### Video Transcript

The scatterplot shows a set of data for which a linear regression model appears appropriate. The data used to produce the scatterplot is given in the table shown. Calculate the equation of the least squares regression line of π¦ on π₯, rounding the regression coefficients to the nearest thousandth.

Essentially, linear regression is a single independent variable thatβs used to predict the value of a dependent variable. So this line will help predict the dependent variable. The equation of this line is π¦ equals π plus ππ₯, where π is equal to π¦ minus ππ₯, where π¦ is the mean π¦ value and π₯ is the mean π₯ value. And π is equal to Sπ₯π¦ divided by Sπ₯π₯. Sπ₯π¦ is the covariance of π₯ and π¦ divided by π and Sπ₯π₯ is a variance of π₯ divided by π.

The formulas for these, Sπ₯π¦ is equal to the sum of π₯ times π¦s minus the sum of π₯ times the sum of π¦ divided by π and then Sπ₯π₯ is equal to the sum of π₯ squareds minus the sum of the π₯s squared divided by π. Letβs go ahead and make a table of everything we need to find. Letβs first begin by finding π. Here are our formulas. So if we take all our π₯s and we square them, we have these answers. And if we take π₯ times π¦, we have these answers. And if we would find the sum of each column, we have these: 18, 45.1, 51, and 78.05.

18 is the sum of the π₯s. 45.1 is the sum of the π¦s. 51 is the sum of the π₯ squares and 78.05 is the sum of π₯ times π¦s. And now weβve plugged them in correctly. After multiplying and dividing, we have 78.05 minus 101.475 divided by 51 minus 40.5, which is equal to negative 23.475 divided by 10 and a half which equals negative 2.236. This is the value of π.

So for our equation, π¦ equals π plus ππ₯, we have π¦ equals π minus 2.236π₯. So now we need to find π. π was equal to the mean value of π¦ minus π times the mean value of π₯. To find the mean, you take the sum and divide by, in this case, eight since thereβs eight π₯s and eight π¦s. After plugging in, this results in 10.669. Therefore, the equation of the least squares regression line of π¦ on π₯ will be π¦ equals 10.669 minus 2.236π₯.

Now remember, depending on how you rounded, for example, when you found the π₯ times π¦s, we rounded three decimal places right away. So therefore, keep in mind that your final answer maybe just a little bit different depending on how far you rounded throughout your work.