Which Regression Equation Best Fits These Data

Learning Outcomes

Create and interpret a line of all-time fit

Data rarely fit a direct line exactly. Unremarkably, yous must exist satisfied with rough predictions. Typically, y'all have a ready of information whose scatter plot appears to "fit" a straight line. This is called aLine of Best Fit or Least-Squares Line.

Example

A random sample of 11 statistics students produced the following data, where10 is the 3rd exam score out of 80, and y is the final exam score out of 200. Tin can you lot predict the concluding exam score of a random student if you know the third exam score?

10 (third exam score)	y (terminal exam score)
65	175
67	133
71	185
71	163
66	126
75	198
67	153
seventy	163
71	159
69	151
69	159

Table showing the scores on the final exam based on scores from the tertiary test.

This is a scatter plot of the data provided. The third exam score is plotted on the x-axis, and the final exam score is plotted on the y-axis. The points form a strong, positive, linear pattern.

Scatter plot showing the scores on the final exam based on scores from the 3rd examination.

try it

SCUBA defined take maximum dive times they cannot exceed when going to different depths. The information in the tabular array show different depths with the maximum dive times in minutes. Use your calculator to find the least squares regression line and predict the maximum dive time for 110 anxiety.

10 (depth in anxiety)	Y (maximum dive time)
50	eighty
60	55
seventy	45
80	35
90	25
100	22

[latex]\displaystyle\hat{{y}}={127.24}-{1.eleven}{ten}[/latex]

At 110 feet, a diver could dive for only 5 minutes.

The third exam score,10, is the contained variable and the last exam score, y, is the dependent variable. We will plot a regression line that best "fits" the data. If each of you were to fit a line "past eye," you would draw unlike lines. We can utilize what is called aleast-squares regression line to obtain the best fit line.

Consider the following diagram. Each point of information is of the the form (x, y) and each betoken of the line of best fit using least-squares linear regression has the form [latex]\displaystyle{({x}\lid{{y}})}[/latex].

The [latex]\displaystyle\lid{{y}}[/latex] is read " y hat" and is theestimated value of y . It is the value of y obtained using the regression line. It is not generally equal to y from data.

The scatter plot of exam scores with a line of best fit. One data point is highlighted along with the corresponding point on the line of best fit. Both points have the same x-coordinate. The distance between these two points illustrates how to compute the sum of squared errors.

The term [latex]\displaystyle{y}_{0}-\chapeau{y}_{0}={\epsilon}_{0}[/latex] is called the "error" or residue. It is non an error in the sense of a fault. The absolute value of a remainder measures the vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data indicate and the predicted betoken on the line.

If the observed data point lies above the line, the residuum is positive, and the line underestimates the actual information value fory. If the observed data indicate lies below the line, the residual is negative, and the line overestimates that bodily data value for y.

In the diagram above, [latex]\displaystyle{y}_{0}-\hat{y}_{0}={\epsilon}_{0}[/latex] is the balance for the point shown. Here the point lies in a higher place the line and the residuum is positive.

ε = the Greek letter epsilon

For each data betoken, you can calculate the residuals or errors,
[latex]\displaystyle{y}_{i}-\lid{y}_{i}={\epsilon}_{i}[/latex] for i = i, ii, three, …, xi.

Each |ε| is a vertical distance.

For the instance about the third exam scores and the final examination scores for the xi statistics students, there are 11 data points. Therefore, there are elevenε values. If you foursquare each ε and add, you get

[latex]\displaystyle{({\epsilon}_{{1}})}^{{two}}+{({\epsilon}_{{ii}})}^{{2}}+\ldots+{({\epsilon}_{{11}})}^{{two}}={\stackrel{{eleven}}{{\stackrel{\sum}{{{}_{{{i}={1}}}}}}}}{\epsilon}^{{2}}[/latex]

This is called theSum of Squared Errors (SSE).

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

[latex]\displaystyle\hat{{y}}={a}+{b}{x}[/latex]

where
[latex]\displaystyle{a}=\overline{y}-{b}\overline{{x}}[/latex]

and

[latex]{b}=\frac{{\sum{({x}-\overline{{ten}})}{({y}-\overline{{y}})}}}{{\sum{({x}-\overline{{x}})}^{{ii}}}}[/latex].

The sample ways of the
x values and the y values are [latex]\displaystyle\overline{{ten}}[/latex] and [latex]\overline{{y}}[/latex].

The gradient
b can exist written as [latex]\displaystyle{b}={r}{\left(\frac{{s}_{{y}}}{{due south}_{{x}}}\right)}[/latex] where s _y = the standard deviation of they values and s _x = the standard departure of the 10 values. r is the correlation coefficient, which is discussed in the next section.

Least Squares Criteria for Best Fit

The process of fitting the all-time-fit line is calledlinear regression. The idea behind finding the all-time-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 pocket-sized as possible. Any other line you lot might choose would have a college SSE than the best fit line. This best fit line is chosen the least-squares regression line.

Note

Computer spreadsheets, statistical software, and many calculators can rapidly summate the best-fit line and create the graphs. The calculations tend to exist deadening if done by mitt. Instructions to use the TI-83, TI-83+, and TI-84+ calculators to notice the best-fit line and create a scatterplot are shown at the end of this section.

Example

Third Test vs Final Exam Example

The graph of the line of all-time fit for the third-examination/terminal-exam case is as follows:

The scatter plot of exam scores with a line of best fit. One data point is highlighted along with the corresponding point on the line of best fit.

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

[latex]\displaystyle\hat{{y}}=-{173.51}+{4.83}{x}[/latex]

Call up, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, and so it is reasonable to apply a best fit line to brand predictions for y given x within the domain of 10-values in the sample data, merely not necessarily for x-values exterior that domain. Y'all could use the line to predict the final examination score for a student who earned a grade of 73 on the third exam. You should NOT employ the line to predict the terminal exam score for a educatee who earned a grade of 50 on the third exam, because 50 is not inside the domain of the 10-values in the sample information, 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 gradient of the line in the context of the state of affairs represented past the data. You should be able to write a sentence interpreting the slope in plainly English.

Interpretation of the Slope: The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit of measurement increase in the independent (10) variable, on boilerplate.

Third Exam vs Concluding Test Example: Slope: The gradient of the line is b = iv.83.

Interpretation: For a one-betoken increase in the score on the 3rd exam, the final examination score increases by four.83 points, on average.

Using the Linear Regression T Exam: LinRegTTest

In the STAT listing editor, enter the X information in list L1 and the Y information in list L2, paired and then that the corresponding (x,y) values are adjacent 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 take a different item chosen LinRegTInt.)
On the LinRegTTest input screen enter: Xlist: L1 ; Ylist: L2 ; Freq: i
On the next line, at the prompt β or ρ, highlight "≠ 0" and press ENTER
Go out the line for "RegEq:" blank
Highlight Calculate and press ENTER.

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

The second line saysy = a + bx. Scroll down to notice the values a = –173.513, and b = four.8273; the equation of the best fit line is ŷ = –173.51 + 4.83xThe two items at the bottom are rtwo = 0.43969 and r = 0.663. For at present, merely notation where to find these values; nosotros will hash out them in the next two sections.

Graphing the Scatterplot and Regression Line

We are assuming your X data is already entered in listing L1 and your Y data is in list L2
Printing 2nd STATPLOT ENTER to utilize Plot 1
On the input screen for PLOT i, highlightOn, and press ENTER
For TYPE: highlight the very first icon which is the scatterplot and press ENTER
Point Xlist: L1 and Ylist: L2
For Mark: it does not matter which symbol y'all highlight.
Press the ZOOM key and then the number 9 (for menu particular "ZoomStat") ; the calculator volition fit the window to the information
To graph the best-fit line, press the "Y=" key and type the equation –173.5 + iv.83X into equation Y1. (The X key is immediately left of the STAT key). Press ZOOM 9 once more to graph it.
Optional: If yous want to change the viewing window, printing the WINDOW cardinal. Enter your desired window using Xmin, Xmax, Ymin, Ymax

Annotation

Another way to graph the line subsequently you create a scatter plot is to use LinRegTTest. Brand sure you have washed the besprinkle plot. Check it on your screen.Go to LinRegTTest and enter the lists. At RegEq: press VARS and arrow over to Y-VARS. Press ane for 1:Role. Press ane for one:Y1. Then arrow downward to Calculate and do the calculation for the line of all-time fit.Printing Y = (yous will come across the regression equation).Press GRAPH. The line will be drawn."

The Correlation Coefficient r

Besides looking at the scatter plot and seeing that a line seems reasonable, how can you tell if the line is a adept predictor? Utilize the correlation coefficient as another indicator (besides the scatterplot) of the strength of the relationship betweenx and y.

Thecorrelation coefficient, r , developed by Karl Pearson in the early 1900s, is numerical and provides a mensurate of strength and direction of the linear association between the independent variable x and the dependent variable y.

The correlation coefficient is calculated as [latex]{r}=\frac{{ {due north}\sum{({x}{y})}-{(\sum{x})}{(\sum{y})} }} {{ \sqrt{\left[{n}\sum{x}^{2}-(\sum{ten}^{2})\right]\left[{northward}\sum{y}^{2}-(\sum{y}^{2})\right]}}}[/latex]

wheren = the number of data points.

If you doubtable a linear relationship between10 and y, and so r can measure how potent the linear relationship is.

What the VALUE of r tells united states: The value of r is always between –1 and +1: –1 ≤ r ≤ ane. The size of the correlation rindicates the strength of the linear relationship betwixt ten and y. Values of r close to –1 or to +1 indicate a stronger linear human relationship between x and y. If r = 0 there is absolutely no linear relationship between x and y (no linear correlation). If r = one, at that place is perfect positive correlation. If r = –i, in that location is perfect negativecorrelation. In both these cases, all of the original information points lie on a straight line. Of course,in the real earth, this volition non generally happen.

What the SIGN of r tells us: A positive value of r means that when x increases, y tends to increment and when x decreases, y tends to decrease (positive correlation). A negative value of r means that when x increases, y tends to decrease and when x decreases, y tends to increase (negative correlation). The sign of r is the same as the sign of the gradient,b, of the all-time-fit line.

Note

Strong correlation does non advise thatx causes yor y causes ten. We say "correlation does non imply causation."

(a) A scatter plot showing information with a positive correlation. 0 < r < 1

(b) A scatter plot showing information with a negative correlation. –1 <r < 0

The formula forr looks formidable. All the same, computer spreadsheets, statistical software, and many calculators can quickly calculate r. The correlation coefficient ris the bottom particular in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ figurer (meet previous section for instructions).

The Coefficient of Conclusion

The variable rii is called the coefficient of determination and is the square of the correlation coefficient, merely is usually stated every bit a per centum, rather than in decimal grade. It has an interpretation in the context of the data:

r ², when expressed as a percent, represents the percent of variation in the dependent (predicted) variable y that tin be explained by variation in the independent (explanatory) variable x using the regression (best-fit) line.
1 – r ², when expressed as a percentage, represents the per centum of variation in y that is NOT explained past variation in x using the regression line. This tin exist seen every bit the scattering of the observed information points about the regression line.

The line of best fit is [latex]\displaystyle\hat{{y}}=-{173.51}+{four.83}{10}[/latex]

The correlation coefficient isr = 0.6631The coefficient of determination is r ⁱⁱ = 0.66312 = 0.4397

Interpretation of r ² in the context of this instance: Approximately 44% of the variation (0.4397 is approximately 0.44) in the final-exam grades can be explained past the variation in the grades on the 3rd test, using the all-time-fit regression line. Therefore, approximately 56% of the variation (1 – 0.44 = 0.56) in the final exam grades can Not be explained past the variation in the grades on the 3rd exam, using the all-time-fit regression line. (This is seen every bit the scattering of the points about the line.)

Concept Review

A regression line, or a line of best fit, can be fatigued on a scatter plot and used to predict outcomes for thex and y variables in a given data set or sample data. There are several ways to discover a regression line, but usually the least-squares regression line is used because information technology creates a uniform line. Residuals, also called "errors," measure the distance from the actual value of y and the estimated value of y. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of all-time fit. Regression lines tin be used to predict values inside the given gear up of data, but should non be used to make predictions for values outside the fix of data.

The correlation coefficientr measures the strength of the linear association betwixt x and y. The variable r has to be between –ane and +1. When r is positive, the x and y volition tend to increase and subtract together. When r is negative, ten will increase and y will decrease, or the opposite, x will subtract and y will increment. The coefficient of determination r2, is equal to the foursquare of the correlation coefficient. When expressed as a pct, r2 represents the percent of variation in the dependent variable y that tin be explained by variation in the contained variable 10 using the regression line.