we took).
Since this mean (10.3) lies outside our acceptable region (9.35 to 10.25), our decision would be to Reject the Null Hypothesis.

Meaning, from the sample we took we found sufficient evidence to say that the claim of 9.8mg for the entire population was not good.

This is the Critical Value Method of making decisions on the Hypothesis and hence known as Hypothesis testing.

Only thing we missed was the calculation of Zc.
In the next post we will discuss on how to calculate the value of Zc