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[MUSIC]

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Okay, so we've talked about this simple
linear regression model, just a line.

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And then we talked about how our

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goal in fitting this model is gonna be
to search over all possible lines, and

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find the line that minimizes
the residual sum of squares.

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We haven't talked about how we're
gonna do that, that's gonna come next.

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We're gonna talk about
specific algorithms for

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searching over all possible lines to
minimize residual sum of squares.

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But right now, let's just assume
that we have some fitted line and

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let's talk about how we're gonna use it
and how we can interpret the parameters.

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Let's start by contrasting
the model from the fitted line.

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So, our model,
which I've written again here,

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is in terms of parameters.

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These are parameters,

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they're unknown

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variables of our model.

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And our goal is to estimate these
parameters, fix values of these

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variables based on our data, and
so that's what w hat represents.

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These are estimated parameters,

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so these take actual values.

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So for example, maybe our

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intercept is -44,850 and

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our slope is 280.76.

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And these aren't random
numbers that I generated here,

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this comes from the first course.

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If you look at the notebook for doing
the regression fade of sales price on

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square feet, these were the numbers
that came out for these two parameters.

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Okay, so these estimated
parameters define a specific line.

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Okay, that's what this is.

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This line represents -44,850

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+ 280.76 times x.

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So, a model is in terms
of sum parameters and

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a fitted line is a specific
example within that model class.

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So, it's a specific line.

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Now let's talk about how we can
think about using our fitted line.

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So one thing that we can ask is

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to predict the value of this house
that we'd like to list for sale.

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So this house has some
number of square feet, and

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we'd like to guess the price, and
how are we gonna guess the price?

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Well, it's very straightforward.

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We're just gonna plug
into our fitted line.

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And so we're gonna take the square feet
of this house, which is represented here.

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Plug that into the equation of this line,
and

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that's gonna produce our estimated
value of the house right here.

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I want to mention one more thing here.

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So I wanna mention that y hat,
that's our predicted value of the house.

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Y hat is exactly equal to f hat(x), okay?

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Our model, remember our model said that

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yi is approximately equal to f(xi).

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But when we go to do a prediction, y hat.

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That's exactly equal to f hat(x).

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And why is that?

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That's because the error is equally
likely to be above or below the line.

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And f hat represents our best
guess of the line given our data.

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And so
if we wanna guess the value of the house,

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we're unsure if that error for
that specific house was above or

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below the line, so our best guess
is to put it exactly on the line.

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Well I can use this fitted
line also if I'm a buyer,

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not just if I'm a seller just
in listing my house for sale.

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So for example,
what I mean here is that I might have

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some amount of money that
I can spend on a house.

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And I want to know how a big of
a house can I expect to purchase

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with that amount of money.

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Well instead of estimating or predicting

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the value of the house,
I can think of predicting the square feet.

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So just using this equation in reverse.

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So I have some amount
of money in the bank.

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And that's gonna be $ in bank is this.

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And then I'm gonna look
at how many square feet,

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I believe that I can purchase
with that amount of money.

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Okay, so
let's just go through a concrete example.

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Here's the fitted regression
line I talked about before.

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So it's -44,850 + 280.76 times x.

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And what I'd like to do is
predict the value of a house that

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has 2,640 square feet.

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So I do, it's just plug in
2,640 into this equation.

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That's my y hat.

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And that comes out to be,
if you do that calculation, $696,356.

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Likewise, I can say, well,

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I wanna predict how many square
feet of house I can purchase

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if I have $859,000 to spend on the house.

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So if you do this calculation,
you get that you can afford a house or

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you expect to buy a house that's
roughly 3,219 square feet.

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[MUSIC]