[MUSIC] 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. [MUSIC]