6.4 Linear regression and correlation: the regression equation  (Page 2/2)

$\epsilon$ = the Greek letter epsilon

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

Each $\left|\epsilon \right|$ 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

$({\epsilon }_1{)}^2+({\epsilon }_2{)}^2+\text{...}+({\epsilon }_{11}{)}^2=\stackrel{11}{\underset{\text{i = 1}}{\Sigma }}{\epsilon }^2$

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:

where $a=\overline{y}-b\cdot \overline{x}$ and $b=\frac{\Sigma (x-\overline{x})\cdot (y-\overline{y})}{{\Sigma (x-\overline{x})}^2}$ .

$\overline{x}$ 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 $(\overline{x},\overline{y})$ .

The slope $b$ can be written as $b=r\cdot \left(\frac{s_y}{s_x}\right)$ 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

Third exam vs final exam example:

The least squares regression line (best fit line) for the third exam/final exam example has the equation:

Understanding slope

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.

Using the ti-83+ and ti-84+ calculators

Using the linear regression t test: linregttest

1. In the STAT list editor, enter the X data in list L1 and the Y data in list L2, paired so that the corresponding (x,y) values are next to each other in the lists. (If a particular pair of values is repeated, enter it as many times as it appears in the data.)
2. On the STAT TESTS menu, scroll down with the cursor to select the LinRegTTest. (Be careful to select LinRegTTest as some calculators may also have a different item called LinRegTInt.)
3. On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1
4. On the next line, at the prompt β or ρ, highlight "≠ 0" and press ENTER
5. Leave the line for "RegEq:" blank
6. Highlight Calculate and press ENTER.

The output screen contains a lot of information. For now we will focus on a few items from the output, and will return later to the other items.

• The second line says y=a+bx. Scroll down to find the values a=-173.513, and b=4.8273 ; the equation of the best fit line is $\hat{y}=-173.51+\text{4.83}x$
• The two items at the bottom are $r^2$ = .43969 and $r$ =.663. For now, just note where to find these values; we will discuss them in the next two sections.

Graphing the scatterplot and regression line

1. We are assuming your X data is already entered in list L1 and your Y data is in list L2
2. Press 2nd STATPLOT ENTER to use Plot 1
3. On the input screen for PLOT 1, highlight On and press ENTER
4. For TYPE: highlight the very first icon which is the scatterplot and press ENTER
5. Indicate Xlist: L1 and Ylist: L2
6. For Mark: it does not matter which symbol you highlight.
7. Press the ZOOM key and then the number 9 (for menu item "ZoomStat") ; the calculator will fit the window to the data
8. To graph the best fit line, press the "Y=" key and type the equation -173.5+4.83X into equation Y1. (The X key is immediately left of the STAT key). Press ZOOM 9 again to graph it.
9. Optional: If you want to change the viewing window, press the WINDOW key. Enter your desired window using Xmin, Xmax, Ymin, Ymax

**With contributions from Roberta Bloom