Part $4$: Linear Regression Model

In this part, we will create our linear regression model by calculating the model coefficients hat$({)}_0$ and hat$({)}_1.$ Before we can do so$,$

we have one last quantity to calculate, which we will denote by $S_{xY}.$ The formula for this value is provided below.

$S_{xY}={\displaystyle \underset{i=1}{\overset{n}{}}}(x_i-(\stackrel{}{x}))(y_i-(\frac{?}{\text{b}\text{a}\text{r}}(y)))=(x_1-(\stackrel{}{x}))(y_1-(\frac{?}{\text{b}\text{a}\text{r}}(y)))+(x_2-(\stackrel{}{x}))(y_2-(\frac{?}{\text{b}\text{a}\text{r}}(y)))+$cdots$+(x_n-(\stackrel{}{x}))(y_n-(\frac{?}{\text{b}\text{a}\text{r}}(y)))$

Create a markdown cell with a level $2$ header that reads "Part $4$: Linear Regression Model".

In the same cell, add unformatted text explaining that in this part we will calculate $S_{xY},$ which we will then use to find the

coefficients for our linear regression model.

Calculate $S_{xY},$ storing the result in a variable named Sxy$.$ You will need to use a loop to perform this task. For the sake of

efficient memory usage, you should not create any additional lists when performing this calculation.

Print the result with text output as shown below. Round the displayed value of Sxy to $2$ decimal places. Do not round

the actual values stored within the variables themselves, just the values displayed.

$Sxy=$

We are now ready to calculate the model coefficients hat$({)}_0$ and hat$({)}_1$ using values that we have previously calculated. Formulas

for these coefficients are provided below.

hat$({)}_1=\frac{S_{xY}}{s_{xx}}$ and hat$({)}_0\frac{?}{b}=ar(y)-$hat$({)}_1*\stackrel{}{x}$

Create a markdown cell with unformatted text explaining that we will now be calculating the coefficients of our model.

Calculate the values of hat$({)}_0$ and hat$({)}_1$ storing the results in variables named beta$\_$ and beta$\_1.$ Print the results with text

output as shown below. Round the displayed value of beta$\_$ to $4$ decimal places and round the displayed value of

beta$\_1$ to $8$ decimal places. Do not round the actual values stored within the variables themselves, just the values

displayed.

$bet{a}_0=$

$bet{a}_1=xxxx$