Making a Decision - Reject or Fail to Reject Null Hypothesis

Till now we have covered that how we create null and alternate hypothesis.

In this chapter we will see how we can create a decision using those hypotheses.

Either we reject the null hypothesis or we fail to reject null hypothesis.

So, to understand all these imagine that you have a friend who claims that his average score in cricket is 70, which means based on his old past games, he must have scored 70 runs or more than that in each of his innings.

So, he's saying that my score is 70 on an average.

So, this is our one claim.

This is the start of my hypothesis.

Now if you do not trust him, so you played few numbers of games with him.

Imagine that you played 5 games with that person.

After paying 5 games, suppose his average score came to 20.

So, obviously that is very less than 70.

So, you, believing him that his claim is right, you will not be trusting his claim.

But even this is possible that his average score comes out to be 65, so your chances of believing his claims will increase.

Now it is a simple thing, either his score come to 20 or 65 or any score in between that.

 So, which is the boundary on which I can believe, I can accept the claim, that particular decision that is called my critical point.

Which means such a point or such a boundary on which we believe that our claim is correct.

It can be between 20 and 65 or its score can be even less than 20 as well.

It is proved that his current performance is not that great or it can even be more than 65 if you play five games, so it means that your friend is under estimating himself.

In this situation, let's assume that our population mean is 70 which means my claim that my average score in cricket will be 70, that would be my population mean.

Now in this situation it is also possible that the 5 games that we have played, its score is 20 or it is 65 or any other number.

So, let us take that its sample mean is 50, which means my ex-bar’s value I have assumed to be 50.

But is it necessary that the sample mean is equivalent to my population mean? We cannot guarantee this, why? Because we have played the game only five times.

It is possible that if we play this game for more number of times than my sample mean is equal to population mean.

So, we have seen the entire scenario, where my population mean which means mew is 70 and my sample mean is 50.

So, this sample mean 50, that can lie inside the critical region or it can lie outside of the critical region.

So, basically the curve that we see on the left side, there we have got a region on the left which we call as lower critical point or we call it lower critical value.

And there is a region on the right which we call as critical region but since it is on the right side, we call it upper critical value.

UCB and LCV.

So, if my sample mean 50, comes in lower critical region, this means that I'm rejecting my null hypothesis.

Why? Because it was our null hypothesis that its value will come as 70.

But it was less than 70 so our null hypothesis got rejected.

The area apart from my critical region, we call it as acceptance region, which means the remaining area after left critical value and right critical value is called as the acceptance region.

Which means, let's imagine that my sample mean is 50 lies in the acceptance region.

So, this means that we are failing to reject our null hypothesis.

This way in different scenarios, if we compare our sample mean to the population mean, and we see that how our sample mean’s value lie in the critical region or in the acceptance region.

So, how do we make the decision? If my sample mean’s value lies in the critical region then we reject the null hypothesis.

But if my sample mean lies in the acceptance region, then we fail to reject the null hypothesis.

Like here with the blue area we have marked the critical region and the centre part is the acceptance region.

In this we fail to reject the null hypothesis and we reject right and left area.

So, we have seen that how we create our decision based on the region but using null and alternate hypothesis and by defining the critical regions position.

We can perform different types of tests.

Those tests we can generally simply denote on the basis of signs.

We divide those tests in three different parts, which means if we see the sign of our alternate hypothesis, either it is greater or smaller or it is not equal to, which means if in my H1 not equal to comes, then it will be called my two tailed test, which means my rejection region will lie on both the sides of the distribution.

This particular two tail test is used in many pharmaceutical companies.

Suppose if there is a medicine in which paracetamol is a very important component and if that medicine has to work exactly on a patient, then the ideal amount of paracetamol in it should be 200 MG.

If paracetamol’s amount is less than 200.

So, basically, this simply means that it will not treat our symptoms.

It means that those medicines are not very useful for us.

But if its amount is more than 200 MG then in that case, it is possible that the medicine creates any side effect, so in this way we define our two tailed test where our rejection region lies on both the sides.

If on my H1 or on alternate hypothesis less than symbol lies then it will be called as my lower tailed symbol, which means the rejection region or the critical region will lie to the left of the distribution.

If it is greater than symbol in alternate hypothesis then that would be called my upper tailed test.

Which means that the injection region will lie on the right side of the distribution.

So, if you see its example like in Maggi, the average lead content should be less than 2.5 ppm.

So, we can use here lower tailed test.

Why? Because I'm more focusing on the left side.

So, we will see this in a graphical presentation.

If my sign is of not equal to, then it will be my total taste.

Where my rejection regions will lie on both the sides.

If in my alternate hypothesis less than symbol comes, then this would be my lower tailed test where the rejection region will lie on the left side, like we can see in this graph, that my rejection region lies on the left side.

If in alternate hypothesis there is a greater than symbol in that case this would be called my upper tailed test and my rejection region will lie on the right of the curve.

So, in this way, we perform different types of tests and we get to know that how valid is our hypothesis testing and how we can use them in different scenarios.

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