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
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:
$\hat{y}=a+\text{bx}$
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
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 .
Computer spreadsheets, statistical software, and many calculators can quickly
calculate the best fit line and create the graphs. The calculations tend to be tedious if done by hand. Instructions to use the TI-83, TI-83+, and TI-84+ calculators to find the best fit line and create a scatterplot are shown at the end of this section.
Third exam vs final exam example:
The graph of the line of best fit for the third exam/final exam example is shown below:
The least squares regression line (best fit line) for the third exam/final exam example has the equation:
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.
Using the ti-83+ and ti-84+ calculators
Using the linear regression t test: linregttest
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.)
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.)
On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: 1
On the next line, at the prompt β or ρ, highlight "≠ 0" and press ENTER
Leave the line for "RegEq:" blank
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
$\hat{y}=-173.51+\text{4.83}x\phantom{\rule{20pt}{0ex}}$
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
We are assuming your X data is already entered in list L1 and your Y data is in list L2
Press 2nd STATPLOT ENTER to use Plot 1
On the input screen for PLOT 1, highlight
On and press ENTER
For TYPE: highlight the very first icon which is the scatterplot and press ENTER
Indicate Xlist: L1 and Ylist: L2
For Mark: it does not matter which symbol you highlight.
Press the ZOOM key and then the number 9 (for menu item "ZoomStat") ; the calculator will fit the window to the data
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.
Optional: If you want to change the viewing window, press the WINDOW key. Enter your desired window using Xmin, Xmax, Ymin, Ymax
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