Z-Test: P-Value Method

In this unit we would see like critical value method, which are the other methods by using which we can create the decision of hypothesis.

From that very important and widely used method in the industry is p value method.

Before starting this module, I just want to tell that we will take the same examples, which means the AC sales problem and average lifecycle of a product, through both these examples, we will understand p value method.

In that we already have more than half the knowledge that what will be null hypothesis or what will be the alternate hypothesis and we will see if in the place of critical value method, if I use p value method.

So, will my answer be different or it will be same.

With this you can also create the difference that the p value method can be easily used or critical value method.

 So, come let's start understanding the p value method in our AC sales problem example.

So, if you would remember, in our AC sales problem, the null hypothesis which was there, what was it simply? It was simply mew was my 350 units, which means if I say that in my every store, every month 350 units AC are sold.

So, this would be my null hypothesis and what would be alternate hypothesis.

That mew is not equal to 350.

Now, after normally creating null and alternate hypothesis.

What is the next important part? That we have to create one decision and to create that decision what we will do first is we will find one Z value and we will find a Z value for a given sample mean.

So, I will remind you that our sample mean was 370.16, population mean was 350.

Population’s standard deviation with us was 90, from that we had found a standard error, which means sample’s standard deviation was found.

Which was sigma upon under root n, which means, 90 upon under root 36 if you do.

Then you will get sigma x bar which is 50.

So, all these things we have already seen in the previous model and in this also we have revised that how we have calculated different values.

One very important difference in the p value method and critical value method.

It is that even in critical value method we were calculating one Z score, but the ZC we were calculating it for critical points, which means for UCV and LCV we calculate our Z score then we use critical value method but if we calculate Z score in a normal way of the sample mean then that would be called my P value method.

So, come let's see, how do we calculate Z score for P value? Now, simply you must be remembering Z score’s formula.

So, my Z score’s formula that is there, in that we have put our sample mean’s value, 370.16 minus my population mean which is mew X bar, that we have put as 350 and we divided it by sigma X bar, which means 50.

So, I directly get value of Z directly which is 1.34, if you see in that example, in critical value one.

Instead of simply getting its value we were calculating UCVs value, we were subtracting it form critical regions.

But here we normally apply a Z scores formula and we find our Z’s value.

Now, when we will read the Z table, that will be in reverse.

Which means in his method we directly got a Z score which is 1.34.

Corresponding to 1.34, my whichever value comes, that will be my cumulative probability, any values below it will have this probability.

 Now if I read the values in it, in 1.3 horizontal line and corresponding to 0.04 vertical line.

Whatever is my value, that will give me one cumulative probability that Z is less than equal to 1.34.

Why? Because we find cumulative probability that its sum should be 1 but below a particular value, the area which is there under the curve that will be my cumulative probability.

If the Z’s value was 1.34 then the probability of Z less than equal to 1.34 will be 0.9099, which we can find with the Z table.

 In

the first step we calculated the Z score, we calculated its cumulative

probability.

Next, now you must be thinking for so long that you are reading the P value method, what is P value? P value is very widely used, we call it one probability.

Which means that my null hypothesis will not be rejected.

Which means we find such a probability where our main motive is that our null hypothesis should not be rejected.

So, this means that if your P value is higher, that means if your P value is high, then whatever is your observed data point that will lie along with the population mean.

And how much ever it will lie along with the population mean, it will go that much in the acceptance area.

And we will get higher chances that we cannot reject null hypothesis.

So, this means that if your p value is higher, so there are chances that we fail to reject our null hypothesis.

But if your p value is less, than that area will get further away from the population mean.

And it will start to lie in the critical region.

We had seen in the previous module that whichever value lies in the critical region, we reject null hypothesis in it.

So, if I simply see in this curve where I want to tell that from where do we exactly locate P value and from where.

So, you must be seeing one observed result, which is pointed with red.

If that observed result, basically we will call it sample mean when we will calculate our hypothesis.

How much more it will go towards my centre.

That much is the thing better for me because that will go in the acceptance region.

How much ever it will go towards the blue region, which means it will go to the right.

It is that much unlikely observation for me.

And I will reject my null hypothesis there.

So, we have seen what is the P value.

Now, how to calculate it, lets see that.

In this particular scenario, I have taken alpha as 0.05.

Which means, I'm considering 5% probability of committing an error, in this case, the p value that we are calculating that is one minus Z score.

So, here we have calculated a cumulative probability which was 0.9099 but this is my left side probability.

But I want to find the p value that it lies in the critical regions, which means that we will subtract 0.9099 from one and whatever value we'll get it we'll consider it the P value but there is a catch even here, it is that if your test is two tailed tests, whatever is the P value that comes, we will multiply it by two.

If it's a one tailed test, we will consider the value as it is.

Since it is my two tailed test, we will multiply p value with 2.

With this the value that comes is 0.1802, which means 18.02% Now, here we have to make one decision, this is our final step, what happened in this, my p value came as 0.18 and the Alpha value was 0.05.

18% is greater than 5%.

So, simply our value, our observed data point lies in the acceptance region.

And we can say that we fail to reject the null hypothesis.

So, in this way in different scenarios, we can use p value method which is widely accepted and is used easily.

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