# 3.5 Bivariate descriptive statistics: the regression equation  (Page 2/2)

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$\epsilon$ = the Greek letter epsilon

For each data point, you can calculate the residuals or errors, ${y}_{i}-{\stackrel{^}{y}}_{i}={\epsilon }_{i}$ for $i=\text{1, 2, 3, ..., 11}$ .

Each $|\epsilon |$ is a vertical distance.

For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points. Therefore, there are 11 $\epsilon$ values. If you square each $\epsilon$ and add, you get

$\left({\epsilon }_{1}{\right)}^{2}+\left({\epsilon }_{2}{\right)}^{2}+\text{...}+\left({\epsilon }_{11}{\right)}^{2}=\stackrel{11}{\underset{\text{i = 1}}{\Sigma }}{\epsilon }^{2}$

This is called the Sum of Squared Errors (SSE) .

Using calculus, you can determine the values of $a$ and $b$ that make the SSE a minimum. When you make the SSE a minimum, you have determined the points that are on the line of best fit. It turns out thatthe line of best fit has the equation:

$\stackrel{^}{y}=a+\text{bx}$

where

• $a=\overline{y}-b\cdot \overline{x}$
• $\overline{x}$ and $\overline{y}$ are the sample means of the $x$ values and the $y$ values, respectively. The best fit line always passes through the point $\left(\overline{x},\overline{y}\right)$ . and
• $b=r\cdot \left(\frac{{s}_{y}}{{s}_{x}}\right)$
• where ${s}_{y}$ = the standard deviation of the $y$ values and ${s}_{x}$ = the standard deviation of the $x$ values. $r$ is the correlation coefficient which is discussed in the next section.

## Least squares criteria for best fit

The process of fitting the best fit line is called linear regression . The idea behind finding the best fit line is based on the assumption that the data are scattered about a straight line. The criteria for the best fit line is that the sum of the squared errors (SSE) is minimized, that is made as small as possible. Any other line you might choose would have a higher SSE than the best fit line. This best fit line is called the least squares regression line .

Using the summary statistics and correlation coefficient for the relationship between third exam score and final exam score we will calculate the regression equation, the line of best fit.

Start by calculating the slope of the line.

• $b=r\cdot \left(\frac{{s}_{y}}{{s}_{x}}\right)=\mathrm{\left(0.6632\right)}\cdot \left(\frac{\mathrm{20.8008}}{\mathrm{2.85721}}\right)=\mathrm{4.827}$

Using the slope find the u-intercept.

• $a=\overline{y}-\mathrm{\left(b\right)}\cdot \left(\overline{x}\right)=160.4545-\left(4.827\right)\left(69.1818\right)=-173.49$

The least squares regression line (best fit lint) for the third exam/final exam example has the equations: $\stackrel{^}{y}=-\mathrm{173.49}+4.83x$

## Third exam vs final exam example:

The graph of the line of best fit for the third exam/final exam example is shown below:

• Remember, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship betweenthe variables, then it is reasonable to use a best fit line to make predictions for $y$ given $x$ within the domain of $x$ -values in the sample data, but not necessarily for $x$ -values outside that domain.
• You could use the line to predict the final exam score for a student who earned a grade of 73 on the third exam.
• You should NOT use the line to predict the final exam score for a student who earned a grade of 50 on the third exam, because 50 is not within the domain of the x-values in the sample data, which are between 65 and 75.

## Understanding slope

The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.

INTERPRETATION OF THE SLOPE: The slope of the best fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.

## Third exam vs final exam example

• Slope: The slope of the line is b = 4.83.
• Interpretation: For a one point increase in the score on the third exam, the final exam score increases by 4.83 points, on average.

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