6.1.1 How can I make predictions?

Transcription

1 CCA Ch 6: Modeling Two-Variable Data Name: Team: 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:

2 d. What does the slope of your line mean (in context)? What does the y-intercept of your line mean (in context)? 6-. Predict how wide (in yards) Robbie s field of view will be at the south end of the field. Show work: yds = in Predict how wide (in yards) Robbie s field of view will be at the north end of the field. Show work: Show work: yds = in 6-3. a. Label the dimensions on the football field picture on the next page. Shade the part of the field that Robbie can see. Calculate the area of Robbie s field of view.

3 b. How large is the whole field? What percent of the field will Robbie be able to see? (Show calculations.) c. How far is Robbie from the far goal line (in inches)? How much of the far goal line can Robbie see? What is the probability that Robbie will see the final touchdown? (Show calculations.)

4 Closure: How do we describe scatterplots? Why do we use a line of best fit (or trend line)? Are all models of situations perfect? Interpretation of slope: Interpretation of y-intercept: 6.1. How close is the model? Residuals Work Space: Equation:

5 Dear Battle Creek Cereal Executives, Sincerely,

6 6-11. What is the residual for 60 in? What is the difference between a positive and a negative residual (in context)? 6-1. Residual for 471 in? Mark this residual on the scatterplot (include units) a. What is the actual weight for a 600 in 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?

7 6-14. Let s = the amount of sugar (grams) per cup ; let c = # of calories per cup ; s c 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 c b. Meaning of slope (in context): Meaning of y-intercept (in context): Learning Log (Closure) RESIDUALS How do you calculate a residual? How do you draw a residual on a scatterplot? What does a positive residual mean? What does a negative residual mean? What is extrapolation?

8 6.1.3 What are the bounds of my predictions? Upper and Lower Bounds 6-. What information should you gather to answer the question? How could you gather the necessary information? Discuss with your team what statistical information you should include in your report to the anthropologist. Be sure to include all of that information below. Let x = Let y = Work Space: Graph and table are on the next page Equation: Dear Anthropologist, Sincerely,

9 185 x y

10 6-3. a. What point is the farthest from your line of best fit? (, ) What is the residual for that point? b. Draw a dashed line parallel to your line of best fit through the farthest point. Draw another dashed 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 6.4 cm: Between and. (Include units in your answers.) d. Closure:

11 6.1.4 How can we agree on a line of best fit? Least Squares Regression Line Graph is copied from Lesson Resource Page a. Draw a line of best fit. Calculate the equation for that line. Show your work: Equation:

12 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 (x = 103): 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? 6-3. SKIP Use applet in 6-34 instead What is the sum of all the residuals? Why would we need to square the residuals?

13 6-33. Using your Graphing Calculator BEFORE DOING THIS SECTION, RESET YOUR MEMORY. [ nd ] [+] 7:Reset 1:All RAM :Reset Your screen should say RAM cleared. a. How to make a Scatterplot: Enter data: [STAT] 1:Edit Enter data into L1 (x-values) and L (y- values). Check to see if data is entered correctly: [STAT] CALC 1:1-Var Stats [ nd ] [STAT] select the list you wish to check [ENTER] x = should match the checksum value for the list you chose. Graph your data: [ nd ] [Y=] 1:Plot1 [ENTER] On [ZOOM] 9:ZoomStat Sketch your scatterplot as accurately as possible. Label the x and y axis with names and label the min and max values from the WINDOW. How does your graph compare to the El Toro Basketball graph in problem 6-30? b. How to 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. Write these values down. Round to two decimal places unless directed otherwise. Graphing your line with your scatterplot: [Y=] enter the equation you wrote down in slope-intercept form What is the equation of your line? Sketch it onto the scatterplot you drew in part (a). c. How to see the Residuals in your calculator: Your calculator automatically creates a list named RESID every time it does a regression calculation. It will re-calculate this list every time you do [STAT] CALC 4:LinReg(ax+b) To set up the table with L1, L and RESID: [STAT] 5:SetUpEditor [ nd ] [STAT] select L1 [, ] [ nd ] [STAT] select L [, ] [ nd ] [STAT] select RESID [ENTER] [ENTER] To see L1, L and RESID data: [STAT] 1:Edit ** NOTE: The RESID list does not automatically change when you change L1 and L. When you change the values in L1 and L, you need to re-do STAT CALC 4: to get the updated residuals.

14 d. Use your LSRL to predict how many points Antonio Kusoc made (x = 103): e. Meaning of slope (in context): Meaning of y-intercept (in context): Why is this regression equation not reasonable for players playing less than 354 minutes? THIS PROBLEM COULD BE DONE TOGETHER AS A CLASS. If doing this problem on your own, go to (There is a link on Mrs. Fruchter s web page under Chapter 6, under section ) a. On the left side of the screen, click the circle by the #3. Position red line by grabbing and moving the red dots so that you think the line best fits the data. b. On the left side of the screen, click the circle by the #1 to show the residuals and their sum. (The sum is shown by #79 on the left.) Now re-position the line to reduce the sum of the residuals. What is the smallest sum of residuals you can get (Round to two decimals.)? c. Since there is sometimes more than one line that has the least sum of residuals, mathematicians minimize the sum of the squares of the residuals instead. Un-click the circle by #1. Click the circle by #0. Re-position the line again to make the squares as small as possible and reduce the sum of the squares residuals. (The sum is shown by #78 on the left.) What is the smallest sum of squares you can get? d. Click the circle by #70 to show the one line that minimizes the sum of squares. This is called the Least Squares Regression Line (or LSRL). How close did you get? Closure: Review steps for analyzing data.

15 6..1 When is my model appropriate? Residual Plots a. Let x = ( ) Let y = ( ) LSRL Equation: Graph: b. Follow directions in the book. Would you consider this point an outlier? Why? c. What is the impact of the outlier? Will Amy s predictions for the field of view be too large or too small? Explain.

16 6-48. a. Discuss with your team what elements a statistical analysis report should contain. b. Statistical Report: 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? a. On the graph you created in problem 6-48, sketch in the regression line. Then 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? Explain. b. Complete the following table using your calculator. Round residuals to two decimals. L L RESID What is the sum of your residuals? Why does that make sense?

17 c. How to make a Residual Plot: Data should already be entered into L1 and L. The LSRL should have already been calculated using STAT CALC 4: Graph your residual data: [ nd ] [Y=] 1:Plot1 [ENTER] On Xlist: L1 Ylist: RESID [ZOOM] 9:ZoomStat NOTE: To paste the RESID list into the Ylist prompt, go to [ nd ] [STAT]. All of your lists are there. NOTE: When you do [ZOOM] 9: you should see a horizontal line across the screen. That represents the LSRL, just rotated. If you do not see that horizontal line, go to [ nd ] [ZOOM] AxesOn. (If you have a TI-84 Color, the Axes: prompt should have the name of a color.) Is a linear model a good fit? How do you know? a. Scatterplot and LSRL: b. Residual Plot: b. What does the residual plot tell you? 6-5. a. Do you think a linear model is appropriate? Why or why not? b. Predicted number of farms: c. Actual number of farms: SKIP

18 6-54. LEARNING LOG Residual Plots What is the purpose of a residual plot? How do you interpret a residual plot? 6.. How can I measure my linear fit? Correlation Before starting this lesson, go to [ nd ] [0] and scroll down until you see DiagnosticOn. Hit [ENTER] twice your calculator should say Done CORRELATION COEFFICIENT r = r = r = r = r = r = r = Create your own scatterplot r = Create your own scatterplot r = Create your own scatterplot

19 What happens to r as the points get closer to a straight line? What is the largest possible value for r? What does that graph look like? What is the smallest possible value for r? What does that graph look like? Can r ever be zero? What does that graph look like? What does that tell you about the association between the variables? Further Guidance SKIP r 0.9 : r 0.6 : r 0.1: r 0.6 : LEARNING LOG Correlation Coefficient or r How does the value of r help you numerically describe the strength and direction of an association? a. r = Is the association strong or weak? b. Direction: Strength: Form: Outliers: c. Form: Direction: Strength:

20 6-7. EXTENSION If you are doing this on your own, go to (There is a link on Mrs. Fruchter s web page under Chapter 6, under section 6...) a. Grab the green dots on the graph and move them around to create scatterplots with the associations listed below. Sketch your graph in the space provided and record the values of r. Strong positive linear association: Weak positive linear association: r = Strong negative linear association: No linear association: r = r = r = b. First, re-create a scatterplot with a strong negative linear association (so that r 0.95 ). Then drag one of the points around to observe the effect one point can have on slope and y-intercept. Is it possible to change a strong negative linear association to a positive association by moving only one point? If so, sketch the before and after scatterplots below and record the values of r. Strong negative linear association: Change one point to make a positive association: r = r = Closure: What is the largest r value? What does that graph look like? What is the smallest r value? What does that graph look like? How is the value of r helpful to us?

21 6..3 Why can t studies determine cause and effect? Association is Not Causation a. What does the residual plot tell you? b. Worst prediction: (, ) Explain both the x- and y-coordinate (in context): c. Make a guess (and sketch) what you think the original scatterplot might have looked like. Label both axes a. Describe association: b. Why did the mayor say what he said? Do you agree with the mayor? Explain: a. Do you agree with the newspaper and the statements made? Explain. b. What could explain the association other than spinach makes you stronger?

22 6-8. Possible lurking variable(s): a. b. c. d. e Reasonable Statement: Misinterpretation of Association:

23 6..4 What does the correlation mean? Interpreting Correlation in Context a. R Use the example in the Math Notes box to write a complete sentence about what this value means in the context of the problem: b. What other factors can go into determining someone s height? 6-9. a. r = Why is the data unusual? What does r mean in context? b. What can Alyse say about the variability in height (use the complete sentence format)? What can she say about predicting height for a student? 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: a. Least Squares Regression Line: y-intercept (in context): b. r R

24 c. Improved Statistical Report: a. Least Squares Regression Line: meaning of slope (in context): meaning of y-intercept (in context): b. r R Sentence explaining variation (in context): Explain the difference between this R and the one in the previous problem: Interpretation of researcher s results: Does watching TV help you live longer? Explain:

25 6-98. LEARNING LOG Completely Describing Association 6..5 What if a line does not fit the data? Curved Best-Fit Models a & b. Scatterplot and LSRL: c. Linear Residual Plot: LSRL: c. Conclusions about a linear model: d. r R Sentence explaining variation (in context):

26 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 Quadratic R d. Which model is a better fit? a. Scatterplot and LSRL: Residual Plot: LSRL: 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: Sketch this curve on the scatterplot above. d. Based on the scatterplot, which model is a better fit?

27 e. Exponential Residual Plot: Appropriateness of the exponential model: a. Scatterplot and Quadratic Model: b. Residual Plot: a. Quadratic regression equation: 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: g. Which model is most useful? Justify your choice: SKIP