Algebra I Name Unit #2: Linear and Exponential Functions Lesson #13: Linear & Exponential Regression, Correlation, & Causation Day #1 Period Date When a table of values increases or decreases by the same each time, it is. The general form for the equation of a line is, where m = and b =. #1. On the set of axes, graph the following table of values. x y -2-7 -1-4 0-1 1 2 2 5 3 8 Do the points appear to form a perfect line? What is the y-intercept of the line? What is the slope of the line? So, the equation of this line is But what do we do if we don t have a PERFECT line? Let s find out.
#2. On the set of axes, graph the following table of values. x y -4 10-2 4-1 3 0 1 2-2 3-5 4-6 Do the points appear to form a perfect line? Does it kinda sorta look like it forms a line? Since it doesn t form a perfect line, we can t use the same techniques we ve used in the past to find the equation for that line. We MUST use our calculator to determine an equation that is the best fit for our data. The method we use to find such an equation is called. If your data appears to look, you need to perform a linear regression. The equation you get is called the line of If your data appears to look, you need to perform an exponential regression. The equation you get is called the curve of #3 Which type of function (linear or exponential) would best model the data in each of the scatter plots shown below?
So how do we determine the equation for our data? We ve talked about performing regression before, back when we were writing the equation of exponential growth and decay functions. This CANNOT be done without a calculator! HOW TO PERFORM A LINEAR REGRESSION Step 1: Press 2 nd 0 (CATALOG) Step 2: Move the cursor down to DiagnosticOn Step 3: Press ENTER to select and press ENTER again to execute the command. Your calculator should say DONE Step 4: From the home screen, select STAT. Step 5: Click ENTER from the Edit option of the menu. Step 6: Enter the values of x in L1 and the values of y in L2. Step 7: Press STAT. Step 8: With the arrows, move the top cursor over to the option CALC and move the down cursor to 4: LinReg(ax + b), and then click ENTER. Step 9: With LinReg(ax + b) on the screen, move the cursor down to CALCULATE and press ENTER Step 10: The value of r, the correlation coefficient, should appear on the screen. #4 The table below shows the number of grams of carbohydrates, x, and the number of Calories, y, of six different foods. Which equation best represents the line of best for for this set of data? [August 2014 #21] (1) y = 15x (2) y = 0.07x (3) y = 0.1x 0.4 (4) y = 14.1x + 5.8
#5 A nutritionist collected information about different brands of beef hot dogs. She made a table showing the number of Calories and the amount of sodium in each hot dog. a) Write the linear regression equation for the line of best fit. Round all values to the nearest hundredth. b) Using this equation, predict the milligrams of sodium, to the nearest whole number, of a hot dog containing 230 calories.
Day #2 One of the reasons we even bother for perform regressions is to create an equation that best fits the data. Why bother? Once you find an equation that fits your data, and assuming that the trend continues, we can predict what future values might be. This is helpful in many real life applications. #1 Erica, the manager at Stellarbeans, collected data on the daily high temperature and revenue from coffee sales. Data from nine days this past fall are shown in the table below. State the linear regression function,, that estimates the day's coffee sales with a high temperature of t. Round all values to the nearest integer. Using your equation, predict the coffee sales, to the nearest dollar, on a day with a high temperature 32 degrees. Why do you think knowing an equation that relates temperature to coffee sales is important? Who might benefit from knowing this information and how might it be used?
#2 The table below shows the attendance at a museum in select years from 2007 to 2013. [Aug 2015 #36] State the linear regression equation represented by the data table when x = 0 is used to represent the year 2007 and y is used to represent the attendance. Round all values to the nearest hundredth Using this equation, predict the attendance (in millions) at the museum in 2017. Round your answer to the nearest tenth #3 The data table below shows the median diameter of grains of sand and the slope of the beach for 9 naturally occurring ocean beaches. [Aug 2016 #33] Write the linear regression equation for this set of data, rounding all values to the nearest thousandth. Using this equation, predict the slope of a beach, to the nearest tenth of a degree, on a beach with grains of sand having a median diameter of 0.65 mm.
Day #3 Last week, we worked on finding the equation for the line of best fit for data that appeared to be linear. But data is NOT always going to be linear! Today we are going to explore what to do with data that appears to be exponential. [Hint: it s pretty much the same thing!] #1 A rapidly growing bacteria has been discovered. Its growth rate is shown in the chart. Hours Since Observation Began Number of Bacteria in Sample 0 20 1 40 2 75 3 150 4 297 5 496 Prepare a scatter plot of the data with hours as the independent variable and the number of bacteria as the dependent variable. Determine which regression model will be the best fit for your data. Explain your reasoning. Write the regression equation for your model, rounding values to the nearest hundredth.
#2 An application developer released a new app to be downloaded. The table below gives the number of downloads for the first four weeks after the launch of the app. [June 2015 #36] Write an exponential equation that models these data. Use this model to predict how many downloads the developer would expect in the 26th week if this trend continues.
Day #4 Another way to look at data is by analyzing a scatter plot. This allows us to make some generalizations about the data we see plotted. Linear relationships can be described as either positive or negative. Below are two scatter plots that display a linear relationship between two numerical variables, x and y. The relationship displayed by Scatter Plot 1 is a positive linear relationship. As the value of the x-variable increases, the value of the y-variables tend to. If you were to describe this relationship using a line, would the line have a positive or negative slope? The relationship displayed by Scatter Plot 2 is a negative linear relationship. As the value of the x-variable increases, the value of the y-variables tend to. If you were to describe this relationship using a line, would the line have a positive or negative slope? In a positive linear relationship, as one variable variable., the other In a negative linear relationship, as one variable variable., the other
Some Linear Relationships are Stronger than Others Ex #1: Below are two scatter plots that show a linear relationship between two numerical variables x and y. Is the linear relationship in Scatter Plot 3 positive or negative? Is the linear relationship in Scatter Plot 4 positive or negative? It is also common to describe the strength of a linear relationship. We would say that the linear relationship in Scatter Plot 3 is weaker than the linear relationship in Scatter Plot 4. Why do you think the linear relationship in Scatter Plot 3 is considered weaker than the linear relationship in Scatter Plot 4?
Strength of Linear Relationships Ex #2: Consider the three scatter plots below. Place them in order from the one that shows the strongest linear relationship to the one that shows the weakest linear relationship. Strongest Weakest Explain how you determined the order of the scatter plots.
The Correlation Coefficient The correlation coefficient is a number between -1 and +1 (including -1 and +1) that measures the strength and direction of a linear relationship. The correlation coefficient is denoted by the letter r. Several scatter plots are shown below. The value of the correlation coefficient for the data displayed in each plot is also given. r = 1.00 r = 0.71 r = 0.32 r = -0.10 r = -0.32 r = -0.63 r = -1.00
The correlation coefficient is positive when as the x-values increase, the y-values also tend to. The correlation coefficient is negative when as the x-values increase, the y-values tend to. Is the linear relationship stronger when the correlation coefficient is closer to 0 or closer to 1? IMPORTANT PROPERTIES OF THE CORRELATION COEFFICIENT 1. The sign of r (positive or negative) indicates the direction of the linear relationship. 2. A value of r = +1 indicates a perfect positive linear relationship, with all points in the scatter plot falling exactly on a straight line. 3. A value of r = -1 indicates a perfect negative linear relationship, with all points in the scatter plot falling exactly on a straight line. 4. The closer the value of r is to +1 or -1, the stronger the linear relationship. Another thing to think about when looking at relationships is causation. Does something cause something else to happen? Ex #3: Beverly did a study this past spring using data she collected from a cafeteria. She recorded data weekly for ice cream sales and soda sales. Beverly found the line of best fit and the correlation coefficient, as shown in the diagram below. Given this information, which statement(s) can correctly be concluded? I. Eating more ice cream causes a person to become thirsty. II. Drinking more soda causes a person to become hungry. III. There is a strong correlation between ice cream sales and soda sales. 1) I, only 2) III, only 3) I and III 4) II and III
Day #5 Earlier, we spent some time with linear regression. We entered all the data in the calculator and were given the best equation for that data. For the next two days, we are going to revisit how to perform a linear regression but this time, we are also going to find the correlation coefficient (hint: it s super easy). CALCULATING THE VALUE OF THE CORRELATION COEFFICIENT Step 1: Press 2 nd 0 (CATALOG) Step 2: Move the cursor down to DiagnosticOn Step 3: Press ENTER to select and press ENTER again to execute the command. Your calculator should say DONE Step 4: From the home screen, select STAT. Step 5: Click ENTER from the Edit option of the menu. Step 6: Enter the values of x in L1 and the values of y in L2. Step 7: Press STAT. Step 8: With the arrows, move the top cursor over to the option CALC and move the down cursor to 4: LinReg(ax + b), and then click ENTER. Step 9: With LinReg(ax + b) on the screen, move the cursor down to CALCULATE and press ENTER Step 10: The value of r, the correlation coefficient, should appear on the screen. Here is the nice thing once you turn your DiagnosticOn, it stays on. You don t need to do this every time but if you do a linear regression, and the r value does not appear, you have to know how to switch it on. Ex #3: Enter the shoe length and height data into your calculator. Find the value of the correlation coefficient between shoe length and height. Round your answer to the nearest tenth Shoe Length (x) Height (y) inches inches 12.6 74 11.8 65 12.2 71 11.6 67 12.2 69 11.4 68 12.8 70 12.2 69 12.6 72 11.8 71
If the value of the correlation coefficient is between You can say that r = 1.0 There is a perfect positive linear relationship. 0.7 r < 1.0 There is a strong positive linear relationship. 0.3 r < 0.7 There is a moderate positive linear relationship. 0 < r < 0.3 There is a weak positive linear relationship. r = 0 There is no linear relationship. 0.3 < r < 0 There is a weak negative linear relationship. 0.7 < r 0.3 There is a moderate negative linear relationship. 1.0 < r 0.7 There is a strong negative linear relationship. r = 1.0 There is a perfect negative linear relationship. That is a lot to read and remember, right? You don t need to completely memorize this Just remember that if the correlation coefficient is close to 1 or -1, the more the data looks like a line. If the correlation coefficient is close to zero, the data looks more random. Ex#2: The table and scatter plot below display the fat content (in grams) and number of calories per serving for 16 fast-food items. Fat Calories (g) (kcal) 2 268 5 303 3 260 3.5 300 1 315 2 160 3 200 6 320 3 420 5 290 3.5 285 2.5 390 0 140 2.5 330 1 120 3 180 Based on the scatter plot, do you think that the value of the correlation coefficient between fat content and calories per serving will be positive or negative? Based on the scatter plot, estimate the value of the correlation coefficient between fat content and calories per serving. Using your graphing calculator, calculate the value of the correlation coefficient. Round your answer to the nearest hundredth.
Ex #5: The same study also collected data on sodium content (in mg) and numbers of calories per serving for the same 16 fast food items. The data is represented in the table and scatter plot below. Sodium (mg) Calories (kcal) 1042 268 921 303 250 260 970 300 1120 315 350 160 450 200 800 320 1190 420 570 290 1215 285 1160 390 520 140 1120 330 240 120 650 180 Based on the scatter plot, do you think that the value of the correlation coefficient between sodium content and calories per serving will be positive or negative? Explain why you made this choice. Based on the scatter plot, estimate the value of the correlation coefficient between sodium content and calories per serving. Using your graphing calculator, calculate the value of the correlation coefficient. Round your answer to the nearest hundredth. For these 16 fast-food items, is the linear relationship between fat content and number of calories stronger or weaker than the relationship between sodium content and number of calories. Does this surprise you? Why or why not?
Day #6 #1 A nutritionist collected information about different brands of beef hot dogs. She made a table showing the number of Calories and the amount of sodium in each hot dog. [Jan 2015 #35] a) Write the correlation coefficient for the line of best fit. Round your answer to the nearest hundredth. b) Explain what the correlation coefficient suggests in the context of this problem. #2 Erica, the manager at Stellarbeans, collected data on the daily high temperature and revenue from coffee sales. Data from nine days this past fall are shown in the table below. [Jan 2016 #35] a) State the linear regression function, f(t), that estimates the day's coffee sales with a high temperature of t. Round all values to the nearest integer. b) State the correlation coefficient, r, of the data to the nearest hundredth. Does r indicate a strong linear relationship between the variables? Explain your reasoning.
#3 The table below shows the attendance at a museum in select years from 2007 to 2013. [Aug 2015 #36] a) State the linear regression equation represented by the data table when x = 0 is used to represent the year 2007 and y is used to represent the attendance. Round all values to the nearest hundredth. b) State the correlation coefficient to the nearest hundredth and determine whether the data suggest a strong or weak association.