CCA Ch 6: Modeling Two-Variable Data Name: 6.1.1 How can I make predictions? Line of Best Fit 6-1. a. Length of tube: Diameter of tube: Distance from the wall (in) Width of field of view (in) b. Make a scatterplot of your data. Be sure to label both axes and set up a reasonable and consistent scale on each axis.
b. Describe your scatterplot (direction, strength, form, outliers): c. Sketch a line of best fit on your scatterplot. Calculate your line of best fit. (Show all work.) Equation: d. What does the slope of your line mean (in context)? What does the y-intercept of your line mean (in context)? 6-2. Predict how wide (in yards) Robbie s field of view will be at the south end of the field. Show work: Predict how wide (in yards) Robbie s field of view will be at the north end of the field. Show work: 6-3. C O L L E G E T E C H N O S
a. Label the dimensions on the football field picture. Shade the part of the field that Robbie can see. Calculate the area of Robbie s field of view. b. What percent of the field will Robbie be able to see? (Show calculations.) c. What is the probability that Robbie will see the final touchdown? (Show calculations.) Closure: How do we describe scatterplots? Why do we use a line of best fit? Are all models of situations perfect? Interpretation of slope: Interpretation of y-intercept:
6.1.2 How close is the model? Residuals 6-10. Work Space: Dear Battle Creek Cereal Executives, Sincerely,
6-11. What is the residual for 260 in 2? What is the difference between a positive and a negative residual (in context)? How could graphing the actual and predicted data points help? 6-12. Residual for 471 in 2? Mark this residual on the scatterplot (include units). 6-13. a. What is the actual weight for a 600 in 2 box when the residual is 1005 g? Work space: b. Why do you suppose the residual is so large? c. Meaning of slope (in context): Meaning of y-intercept (in context): Does the y-intercept make sense (in context)? Why or why not? 6-14. Let s = the amount of sugar (grams) per cup ; let c = # of calories per cup ; s = 16.9 + 0.23c a. What does a negative residual mean (in context)? Would a negative or positive residual be better for Armen s diet? Why?
Let s = the amount of sugar (grams) per cup ; let c = # of calories per cup ; s = 16.9 + 0.23c b. Meaning of slope (in context): Meaning of y-intercept (in context): 6-15. Learning Log Residuals Closure: 6.1.3 What are the bounds of my predictions? Upper and Lower Bounds 6-22. What information should you gather to answer the question? What process could you use to gather this information? What statistical information could/should you report back to the anthropologist?
Let x = Let y = x y Dear Anthropologist, Sincerely,
6-23. a. What point is the farthest from your line of best fit? What is the residual for that point? b. Draw a line parallel to your line of best fit through the farthest point. Draw another parallel line the same distance from your line of best fit on the other side of your line. c. Possible height range for a humanoid with forearm length 26.4 cm: d. Closure: 6.1.4 How can we agree on a line of best fit? Least Squares Regression Line 6-30. Graph on the next page is copied from Lesson 6.1.4 Resource Page a. Draw a line of best fit. Calculate the equation for that line. Show work: b. Which data point is an outlier? (, ) Who is that data for? What is his residual? c. Would a player be more proud of a negative or positive residual? Why? d. Predict how many points Antonio Kusoc made:
6-31. What do you think about when deciding where to place a line of best fit? What makes one line a better model than another line? How can you numerically describe how close the prediction made by the model is to a player s actual total points? Why is thinking about absolute value important in this problem? 6-32. a. What is the sum of the squares of the residuals for your line of best fit? Player Name Actual Points Predicted Points Residual Residual 2 Sordan, Scottie 2491 Lippen, Mike 1496 Karper, Don 594 Shortley, Luc 564 Gerr, Bill 688 Jodman, Dennis 351 Kennington, Steve 376 Bailey, John 5 Bookler, Jack 278 Dimkins, Rickie 216 Edwards, Jason 98 Gaffey, James 182 Black, Sandy 185 Talley, Dan 36
6-33. Using your Graphing Calculator a. Make a Scatterplot: Enter data: [STAT] 1:Edit Enter data into L1 (x) and L2 (y). Check to see if data is entered correctly: [STAT] CALC 2:2-Var Stats Σx = should match the checksum value for L1 data. Σy = should match the checksum value for L2 data. Graph your data: [ZOOM] 9:ZoomStat b. Calculate your Least Squares Regression Line (LSRL): Calculate slope and y-intercept: [STAT] CALC 4:LinReg(ax+b) [ENTER] In the model (ax+b), a is the slope and b is the y-intercept. On your screen you will be given values for a and b. Write these values down and round them as directed. Then go to Y1 and enter your equation. Calculate slope and y-intercept AND have your calculator automatically save the equation into Y1: [STAT] CALC 4:LinReg(ax+b) [VARS] Y-VARS 1:Function 1:Y1 [ENTER] c. Residuals: Your calculator automatically creates a list named RESID every time it does a regression calculation. To see this list in your table with L1 and L2: [STAT] 5:SetUpEditor [2 nd ] [1] [, ] [2 nd ] [2] [, ] [2 nd ] [STAT] scroll down until you see RESID then hit [ENTER] [ENTER] d. Antonio Kusoc s predicted points: e. Meaning of slope (in context): Meaning of y-intercept (in context): Why is this regression equation not reasonable for players playing less than 350 minutes? 6-34. a. Go to http://sambaker.com/courses/j716/demos/leastsquares/leastsquaresdemo.html There is a link on Mrs. Fruchter s web page under Chapter 6, under section 6.1.4. b. On the right side of the screen, click on the boxes labeled Show Residuals and Residuals Sum. c. Drag the line up and down on either end to reduce the sum of residuals. What is the lowest sum of residuals you can get?
d. Click on the boxes labeled Hide Residuals and Hide Residuals Sum. Then click on the boxes labeled Mean Line, Show Squares and Show Squares Sum. e. Drag the line up and down on either end again to reduce the sum of the squares. What is the lowest sum of squares you can get? f. Click on the box labeled LS Line. How close were you to this line? 6.2.1 When is my model appropriate? Residual Plots 6-47. a. Let x = Let y = LSRL Equation: Graph: b. Follow directions in the book. Would you consider this point an outlier?
c. What is the impact of the outlier? Will Amy s predictions for the field of view be too large or too small? Explain how you know: 6-48. a. Discuss with your team what elements a statistical analysis report should contain. b. Statistical Report: 6-49. a. Scatterplot I: Residual Plot Scatterplot II: Residual Plot Scatterplot III: Residual Plot b. Which of the scatterplots does a linear model fit the data best? How do you know? 6-50. a. Draw a rough sketch of the scatterplot and its regression line from problem 6-48. Sketch the vertical residuals from each point to the regression line. If you want to purchase an inexpensive pizza, should you choose a pizza with a positive or negative residual? b. What is the sum of your residuals? Are you surprised? c. To create a residual plot: 2 nd Y= 1:Plot 1 On Type: Xlist: L1 Ylist: RESID ZOOM 9:ZoomStat If there is no horizontal line across the middle of the graph: 2 nd ZOOM AxesOn Is a linear model a good fit?
6-51. a. Scatterplot and LSRL: b. Residual Plot: b. What does the residual plot tell you? 6-52. a. Do you think a linear model is appropriate? Why or why not? b. Predicted number of farms: c. Actual number of farms: 6-53. Dear Sophie and Lindsey, Sincerely, 6-54. LEARNING LOG Residual Plots What is the purpose of a residual plot? How do you interpret a residual plot? How do you create a residual plot?
6.2.2 How can I measure my linear fit? Correlation 6-67. CORRELATION COEFFICIENT r = r = r = r = r = r = r = r = r = What happens to r as the points get closer to the line of best fit? What is the largest possible value for r? What is the smallest possible value for r? Can r ever be negative? Can r ever be zero? What does that mean? What does the value of r reveal about the strength of the linear association?
6-68. Further Guidance SKIP 6-69. r = 0.9 : r = 0.6 : r = 0.1 : r = 0.6 : 6-70. LEARNING LOG Correlation Coefficient or r How does the value of r help you numerically describe the strength and direction of an association? 6-71. a. r = Is the association strong or weak? b. Direction: Strength: Form: Outliers: c. Form: Direction: Strength: 6-72. EXTENSION a. Go to http://illuminations.nctm.org/lessondetail.aspx?id=l456#qs. There is a link on Mrs. Fruchter s web page under Chapter 6, under section 6.2.2. b. Create a scatterplot by clicking points onto the graph. After adding several points, click the Show Line box. This will reveal the LSRL, the number of points on the graph, the value of r, and the LSRL equation. To delete points, Ctrl-click on a point. To move points, Shift-click then drag the point. c. Strong positive linear association: Weak positive linear association: r = r =
Strong negative linear association: No linear association (random scatter): r = d. 5 points for strong negative linear association: r = Change one point to make a positive association: r = r = 6.2.3 Why can t studies determine cause and effect? Association is Not Causation 6-79. a. What does the residual plot tell you? b. Worst prediction: (, ) Difference between actual and prediction: c. Make a guess (and sketch) what you think the original scatterplot might have looked like. Label both axes. Explain why this is the worst prediction (in context): 6-80. a. Describe association:
b. Why did the mayor say what he said? Do you agree with the mayor? Explain: 6-81. a. Do you agree with the newspaper and the statements made? b. What could explain the association other than spinach makes you stronger? 6-82. Possible lurking variable: 6-83. a. b. c. d. e. 6-84. Reasonable Statement: Misinterpretation of Association:
6.2.4 What does the correlation mean? Interpreting Correlation in Context 6-91. a. b. Other factors: 6-92. a. r = Why is the data unusual? What does r mean in context? b. What can Alyse say about the variability in height? What can she say about predicting height for a student? 6-93. a. What do you notice about the pattern? Guess for r? Sentence about variability: b. What would the line of best fit look like? Equation: 6-94. a. Least Squares Regression Line: y-intercept (in context): b. r = r 2 = c. Improved Statistical Report:
6-95. a. Least Squares Regression Line: meaning of slope (in context): meaning of y-intercept (in context): b. r = r 2 = Sentence explaining variation (in context): Explain the difference between this r 2 and the one in the previous problem: 6-96. Interpretation of researcher s results: 6-97. Does watching TV help you live longer? Explain: 6-98. LEARNING LOG Completely Describing Association
6.2.5 What if a line does not fit the data? Curved Best-Fit Models 6-105. a & b. Scatterplot and LSRL: c. Linear Residual Plot: c. Conclusions about a linear model: d. r = r 2 = Sentence explaining variation (in context): 6-106. a. Why is it reasonable to assume that a quadratic model is better? b. Quadratic Regression Equation: b & c. d. Quadratic Residual Plot: Linear r 2 = Quadratic r 2 = d. Which model is a better fit?
6-107. a. Scatterplot and LSRL: Residual Plot: a. What does the residual plot tell you about Giulia s data? b. What type of function might fit this data better? c. Exponential regression equation: d. Based on the scatterplot, which model is a better fit? e. Exponential Residual Plot: Appropriateness of the exponential model: 6-108. a. Scatterplot and Quadratic Model: b. Residual Plot:
b. Appropriateness of the quadratic model? c. What model might fit this data better? d. Exponential regression equation: e. Based on the scatterplot, which model is a better fit? f. Exponential Residual Plot: Appropriateness of the exponential model: g. Which model is most useful? Justify your choice: 6-109. a. According to your exponential model, how much oil was produced in 1996? b. Why was your prediction so far off? c. Scatterplot and Exponential Model: Does your exponential model continue to fit the data?
d. Scatterplot from 1982 1996: What model do you believe will best fit this data? e. Linear regression equation for 1982 1996: f. Residual Plot from 1982 1996: Comment on the residual plot: g. According to your linear model, how much oil was produced in 1996? d. Piecewise scatterplot: Piecewise equations with domain restrictions: EQ1: Domain: EQ2: Domain: