# Question Video: Calculating the Equation of the Least Squares Regression Line Mathematics

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.

03:13

### 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 thatβs 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 ππ₯, where π is equal to π¦ minus ππ₯, 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. Letβs go ahead and make a table of everything we need to find. Letβs 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 weβve 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 ππ₯, 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 thereβs 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.