In this session, we will discuss about the and different views of the best.
So they have been using the test so that we could compare the population on a sample.
But in a few cases, they cannot use best.
So that does when.
Using Z is not always possible, as we would not have the population barometers always handy.
So in case we don't have any population barometers, so in those scenarios, we will not be able to
use the Zionist, hence the use of the abortion, which is students the disc.
We can use the deepest if something has size below 30.
And it has a population standard deviations.
For example.
We have an assumption that is the data is independent, it is collected randomly and the data is approximately
normally distributed.
I'm the interpretation of the issue is when when we find out that the value you finding out that the
value is just similar as the Z value, the formulas are almost similar and how we interpret the data
is slightly different.
So if the calculated the value in the is less than the value of the table, then if the value is less
than zero point zero five or zero point zero one, the level which is present at zero point zero five
0.01, then the null hypothesis is accepted.
It is just similar to the Z value.
So in the Z value, we had to provide the alpha level in the Z table while we were calculating the value
we originally had to provide all alpha level and based on the alpha level behind the Z value, which
was one point nine six.
In case of zero point zero five, but then piece of detail, we do not have any such alpha.
We have alpha level person that is in the basement.
So what we do is we find out the value which we want to compare and we check if the value is present
in the basement.
I'm in the they will become very from the levels of zero point zero point zero one.
If the value is less, then the value is present at zero point zero five or zero point zero one, then
the null hypothesis is accepted and increase the likelihood that the value is greater than that which
is present in the people at zero point zero five or 0.01.
Then the null hypothesis.
David A..
So let's go for that and see how this actually works.
So there are three types of B this one is the one somebody, just the one somebody.
This is used when we have a single group of data and we want to take this group against against one
when we have some some single group of data.
And we want to check this against an already known value from the population.
And that is the independence of graffitists, this independence, empathy, this is where we have two
independent groups that we have some data of flu samples and these samples belong to different categories,
except one category would be under the category could be Fehmi or one category would be students from
one university.
I'm under the guise of clinical research from the second university or one category could be the same
people from one city and under the category could be people from another city.
So the sample which is taken belongs to two different groups of people.
And if they are, this is when they are checking on the same group, but they are checking out different
conditions, we are taking a different time.
We will see they are checking out one year of difference.
So this is going to hurt somebody.
So now to see one question which would actually explain this, how this is working.
So here we have.
Again, the same Juden question.
So what do we have here is we have a study of the status of critically ill children.
So here again, we have 16 children with the permanency out of these children.
They have 12 of one point, the number of missing beat and the standard deviation is one point nine.
And the actual win is one point four, which is already pretty clear.
Now you can see the definition of when somebody's desk, which is the meaning of a single group against
a wall.
So this is one group which we have where we have 16 children.
It is found that the number of missing will be this one point and the standard deviation is one point
nine and the actual nine.
This already one mean single group against me, so this is the only one point four, so this is being
the only want to find out if they'll mean the same or different.
So how do we do this?
So the hypothesis is that the mean is actually equal to one point for.
And another hypothesis is that the line is not equal to one point for.
And that is the mean of the student are having this critical illness is one point for and here the is
not one point for.
So the data is and is equal to 60.
So that is NWC in the Sanford.
Now, the test statistic is so people estimate of population barometer deriving from this data.
So we have the median value, the average mean value at one point to.
And we are performing the one simple thing that's doing this so that the value will be.
Exper minus meal divided by.
Those standard errors.
In the Z formula, in the formula for Z value, the formula used to be exper minus new, divided by
the standard deviation of the population by here.
This is divided by the standard deviation for the sample, which is the standard error divided by the
number of sample, which is how we calculate the standard deviation.
So we are finding out one point two minus one point four divided by one point nine.
From this outfit, and this is the size of the sample in Swaroop, so we find out the value minus point
for the one.
Now, this is the P-value.
Now we need to check the P in the.
Now, how does the detainee rule in that be able to have these?
So this is one column which is having so numeric values and this has given us a degree of freedom.
What is degree of freedom we will look at just in a few minutes and this rule that Alpro contains the
probability.
That is what is the probability for having the confidence I'm here, we have the.
One day this time, to be honest, that is if they are taking in one direction or we are checking in
both directions.
So when we're talking about.
Meal is not equal to one point four.
This means that you can be less than one point for total and you can be greater than one point for all.
It can have any value.
There is no restriction.
That meal has to be greater than one point for only or has to be less than one point for only.
Here, meal can be any value in any direction.
So then you can have any protection.
This means then it can be a part of any of the two bills.
If we look at this blog.
And so if nobody value is the central idea, then be accepted null hypothesis, if it belongs to any
of these stupid idea, then it is a part of the rejection region.
So this is the data and this is being rejected.
So now the P value will be divided into two parts.
So initially, if we were taking in only one direction, let us see.
The condition would have been mbewe is greater than 40.
If the view is greater than for being this is an addiction criteria, then we will have only one because
we are not checking for the Lordi when we are seeing new is greater than 14, then we are not checking
in this direction because here we are sure that it is more or less than 40.
We are just checking new is greater than for being here.
So it didn't look like an addiction.
Gladia will be only one beat back.
The Godding hypothesis is that new is not equal to forty or one point four in the scenario.
So new is not equal to one point, which means that it has to be significantly far away from one point
for in any of the directions.
So now the problem is all these.
So then the problem would have been in one field, if the problem would have been in one single thing,
then we will have zero point five percent data here.
The critical region would have been zero point five in this area.
But now, because we are talking about both directions, so the zero point five will be split in two
border regions.
So this will have zero point to fight and this will have zero point zero two five.
So the zero point five will be divided into two.
And we will check for zero point two five level here and zero point zero five level here, zero point
zero zero five level here and zero point zero five here.
So this is the value.
So now we will check for.
Zero point five, so zero point five, in case of one day, if there were boobies, then we would have
checked for zero point one zero.
Similarly.
Here we have zero point zero to five in case of one day, and he will we have zero point zero five.
So in this particular case, we will be finding out the values accordingly.
So let us have a look at this.
So here the number of values that we have, the number of values in the sample that we have is 16.
Now, let us survive, this is prominent, so we have to find out the degree of freedom.
This particular column here refers to the degree of freedom.
So how do we find the degree of freedom?
The degree of freedom is the number of values that we can select independently with one restriction.
So when we are taking out the sample, because the mean has to be of value, which is in between, so
we even be selecting different values independently.
So let's say we have total 50 values, which you want to pick.
OK, then we can restrict, say, 16 values.
Then if we restrict this one value here, if then what?
Then how many values can we have?
We can select one to three values independently.
So out of this one, two, three and four values, we can select three values without any hesitation.
We can select any values as these three values.
By selecting this one value, we will have to make sure that the sum will be equal to four.
So this is what degree of freedom is.
So in case we have these 16 values which we are trying to pick out.
So out of these 16 children, we can pick 15 values independently.
But for the one last value, we will have to make sure that the mean comes out to be, which is simple,
which is as a mean of the population.
So here are the different ways how we can find out different degrees of freedom when we are performing
one somebody's test or bayati this, then the degree of freedom will be in minus one.
When we are working on independently deepest, then the degree of freedom will be in minus two.
When we are walking one guy square, then the degree of freedom will be number of rulers, minus one
in the number of columns, minus one.
When we are working with ANOVA, the degree of freedom between the groups will be number of levels or
groups minus one, and the degree of freedom within the group will be number of subject, minus the
number of levels, what these three are.
We will discuss in something.
But for now, you can remember only these two.
That is number four, that the vest itself.
So in one sample do this and 30 this we will use and minus one, I'm for independent because we will
use in minus two.
So here we have one completist.
So in one somebody this we have an equal to 60, so the degree of freedom will be 50 and the value of
the which we have found is minus zero point four to one.
So we will check for zero point for the one.
So the value zero point four to one lies between these two values, which is somewhat higher.
And it is victorious then this particular value at zero point zero five.
Or zero point zero one.
So here you can see that this value is way less than this value.
So it is anyhow a part of the distribution itself.
This means.
That.
This is just by chance that the methane is coming out to be one point to like the actual minus, one
point for only.
So we can easily accept this hypothesis.
So let us go further.
So you can see that this is these are this is the value, this is minus zero point for the one and this
is minus zero point for the one.
This is where minus zero point six nine one would be.
Zero point six nine one.
This is minus zero point six nine one, so it is way below the actual values which we have for critical
region.
The values are way below the values which are expected for the critical region.
So this is about the zero or one point two is actually a part of the same distribution, which is of
one point one one point for me.
So the hypothesis can be rejected.
So here are a few links for the calculator's, which we have used so we can use this table or this table,
and in case you want to calculate it mathematically and using the calculator without looking at the
people, then you can access the calculator.
I did find Lip-Sync.
Next, we will be looking at the independent sample test, so for the independent sample test, the
assumptions is that one assumption of independence, which means that we need two independent variables,
which is that there should be two different categories which should be represented.
That is something like male or female or another kind of categories could be people from two different
cities or people from two different countries being compared together.
So these are independent samples which are completely exclusive from each other.
So another assumption is assumption of normality, which is that the dependent variable should be approximately
normally distributed dependent, but it should also be measured on a continuous scale.
For example, the test score will be the dependent variable, which is being calculated, and it is
completely independent in nature and it is continuous value.
Now we have another example, so we might face two different groups of customer service associates on
a business related test or testing students from two different universities on their English skills.
So these are different examples of independence for this.
So let us try to solve a problem.
So here we have two datasets.
One is set and that is set to be.
So values for these sects are given to be one, two, two, three, three, four, four, five, five,
six.
And this sect is given to with these values.
Now, we want to find out if these belong to different populations or they belong to the same population
that is in the mean of these two are significantly different or significantly.
So how would we do that?
How would we perform the independent beedis?
So we will do the best by somebody, the values in the groups.
So I have added the values in the first group and the values in the second group, the sum of first
one coming out to be thirty five and the sum of the second one coming out to be 50.
Then we will square the sums from the first step, so the Four-Square is one to do five and second one
is twenty five under.
Now, here we are calculating the mean for the two groups, so the mean will be some of these values
divided by then, so 35 divided by then and four divided by 10.
So we will get them to the three point five and five.
Now we will square the individual scores and then add them up.
So this will be one forty five and ninety eight.
Now, we will insert the numbers in this particular formula.
So when we apply this formula, we have this summation of a square summation of the square in which
we have already calculated these are the means which we have already calculated, the summation of a
square and the sum of squares of data has already been calculated and the number of items has already
been given.
So we will insert all the values in this particular formula, which we have.
So this is a simply if you try to understand me here we have the B value, which is calculated by the
difference of the means.
And we are finding out the.
Squared of E divided by minus the summation of values divided by and so this is in the similar terms
of what we have in B, but it is a combination of both the values.
So it is slightly complex formula.
So.
But you don't need to remember this entire formula.
All you need to remember is the concept and the assumptions that we have that which do you need to apply
for it and then you will be directly applying the best from the calculator itself.
So you just need to have the values and know which test you need to apply while applying statistics.
So then we will be finding out the values and we have to find out the degrees of freedom.
So what would be the degrees of freedom?
We have this data set and we have this data set, so from the independent test.
We can see the degree of freedom is the doable in minus two.
So the total end will be then less than 20, minus two, which is 18, so the degree of freedom is 18.
So we will go to the detail and from the detail, we will select the alpha level and we will compare
with respect to the alpha level and for the degree of freedom of the alpha level is coming out to be
two point.
So when we compare the calculated value, which is minus one point six nine with the alpha level two
point one, we can see that the calculated value is less than the cutoff value, two point one zero.
Therefore, the P value is greater than zero point five.
And as the P value is greater than the alpha level, we cannot conclude that there is a difference between
the means.
So we cannot say that there is a significant difference between the mean values.
So these values can actually be about the same population.
So we cannot conclude that these are different scenarios if they can be a part of the same population.
Here you can see we have a from the table also in the table, the degree of freedom has been selected
as a beam and we come it to this devalue zero point zero to five.
So here we have.
Two point one zero, which is being compared with minus one point six nine.
So when we compare it with minus one point six, so minus one point six nine, which is somewhat higher
and the actual value is to win something.
So hence it will be in the same region in the middle of the region and it will not be a part of the
critical region.
So we cannot conclude we will have to.
See that all these samples are belonging to the same data.
Next is the pig, the test, so let us understand what this is.
Now we will choose the beer to decide if the two measurements on the same item or person or thing is
being them.
So if the measurement is being done on the same item, that is some a group of people from the same
city are being tested for two different things.
So we should also choose this test if the two items that are being measured with all unique conditions,
so that is a unique condition which is attached to this particular scenario.
So, for example, we might be mentioning the car safety performance in vehicle research and testing
a subject.
The cars do a series of crash test.
So we are applying different crash tests on different cars and we're testing them.
So the null hypothesis for this would be that the meaning of one is equal to the mean of two.
So in other words, it assumes that the means of equal.
So we will be comparing what the means and finding out if they're significantly different, if they
are significantly different than the means would be different and they would be a of two different populations.
So would the bear be the null hypothesis?
Is that the pairwise difference between the two, this is equal to the.
Newly equal to zero, so this is the difference between the independent, somebody based on the their
deepest that is an independent of somebody.
This we we're finding out that if both the means are equal, that is both the means of the two distributions
that we have are equal or not, if they're belonging to the same population or not.
But here what we will be finding out, we will be finding out that we have these two populations.
So for these two populations, that we have different scenarios.
Is there any difference or not?
So if the difference between both of these should be equal and different in the world of finds, these
are being calculated.
We are calculating same thing at two different intervals of time.
We want to prove that there is no impact of dying on this.
So the difference between these two is equal to Z.
This is what we are trying to work on in the field the best.
So here, let us try to calculate the BRT this, so here we have the steps forward, we have these two
schools given for the same subjects.
So the first step is to subtract each white school from the school.
So we will subtract this school from the initial school so we get minus seven B minus and minus 10.
So such that we will get these scores.
The next step is to add all the values from step one, so we add all the values from step one, the
next step is to square the differences.
So we square these differences and put these here and then find the some of these old.
So we find the differences, find the sum of differences, we find the squares of individual differences
and some of the squares of the individual differences.
After this, we will apply all the values which we have retrieved from the previously.
These sums and the squares of the south, we will put these in this particular formula.
I'm calculate dirty.
And again, using the disvalue, we will be able to calculate the value from the so here the number
of values are pointed to 11.
So there are 11 froster values.
So based on.
Fared the best it has to be in minus one.
So the degree of freedom would be in minus one.
So we will put this one in the formula that is 11 minus one and 11, and we will get the value as minus
two point seven four.
And again, we can compare it with the other values.
So minus two point seven for.
So four minus two point seven four we will have.
The value from the detail now using the degree of freedom, we will get the alpha level, so the degree
of freedom as then that the value is to point to do it now, we will compare it with the detailed value
so that they will value has a value to minus two point seventy four.
So they calculated the value is greater than the people value and the level of zero point zero five.
So the P value is less than the alpha value.
So we can reject the null hypothesis that there is no difference between the mean.
So these are having difference between the mean.
So we can easily see that these two belong to two different distributions.
Here we can see the value for the degree of freedom, then we have minus two point something, so the
two point something value.
Which is minus two point seven for.
So the minus two point seven four is less than this level, so we can reject the null hypothesis, the
value is greater than this particular value, hence it falls in the critical region and we can reject
the hypothesis.