Z-Test: Critical Value Method

In this module, we will see the first type of test which we will call the Z test: critical value method.

In this particular entire module, we will see through different examples how we create a null and alternate hypothesis or after that any decision that we have to make, do we have to use p value test or t value test, how do we have to use different tests, all of those things we will see through the examples.

So, let's imagine that I am an AC store owner.

Which means I have many AC stores in the entire country and with us, every month in every store the AC units that are sold, which means that the number of AC that sells every month and let’s assume that as per the historic demands.

Which means as per the last records, we sell 350 AC units, every month, in every store.

But this time because of the heat wave.

We thought that may be my AC demand will increase, in such situations, where we already have the mean given.

Our assumed mean value is 350 AC per unit.

So, where we have population data given and we have to prove one assumption, So, in this situation we will consider hypothesis testing.

So, what is our assumption? Average units of every month will be different from 350 units, which was the ideal situation.

So, in this situation if we see its graph, distribution of AC units per month, per store, in the summer it would look something like this.

Where on the x axis I have my total sales and on the y axis I have my count.

350 is called our mean.

In this particular example, if I assume that my population standard deviation is 90.

So, we will see that population’s standard deviation and population’s mean is given with me.

If we believe that we have taken random 36 store’s data.

Which means we took the data of 36 stores, after storing that data we calculated its mean and after that we created its sampling distribution.

If you will see the sampling distribution, the knowledge that we have gained in the previous chapters, in this particular case where my n’s value is greater than 30, which means here our sample is of 36 stores, so my distribution will be equivalent to a normal distribution.

Let’s, suppose that the sample mean comes to 370.16 like we can see in this graph that 370.16 is my particular sample mean, so if you will see in this situation, my assumed mean which was there, 350 units per store, which was all the store’s average mean, my sample mean was different from that.

In such situations where sample mean and population mean is given with me, we will form our own hypothesis.

What would that be? First of all the hypothesis will be of this way.

My population mean is 350 which means there is no change in status quo.

This would be my null hypothesis.

And what will be the alternate hypothesis? That my mew’s value is not equivalent to 350.

To work in this case, where we involve the samples, there we generally we calculate the standard error.

Why? Because we have mean of the sample, with it we simply divide it by sample size, we can find our standard error or sampling distribution’s standard deviation.

Now, we have created null and alternate hypothesis.

In this situation if we would see.

Based on null and hypothesis, we can create two conclusions.

What are those? Let's imagine after doing the analysis my sample mean is greater than 350.

Let's see, if my sample mean is 600, in that situation we can reject our null hypothesis and accept alternate hypothesis.

Why? Because we don't have enough evidence that we will support H knot, so we will have to reject it.

Next, what would be the other situation? If my sample mean is closer to 350, which means my value is lying in the acceptance region.

In this section, we don’t have enough evidence to support an alternate hypothesis.

Either we can say that we fail to reject null hypothesis.

 So,

in which way we do this entire analysis, how we use this critical method, we

will see that now.

Now, you must be thinking in critical value method what exactly is a critical value? What is simply critical region? Any such region which lies on both the sides of the mean.

If you will see in this graph, the regions that are marked with red, that is called my critical region.

 Whichever Z score is associated with this, that is called my Z score or Zc.

Now for our sampling distribution we assume that suppose the probability that the error will happen, which means the one that we denote with an Alpha, we consider it to be 0.05.

So, I will assume this that there are chances that in my analysis, there will be an error of 5%.

It's such situation what are the things given with us? With us mew x is given which is the population mean, which is 350.

So, we had population mean given.

If I want to calculate sampling distribution mean, So, what will we do? The central limit theorem that we have learned.

We can use it’s property and simply say that my sampling distribution mean would be equivalent to population mean.

Which means it will be 350.

And sampling distribution’s standard deviation means, the standard error would be sigma upon under root n.

If I keep sigma’s value as 90 and n’s value as 36.

Then I will get a sampling distribution’s standard error which is 15.

We have

plotted the same standard deviation in this graph where 350 is my mean and from

there if we go 15 right or 15 left, then that is my standard deviation.

We have assumed alpha’s value as 0.05.

The values that you see on the both sides of the mean, which means our critical regions, associated to that is my Zc.

Which means critical value’s Z score.

If you pay attention here, my area under the curve in the critical region that we had assumed as alpha.

So, that value is 0.05.

And the acceptance region’s area became 0.95.

Which means if any value comes in acceptance region, then its chance of acceptance is 95%.

And if it comes to critical region than it means that it will go in 5% rejection region.

So, we

have seen that how we have created null hypothesis, how we have created

alternate hypothesis and what is our Zc.

Now we will come to, how critical value is exactly used.

So, we will follow some particular steps to reach one conclusion.

First of all we find the Zc’s value.

If you will see in this particular curve, our critical values, 0.05.

If it is a two tailed test, then that value is divided by two, which means if I see right areas value, that becomes 0.025, if I see the left area value it will also be 0.05 divided by 2, which means it will be 0.025.

So, by using this we will first calculate Zc’s value.

 What will we do to calculate Z c’s value.

What we will do is, the upper critical value that we have.

In the same curve you will see that we have 2 values.

One which we are calling as a critical value, which lies to the right and one LCV, which we call as lower critical value, which lies to the left.

So, in this curve, whichever situation we had, we know mew, we know the standard deviation.

First of all, we have to find that what would be my value of Zc.

To find that out, we will see UCV, which means at that particular end, what will be my upper critical value of Z? How will we find it? If you might remember, what does upper critical value gives us? It gives us cumulative probability of that particular point.

Cumulative probability means whose area under the code is 1.

Now, if I have the value Alpha is 0.025.

So, what will be my cumulative probability? 1 -0.025, which means 0.975 will be called as UCV’s normal cumulative probability.

You can also call it Zc.

You have found your Zc.

.

Now, you

can use the Z table, we will simply find its Z score.

If you might remember, to find the Z score, this table that we have is corresponding to 0.9750, whichever values are assigned to horizontal and vertical.

We call that our Z score.

So, if I see on this curve or on this graph.

Then horizontal 0.9750 lies 1.9.

 And if we touch the vertical line then 0.06 lies.

With this my Z score is 1.96.

Now we have got Z score’s value.

If you remember we had learned a confidence interval.

For which we had assigned a formula for upper critical value and lower critical value.

Critical value’s formula was mew plus, minus, Zc multiplied by sigma X bar.

So, in this formula we will put the mew’s value which is 350.

The Zc’s value which we had just calculated, 1.96, we will put that.

And sigma x bar which is my standard error is 15.

We will put that.

If I put both these values, so that value that I will get of UCV and LCV, that will create a confidence interval.

It will create one range, which lies from 320.6 to 379.4.

But the sample mean’s value that had come when we had initially calculated the value was 370.16.

So, if you notice that our value 370.16, that lies in acceptance region not in critical region.

In this

situation we can say that we cannot reject the null hypothesis, which means we

fail to reject the null hypothesis.

So, in this way by using different values, by calculating different Zc’s values, we use the critical value method.

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