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Okay, so we've talked about this simple linear regression model, just a line.
And then we talked about how our
goal in fitting this model is gonna be to search over all possible lines, and
find the line that minimizes the residual sum of squares.
We haven't talked about how we're gonna do that, that's gonna come next.
We're gonna talk about specific algorithms for
searching over all possible lines to minimize residual sum of squares.
But right now, let's just assume that we have some fitted line and
let's talk about how we're gonna use it and how we can interpret the parameters.
Let's start by contrasting the model from the fitted line.
So, our model, which I've written again here,
is in terms of parameters.
These are parameters,
they're unknown
variables of our model.
And our goal is to estimate these parameters, fix values of these
variables based on our data, and so that's what w hat represents.
These are estimated parameters,
so these take actual values.
So for example, maybe our
intercept is -44,850 and
our slope is 280.76.
And these aren't random numbers that I generated here,
this comes from the first course.
If you look at the notebook for doing the regression fade of sales price on
square feet, these were the numbers that came out for these two parameters.
Okay, so these estimated parameters define a specific line.
Okay, that's what this is.
This line represents -44,850
+ 280.76 times x.
So, a model is in terms of sum parameters and
a fitted line is a specific example within that model class.
So, it's a specific line.
Now let's talk about how we can think about using our fitted line.
So one thing that we can ask is
to predict the value of this house that we'd like to list for sale.
So this house has some number of square feet, and
we'd like to guess the price, and how are we gonna guess the price?
Well, it's very straightforward.
We're just gonna plug into our fitted line.
And so we're gonna take the square feet of this house, which is represented here.
Plug that into the equation of this line, and
that's gonna produce our estimated value of the house right here.
I want to mention one more thing here.
So I wanna mention that y hat, that's our predicted value of the house.
Y hat is exactly equal to f hat(x), okay?
Our model, remember our model said that
yi is approximately equal to f(xi).
But when we go to do a prediction, y hat.
That's exactly equal to f hat(x).
And why is that?
That's because the error is equally likely to be above or below the line.
And f hat represents our best guess of the line given our data.
And so if we wanna guess the value of the house,
we're unsure if that error for that specific house was above or
below the line, so our best guess is to put it exactly on the line.
Well I can use this fitted line also if I'm a buyer,
not just if I'm a seller just in listing my house for sale.
So for example, what I mean here is that I might have
some amount of money that I can spend on a house.
And I want to know how a big of a house can I expect to purchase
with that amount of money.
Well instead of estimating or predicting
the value of the house, I can think of predicting the square feet.
So just using this equation in reverse.
So I have some amount of money in the bank.
And that's gonna be $ in bank is this.
And then I'm gonna look at how many square feet,
I believe that I can purchase with that amount of money.
Okay, so let's just go through a concrete example.
Here's the fitted regression line I talked about before.
So it's -44,850 + 280.76 times x.
And what I'd like to do is predict the value of a house that
has 2,640 square feet.
So I do, it's just plug in 2,640 into this equation.
That's my y hat.
And that comes out to be, if you do that calculation, $696,356.
Likewise, I can say, well,
I wanna predict how many square feet of house I can purchase
if I have $859,000 to spend on the house.
So if you do this calculation, you get that you can afford a house or
you expect to buy a house that's roughly 3,219 square feet.
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