Lesson 76. Linear Regression, Scatterplots. Review: Shormann Algebra 2, Lessons 12, 24; Shormann Algebra 1, Lesson 94

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1 Lesson 76 Linear Regression, Scatterplots Review: Shormann Algebra 2, Lessons 12, 24; Shormann Algebra 1, Lesson 94 Tools required: A graphing calculator or some sort of spreadsheet program, like Excel (Microsoft), Numbers (Apple), or Google Spreadsheets. Definitions linear regression - The process of using statistical formulas to estimate the linear equation that best fits, or models, a set of data. coefficient of determination (R 2 ) - A value between 0 and 1 that indicates how well data fits a statistical model. The closer R 2 is to 1, the better the data fits the statistical model in question. extrapolate - to project, extend, or expand (known data or experience) into an area not known or experienced. 76A Linear Regression In Lesson 24A, you learned to create linear equations from a scatterplot, which means you had to first recognize that a set of data followed a linear pattern. Then, you had to draw the line that you thought best-fit the data, and create it s equation. We had two examples in Lesson 24A, where we also briefly discussed how the linear regression method is a tool that helps you find the one and only line that best fits the data. Linear regression works by using a set of statistical formulas to calculate the line that best fits a set of data. The formulas work much better than making a visual guess as to where the line should go. However, because the formulas that make up the algorithm are complex and tedious to use, we will introduce them later, in Shormann Precalculus. What we will do in Lesson 76 is use computer software programmed with the linear regression algorithms. Example 76.1 Use a graphing calculator or spreadsheet program to perform a linear regression on the following data. Find the coefficient of determination (R 2 ), too. x f(x)

2 solution: There are many possible calculator and computer options available for creating the linear regression, and we won t cover all of them here. We will use 1 spreadsheet option (docs.google.com/spreadsheets), and one graphing calculator option (TI 83/84). Spreadsheet algorithm: 1) Open a new spreadsheet, and type in the data as shown above. The x goes in cell A1, the f(x) in cell B1, the 1 in cell A2, and so on. 2) Click and drag to select all the data, including the x and f(x). If you selected it correctly, it should look highlighted, like this: 3) Look for an icon in the menu that looks like a bar graph, when you hover your mouse over it, an insert chart message should appear, like this: Click the icon. 4) You are now in the Chart Editor. Click Charts, then from the choices on the left, click Scatter, then click the Scatter chart (image with red and blue dots) option. You should see your graph appear to the right, as shown below: 5) Still in the Chart Editor, select Customize, scroll down to Trendline, and click Linear. Scroll down a little more, and next to Label, click Use Equation. Directly under Label, click the box to the left of Show R 2. You should now be able to see the line that best fits the data, it s equation, and it s R 2 value.

3 6) That s it! you can click Insert at the bottom left to insert the graph into your spread sheet. The most important thing though is that you record the linear equation and the R 2 value: y = 2.2x R 2 = The R 2 value for this example is Notice how all the data don t touch the trend line created above. If all the data touched the line, then the R 2 would be closer to 1. An R 2 =1 means there s a 100% chance the data fit a linear model. For our example, there is a 98.4% chance the data fit a linear pattern. Another way to think about it is that 98.4% of the variability in the data can be explained by the linear regression model. Other, unknown factors are responsible for the remaining 1.6% of variability. For linear regression models created from real data, scientists and other researchers consider a linear model to be a good model if R Sometimes, researchers will still use the model if it explains less than 95% of the variability, but they must acknowledge this fact when reporting the data to the public. Texas Instruments 83/84 graphing calculator algorithm: 1) Press STAT, then EDIT. You should see several columns, beginning with L1, and followed by L2. L1 is your x -data column, and L2 is your f(x)-data column. Use the DEL button to delete and numbers that may already be in these columns. Type in your data(not the x and f(x) titles) as it appears in the Example 76.1 table. Type in all your L1 data first, pressing ENTER after typing in each value. Use the arrow keys to scroll over to the L2 column, and type in your L2 data, pressing ENTER after typing in each value. 2) Press STAT again, then CALC, then scroll down to 4: LinReg(ax + b), and hit ENTER twice. You should now see a list of results appear, beginning with the formula y = ax + b, where a is the slope and b is the y-intercept. You should also see a value for r 2 (uses a lower-case R). You now have all the information you need, so record the equation and its R 2 value to 3 or 4 decimal places:

4 y = 2.2x R 2 = Obviously, this was supposed to be the same equation and R 2 as the spreadsheet method calculated, with the only differences due to rounding. NOTE: Did you have an R 2 value appear? If no, then do this: Press 2nd, then press zero (says CATALOG in yellow right above the button), scroll down to DiagnosticOn, and press ENTER twice. It should say Done on the screen. Now repeat Step 2 again, and your R 2 value should appear. Example 76.2 The following is actual data from an experiment on car speed (y-axis) versus time (x-axis). Use a graphing calculator or spreadsheet program to perform a linear regression on the following data. Find the coefficient of determination (R 2 ), too. Then, answer the following questions: a) Would this be considered a good model? Why or why not? b) Use the model to predict the speed at t = 6 s. Is this an appropriate value to use the model for? Why or why not? solution: Use either a spreadsheet or a graphing calculator (it doesn t matter) to find the equation and R 2. MAKE SURE you do this yourself and that you don t just lazily copy down the following results: equation: y = 3.10x , R 2 = This is actually the same data from Ex. 24.2, where we created a best-fit line by hand and the resulting equation was similar, y = 3.1x a) This would be considered a good model, because R b) Considering this as a function of speed vs. time, we could write s(t) = 3.10t s(6) = 3.10(6) = m/s This is not an appropriate value to use because it s an extrapolation. t=6 is a time that is outside the domain of the 0 to 5 seconds over which the model was created. We don t know what happened at t = 6 seconds: the speed could have continued on the same linear trend, or not.

5 Linear regression has the potential to be a very useful tool, but, like just about anything man creates, it can be abused. It is important to always clearly present your R 2 value when using a linear regression model. And, avoid extrapolating at all costs! 76B Scatterplots and Nonlinear Patterns Part 76B focuses on simply identifying scatterplots of some of the graphical forms of functions introduced in Lesson 12. Example 76.3 The following is a scatterplot from a model of mean total phosphorous (TP) as a function of water dilution inputs in Moses Lake in eastern Washington. Name the functional pattern that best describes scatterplot. Then, predict the TP concentration when the dilution input equals 50 x 10 6 m 3. Data is from Figure 6-1 of Restoration and Management of Lakes and Reservoirs, 2nd ed. solution: For the functions shown in the Lesson 12 Rules, the scatterplot best matches an exponential pattern. Note that, in similar problems, we will describe the scatterplot s pattern by its name, not by its symbolic form. Based on the pattern, a good estimate of TP is 83 μg/l for a dilution input of 50. Note scale on the y-axis. You can make an estimate by sketching a best-fit curve through the data, like this:

6 Practice Set 76 (subscripts tell you which lesson each problem came from) Use your best judgment as to when you should use a calculator. Use 3.14 for π Use a graphing calculator or spreadsheet program to perform a linear regression on the following data. Round the slope and y-intercept to 2 d.p. Find the coefficient of determination (R 2 ), too, rounded to 3 d.p. x f(x) The following is a hypothetical scatterplot representing the height of a projectile versus time. Name the functional pattern that best describes scatterplot. Then, predict the projectile s height at t = 1.5 seconds. Round to the nearest meter How would you describe the histogram shown? A) normal B) skewed left C) skewed right D) bimodal H is the standard deviation of a set of numbers {a,b,c,d,e}. For each of the following sets, which one(s) also have a standard deviation equal to H? A) {-a, -b, -c, -d, -e} B) {-2a,-2b,-2c,-2d,-2e} C) {2-a,2-b,2-c,2-d,2-e} D) {a/2,b/2,c/2,d/2,e/2}

7 5 73. Use the first derivative to find the x-location of the global extrema for the quadratic function k(x) = 3x 2-6x + 5. In other words, find x when k (x) = 0. Round to 1 d.p. Identify the extrema as either a max or min Find A, B, C and D for the graph of the sine function shown. Then, write the equation. Make note of the scale on the x and y axes. Angle units are in radians Show cos 2 x + sin x csc x = Which sinusoid function fits the data in the table? x 0 π/6 π/3 π/2 2π/3 5π/6 y A) y = 2 + 2cos x B) y = 3 + 3sin x C) y = 3 + cos x D) y = 4 + sin x Solve the following system. y = 4 x 2 y x = Use the formula developed in Example 59.4 to find the distance, x, from the top of the ship s conning tower to the horizon. Assume the conning tower height = 52.8 ft above the waterline (sea level), and the Earth s radius = 3,960 miles. Round to 1 d.p Find the hydronium ion concentration, [H+], in moles per liter (M), of an aqueous solution with a ph of 2. Write answer in scientific notation, rounded to 1 d.p Solve the following system. 2x 2y z = 15 3x 4y 2z = 33 x + 3y 5z = 24

8 (Accounting) The company s net profit (N) followed the exponential pattern N(t) = N 0 e 0.15t. If initial sales were $200K and expenses were $50k, what was the net profit after 5 years? Assume N = sales - expenses. A) e.75 B) 50e.75 C) 150e.75 D) 200e (Chemistry) The chemical formula for dry ice is CO 2. What is the mass of carbon in 5,000 grams of dry ice? Round to the nearest gram. (O=16, C=12) (CLEP College Mathematics) Each value in data set A is increased by 5 to form data set B. Which of the following is the same for sets A and B? A) mean B) median C) mode D) range Simplify. 12x of 87 equals what number? Simplify the complex fraction shown. h x Simplify Which of the following represents the graph of { x! x 5}? A) B) C) D)