And this video will assess the accuracy of these coefficient between bidone that we have calculated.
I will start by restating the situation that we are in.
In the world, millions of transactions were happening.
We picked a small set or a small sample of 506 such observations and decided to identify the relationship
between house price and number of rooms.
When I used the formula shown in previous video, we get the values of and Beethoven.
This line is minimizing this squared error on the sample of five or six points.
If we had selected some other 506 observations, we may have got some other values of Betaseron be done.
If we take all the million observations of the world and then get a line, this line will be called
population regression line.
This line may or may not be similar as sample regression line.
You can see in this graph, this blue line, it is embolisation line, which is minimizing the error
on these data points, the sample data point.
But this red line with population, the green line, which would have come if I would have saw the list,
could add up for all the data points of the population, these dulaine, the Nazi.
Now, the problem is this, we do not have all the observations.
Therefore, we cannot get population regulation line.
We have sample data so we can get only this sample regression line.
We want to use the completion of simple regression line as an estimate for the population regression
line.
How far off will this ample estimate that is be gap and between gap will be from the population coefficients
which are Betaseron be driven for this, we will use a quantity called standard error of Betaseron between.
The standard of being as unbeatable is calculated using these formulas again and showing you these formulas
and discussing the illusion behind them, you need not memorize them or know their derivation for practical
purposes.
But since these quantities will be part of the output of the model from our software, we need to understand
their meaning.
So in this formula.
Delta Sigma squared, sigma squared, variance of population residuals.
Remember, we discussed vegetables as well as the value, which is the difference of actual weight from
the estimated weight of the population.
Sigma squared is the variance of these values for all the data points of the population.
Since population regression line is not known, the sigma results are not known.
We need to estimate it from our sample data to this estimate is given by this formula, which is Oddisee
is equal to underoos are assessed by and minus two year Oddisee is called residual standard error.
Odyssey's, as did some.
Of course, we discussed this in the last video, which is sum of all the squares.
So for all the data points we get, the errors that that is the residuals.
We square them and we add them that as the Odyssey's.
We divide Odyssey's by and minus two and the number of observations for our dataset.
It is five or six will divided by five or six minus two.
When we find it on the road, we get residuals, standard error.
Will use this the standard edit as a proxy for Sigma.
Also, notice that if X is more spread out in this formula of Bedazzler and vitamin.
Standard of bedazzling be down will be small, which intuitively means that we have more leverage while
estimating the slope in such a case.
Now, what is the practical takeaway from the ethical relations standard, it will be used to give us
a confidence interval, that is, or linear regression.
There is a 95 percent chance that the true value of Beethoven lies in the interval.
We got minus two times standard, Ed.
We don't want to be the one guy plus two times to standard a little bit of an.
So within this interval, we are lower values this and higher values this, we are 95 percent confident
that the actual between lies in this interval.
Similarly, for the 95 percent confidence interval will be the estimated Bredasdorp minus two days standard,
Ed, this.
Standard error of president.
And the higher value will be the zero plus two times standard error of bedazzle, to summarize, what
we have done here is we had two lines, one line we got from reasonable regression that we did.
The other one is a hypothetical line, which is the true line between the population of all the points.
We wanted to show whether we can approximate this Tamburlaine as the population lane.
For that, we found out the confidence interval.
Within which the population coefficient will lay.
So the president of the population line will pay between.
These two values that we get from the regulation and the slope of the population aggression relation
between these two values of the simple equation.
And we have a assigned the probability of how confident we are.
We are saying that there is a 95 percent chance that the population regression coefficients lie within
these intervals that we found out using sample.
Another use of Standard Chartered is to establish that whether X and Y actually have a relationship
or not in a linear model, this relationship is governed by between.
We are saying that life is better one times X plus constant.
So if Beethoven is zero, it means that there is no relationship.
If there is no relationship, then the variable X cannot be used to predict like.
So we need to show that the probability of Beethoven being zero is negligible.
Let me tell you the concept first, and after that, I'll show you the way, how you will find it in
every other statistical book.
The concept is this Biederman has some value that is the most probable value found out by the formula
we saw in the previous videos.
Now we want to see that we are sufficiently confident that it cannot be zero.
It has two parts to it.
First, most valuable value should be far from zero.
And taking the standard of beta, which is giving us the interval in which the rule lies, should be
small.
So basically, we want zero not to lie in this whole interval, in which case I hope you get the idea.
Let's do it the proper way now, this this method is called hypothesis testing.
We will construct two hypotheses.
One is at zero, that is there is no relationship between X and Y.
And its alternative, which is written as it is that there is a relationship between X and Y, so it
will be the one is equal, Luisito, it is between is not equal to zero.
And we want to disprove Ejiro.
To disprove at zero, we will calculate something known as a statistic which is given as being equal
to be Gapminder zero, divided by standard error be double.
So you can see what is what this is representing here, enumerator, it says how far better one is from
zero.
And when you divided by standard at it, you go the distance is how many times I heard it.
So using this formula will get to people, you know, we want this disvalue to be large.
By the way, it is called P-value, because it is based on p distribution, which is similar to normal
distribution.
Basically, the distribution is another probability distribution, and if you have P-value, you can
get the probability of observing any value equal to absolutely or larger.
The probability that you will get from this distribution is called P-value.
Now, this P-value has meaning.
We can interpret P-value as follows.
Small value of B will mean that it is highly unlikely that there is no relationship between pressure
and response.
Which means that we can reject the null hypothesis and declare that there is a relationship between
X and Y.
Typically, we use.
A value like five percent or one percent as the cutoff value for B, that is, if P is less than point
zero one, then the variable X is significantly impacting Y.
Let us look at the results of our sample.
Not that this is a result which we have received from the software package that we are using, and of
course there is a separate video where you will learn how to run this analysis and get this result.
I'm just discussing the result here as an example to all the theory that we just word.
So in the last video, we had just seen the beta one and beta values, this intercept is the president
and this room where evil has this slope, this nine point zero nine is the beta one for this variable.
The first thing that we covered is the standard error.
What does he do?
And we don't.
The formula that I showed you earlier was used to calculate the standard error of zero and be done,
how the standard error can be used, as you can make the statement that for the sample of 506 observations,
the beta that you calculated for minimum error was nine point zero nine.
But you are 95 percent confident that for the global data of placing, the B1 will lay between nine
point zero nine minus two times zero point four one.
Two nine point zero nine plus two times zero point four to.
So this interval is giving you that the estimated Beta's that you have here.
Can be used to make this statement that the true regression coefficients will lie in the interval given
by this standard error.
The next thing we discussed was the value and the p value.
This devaluate P-value, we said that we have one hypothesis, which is saying that there is no relationship.
To disprove that hypothesis, we calculated the devalue which was given by this formula.
So to get the value of this be done, we will probably be the one value divided by the AC.
So nine point zero nine divided by zero point for one, gives you this disvalue.
So corresponding to this disvalue, we calculate a P value, which is written as probability of getting
a P value, which is greater than equal to the speed that we calculated.
This probability value is coming out to be very small.
It's a it's an exponential with the power of minus 16 to this value is very small.
So if this value is very small, we can see that it is unlikely that there is no relationship, which
means that there is some relationship.
And therefore, we are confident that room number variable is impacting the house pricing and the relationship
between them is given by this coalition.
So the takeaway from this lecture is we have one and zero values calculated, for example.
We calculated a standard error, which is helping us in determining two things, one is what is range
in which the true value of Beethoven and be desirable.
And the second thing is whether there is actually a relationship between these predictor and the response
variables.
To establish that there is a relationship we calculated at value, using these two things, using a
disvalue, we calculated a P-value, and if this P value is less than a threshold of one percent or
five percent, whichever you like to use, then we say that there is a relationship.
So far, our model, there is a relationship of house pricing with room number, and that relationship
is this one, which is nine point.