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Section 4.2

Section 4.2. Linear Regression and the Coefficient of Determination. The Least Squares Line. When there appears to be a linear relationship between x and y we attempt to “fit” a line to the scatter diagram. Least Squares Criterion.

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Section 4.2

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  1. Section 4.2 Linear Regression and the Coefficient of Determination

  2. The Least Squares Line • When there appears to be a linear relationship between x and y we attempt to “fit” a line to the scatter diagram. Least Squares Criterion The sum of the squares of the vertical distances from the points to the line is made as small as possible.

  3. Least Squares Criterion d represents the difference between the y coordinate of the data point and the corresponding y coordinate on the line. Thus if the data point lies above the line, d is positive, but if the data point lies below the line, d is negative. As a result, the sum of the d values can be small even if the points are widely spread in the scatter diagram. However, the squares cannot be negative. By minimizing the sum of the squares, we are, in effect, not allowing positive and negative d values to “cancel out” one another in the sum. It is this way that we can meet the least-squares critirion of minimizing the sum of the squares of the vertical distances between the points and the line over all points in the scatter diagram.

  4. Equation of the Least Squares Line ŷ = a + bx a = the y-intercept b = the slope

  5. Finding the Equation of the Least Squares Line • Obtain a random sample of n data pairs (x, y). 1. Using the data pairs, compute Σx, Σy, Σx2, Σy2, and Σxy. Compute the sample means

  6. Finding the Slope • 2. Use the following formula: • Finding the y-intercept

  7. ExampleFind the Least Squares Line

  8. Example cont.Finding the Slope

  9. Example cont.Finding the y-intercept The equation of the least squares line is: ŷ = a + bx ŷ = 2.77 + 1.70x

  10. Graph the least-Squares Line We can use the slope-intercept method of algebra, but may not always be convenient if the intercept is not within the range of the sample data values. It is better to select two x values in the range of the x data values and then use the least-squares line to compute two corresponding y values. The point is always on the least-squares line. To find another point, give x a value and find the y. In our example: = (8.3 , 16.9) Try x = 5. Compute ŷ : ŷ = 2.8 + 1.7(5)= 11.3

  11. Graphing the least squares line • Using two values in the range of x, compute two corresponding y values. • Plot these points. • Join the points with a straight line.

  12. Sketching the Line

  13. Meaning of Slope ŷ = a + bx In the equation , the slope b tell us how many units ŷ changes for each unit change in x. In our example regarding the miles traveled and the time in minutes ŷ = 2.77 + 1.70x The slope 1.70 tell us that a change in one mile takes in average 1.70 minutes. The slope of the least-squares line tells how many units the response variable is expected to change for each unit change in the explanatory variable. The number of units change in the response variable for each unit change in the explanatory variable is called marginal change of the response variable.

  14. Using the Equation of the Least Squares Line to Make Predictions • Choose a value for x (within the range of x values). • Substitute the selected x in the least squares equation. • Determine corresponding value of ŷ.

  15. Predict the time to make a trip of 14 miles • Equation of least squares line: ŷ = 2.8 + 1.7x • Substitute x = 14: ŷ = 2.8 + 1.7 (14) ŷ = 26.6 • According to the least squares equation, a trip of 14 miles would take 26.6 minutes.

  16. Interpolation • using the least squares line to predict ŷ values for x values that are between observed x values in the data set. Extrapolation using the least squares line to predict ŷ values for x values that are beyond observed x values in the data set.

  17. Extreme Data Points • The least squares line can be greatly affected by extreme or influential data points.

  18. The least squares line • Is developed from sample data pairs (x, y). • May not reflect the relationship between x and y for values of x outside the data range. • For example, there is a fairly high correlation between height and age for boys ages 1 year to 10 years. In general the older the boy, the taller the boy. A least-squares line based on such date give good predictions of height for ages 1 to 10. • However, it would be fairly meaningless to use the same linear regression line to predict the height of 20 to 50 years old.

  19. The least squares line • Each different sample data will produce a slightly different equation for the least-squares line. • The least-squares line developed with x as the explanatory variable and y as the response variable can be used only to predict y values from specified x values.

  20. A statistic related to r: • If the sample correlation coefficient is r • The coefficient of determination = r2 How good is the least squares line as an instrument of regression? The answer is the coefficient of determination Coefficient of Determination Is a measure of the proportion of the variation in y that is explained by the regression line using x as the predicting variable

  21. Interpretation of r2 • If r = 0.9753643, then what percent of the variation in minutes (y) is explained by the linear relationship with x, miles traveled? • What percent is unexplained? • If r = 0.9753643, then r2 = .9513355 • Approximately 95 percent of the variation in minutes (y) is explained by the linear relationship with x, miles traveled. • is unexplained (due to the random chance or the probability of lurking variables that influence y). Assignments 7, 8 and 9

  22. Correlation Coefficient r Coefficient of Determination, r 2 (calc) • The correlation coefficient, r, and the coefficient of determination, r2 ,will appear on the screen that shows the regression equation information (be sure the Diagnostics are turned on ---2nd Catalog (above 0), arrow down to DiagnosticOn, press ENTER twice.) • In addition to appearing with the regression information, the values r and r2 can be found under • VARS, #5 Statistics → EQ #7 r and #8 r 2 .

  23. Linear Regression (calc) • A linear regression is also know as the "line of best fit".  • Side note:  Although commonly used when dealing with "sets" of data, the linear regression can also be used to simply find the equation of the line between two points.Find the equation of the line passing through (-1, 1) and (-4,7).Entering the information as described in the example below, we see the following screens: The equation is y = -2x -1.The correlation coefficient is -1 since both point are "on" the line and the line slopes negatively

  24. Linear Regression Model Example (calc) Let's examine an example of the linear regression as it pertains to a "set" of data.  Data:  Is there a relationship between Math SAT scores and the number of hours spent studying for the test?  A study was conducted involving 20 students as they prepared for and took the Math section of the SAT Examination. Let x be the Hours Spent Studying and y be Math SAT Score x y x y x y 4 390 22 790 10 690 9 580 1 350 11 690 10 650 3 400 16 770 14 730 8 590 13 700 4 410 11 640 13 730 7 530 5 450 10 640 12 600 6 520

  25. Linear Regression Model Example cont. • Task: a) Determine a linear regression model equation to represent this data.   b) Graph the new equation.   c) Decide whether the new equation is a "good fit" to represent this data.   d) Interpolate data:  If a student studied for 15 hours, based upon this study, what would be the expected Math SAT score? e) Interpolate data:  If a student obtained a Math SAT score of 720, based upon this study, how many hours did the student most likely spend studying?   f) Extrapolate data:  If a student spent 100 hours studying, what would be the expected Math SAT score?  Discuss this answer. Any answers in relation to this problem are to be rounded to the nearest tenth.If rounding is not indicated in a problem, leave the full calculator entries as answers

  26. Linear Regression Model Example cont. • Step 1.  Enter the data into the lists.  • Step 2.  Create a scatter plot of the data.       Go to STATPLOT (2nd Y=) and choose the first plot.  Turn the plot ON, set the icon to Scatter Plot (the first one), set Xlist to L1 and Ylist to L2 (assuming that is where you stored the data), and select a Mark of your choice. • Step 3.  Choose Linear Regression Model.     Press STAT, arrow right to CALC, and arrow down to 4: LinReg (ax+b).  Hit ENTER.  When LinReg appears on the home screen, type the parameters L1, L2, Y1.  The Y1 will put the equation into Y= for you.        (Y1 comes from VARS → YVARS, #Function, Y1)

  27. Linear Regression Model Example cont. • Step 4.  Graph the Linear Regression Equation from Y1.     ZOOM #9 ZoomStat to see the graph. (answer to part b) • Step 5.  Is this model a "good fit"?     The correlation coefficient, r, is .9336055153 which places the correlation into the "strong" category.  (0.8 or greater is a "strong" correlation)     The coefficient of determination, r 2, is .8716192582 which means that 87% of the total variation in y can be explained by the relationship between x and y.  The other 13% remains unexplained.     Yes, it is a "good fit".          (answer to part c)

  28. Linear Regression Model Example cont. Step 6.  Interpolate:  (within the data set)If a student studied for 15 hours, based upon this study, what would be the expected Math SAT score?From the graph screen, hit TRACE, arrow up to obtain the linear equation at the top of the screen, type 15, hit ENTER, and the answer will appear at the bottom of the screen.                          (answer to part d --                        Math SAT score of 733.1)

  29. Linear Regression Model Example cont. • Step 7.  Interpolate:  (within the data set) If a student obtained a Math SAT score of 720, based upon this study, how many hours did the student most likely spend studying?  Go to TBLSET (above WINDOW) and set the TblStart to 13 (since 13 hours gives a score of 700).  Set the delta Tbl to a decimal setting of your choice.  Go to TABLE and arrow up or down to find your desired score of 720, in the Y1 column          • (answer to part e --  approx. 14.5 hours)

  30. Linear Regression Model Example cont. • Step 8. Extrapolate data:  (beyond the data set)If a student spent 100 hours studying, what would be the expected Math SAT score?      Discuss this answer.            • With your linear equation in Y1, go to the home screen and type Y1(100).  Press ENTER. (Y1 comes from VARS → YVARS, #Function, Y1(100)) • Our equation shows that if a student studies 100 hours, he/she should score 2885.8 on the Math section of the SAT examination.  The only problem with this answer is that the highest score that can be obtained is 800.  So why is this score so outrageous?   ANSWER:  When you extrapolate data, the further you move away from the data set, the less accurate your information becomes.  In this problem, the largest number of hours in the data set was 22 hours, but the extrapolation tried to jump to 100 hours. (answer to part f)

  31. ExampleLinear Regression with Biological Data(or the realities of working with real-life data) Pierce (1949) measured the frequency (thenumber of wing vibrations per second) of chirps made by a ground cricket, at various ground temperatures.  Since crickets are ectotherms (cold-blooded), the rate of their physiological processes and their overall metabolism are influenced by temperature.  Consequently, there is reason to believe that temperature would have a profound effect on aspects of their behavior, such as chirp frequency.

  32. Example cont. Chirps/Second Temperature (º F) 20.088.6 16.071.6 19.893.3 18.484.3 17.180.6 15.575.2 14.769.7 17.182.0 15.469.4 16.283.3 15.078.6 17.282.6 16.080.6 17.083.5 14.176.3

  33. Example cont. Task: • Determine a linear regression model equation to represent this data   • Graph the new equation.   • Decide whether the new equation is a "good fit" to represent this data.   • Extrapolate data:  If the ground temperature reached 95º, then at what approximate rate would you expect the crickets to be chirping?   • Interpolate data:  With a listening device, you discovered that on a particular morning the crickets were chirping at a rate of 18 chirps per second.  What was the approximate ground temperature that morning?    f) If the ground temperature should drop to freezing (32º F), what happens to the cricket's chirping?   Answers in this problem are to be rounded to the nearest thousandth.

  34. Example cont. Step 1.  Enter the data into the lists.  Step 2.  Create a scatter plot of the data.       Go to STATPLOT (2nd Y=) and choose the first plot.  Turn the plot ON, set the icon to Scatter Plot (the first one), set Xlist to L1 and Ylist to L2 (assuming that is where you stored the data), and select a Mark of your choice.Obviously, there is some scatter to this data. This variability is the norm, rather than the exception, when working with biological data sets.  Real life data seldom creates a nice straight line. Step 3.  Choose the Linear Regression Model.     Press STAT, arrow right to CALC, and arrow down to 4: LinReg (ax+b).  Hit ENTER.  When LinReg appears on the home screen, type the parameters L1, L2, Y1.  The Y1 will put the equation in to Y= for you.             (Y1 comes from VARS → YVARS, #Function, Y1)

  35. Example cont. Step 4.  Graph the Linear Regression Equation from Y1.     ZOOM #9 ZoomStat to see the graph. (answer to part b) Step 5.  Is this model a "good fit"?     The correlation coefficient, r, is .8364792791 which just barely places the correlation into the "strong" category.  (0.8 or greater is a "strong" correlation)     The coefficient of determination, r 2, is .6996975844 which means that 70% of the total variation in y can be explained by the relationship between x and y.  The other 30% remains unexplained.     Yes, it is somewhat of a "good fit". (answer to part c)

  36. Example cont. • Step 6.  Extrapolate:  (beyond the data set)If the ground temperature reached 95º, then at what approximate rate would you expect the crickets to be chirping?Go to TBLSET (above WINDOW) and set the TblStart to 20 (since the highest temperature in the data set had 19.8 chirps/second).  Set the delta Tbl to a decimal setting of your choice.  Go to TABLE (above GRAPH) and arrow up or down to find your desired temperature, 95º, in the Y1 column.            (answer to part d --  approx. 21.265 chirps per second)

  37. Example cont. • Step 7.  Interpolate:                            (within the data set)With a listening device, you discovered that on a particular morning the crickets were chirping at a rate of 18 chirps per second.  • What was the approximate ground temperature that morning?From the graph screen, hit TRACE, arrow up to obtain the power equation, type 47, hit ENTER, and the answer will appear at the bottom of the screen. (answer to part e --  the ground temperature will be approx. 84.407º F)

  38. Example cont. • Step 8. If the ground temperature should drop to freezing (32º F), what happens to the cricket's chirping? • The TABLE tells us that at 32º F there are 1.85 chirps per second.  So, what does this really mean?  Are the crickets cold? • These findings are a bit deceiving.  At 32º F, the crickets are dead.  The lifespan of a cricket in a cold climate is very short.  The crickets spend the winter as eggs laid in the soil.  These eggs hatch in late spring or early summer, and tiny immature crickets called nymphs emerge.  Nymphs develop into adults within approximately 90 days. The adults mate and lay eggs in late summer before succumbing to old age or freezing temperatures in the fall. • Also, remember that the further you extrapolate away from the data set, the less reliable the information will be.

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