Summer 2006 I2T2 Probability & Statistics Page 122

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6 Summer 2006 I2T2 Probability & Statistics Page 126 S6 - Stats on the TI-84 Normal Distribution Problem: Use Prob. Sim.: Roll a die 1,000 times. Record the results and draw a histogram displaying the data Problem: Roll a pair of dice 1,000 times. Record the results and draw a histogram displaying the data. L 1 { } L 2 { }

7 Summer 2006 I2T2 Probability & Statistics Page 127 Find the following probabilities when rolling a pair of dice. 1. P(2) = 2. P(3) = 3. P(4) = 4. P(5) = 5. P(6) = 6. P(7) = 7. P(8) = 8. P(9) = 9. P(10) = 10. P(11) = 11. P(12) = Complete the chart and construct a histogram of the expected frequencies for each outcome when a pair of dice is rolled 1,000 times. L 1 { } L 2 { }

8 Summer 2006 I2T2 Probability & Statistics Page 128 L 1 { } L 2 { } Normal Distribution: mean = mode = median " Bell- Shaped Curve " The Rule

9 Summer 2006 I2T2 Probability & Statistics Page 129 Example: 1. Suppose the IQ scores of 12-year-olds are normally distributed with a mean of 100 and a standard deviation of 16. Jessica is a 12-year-old and has an IQ score of 132. What proportion of 12-year-olds have IQ scores less than Jessica's score of 132? normalcdf!"-1 99, 132, 100, 16#$ Example: 2. The finishing times for swimmers performing the 100-meter butterfly are normally distributed with a mean of 55 seconds and a standard deviation of 5 seconds. The sponsors decide to give certificates to all those swimmers who finish in under 50 seconds. If there are 48 swimmers entered in the 100-meter butterfly, approximately how many certificates will be needed?

10 Summer 2006 I2T2 Probability & Statistics Page 130 Problems: 1.) The scores on a 100-point exam are normally distributed with a mean of 80 and a standard deviation of 6. A student's score places him between the 69th and 70th percentile. Which of the following best represents his score? (1.) 66 (2.) 81 (3.) 84 (4.) 86 2.) A survey of the soda drinking habits of the population in a high school revealed the mean number of cans of soda consumed per person per week to be 20, with a standard deviation of 3.5. If a normal distribution is assumed, find an interval that contains the total number of cans per week approximately 95% of the population of this school will drink. Explain why you selected that interval.

11 Summer 2006 I2T2 Probability & Statistics Page ) Homes in Chicago, Illinois, in the price range of $95,000 to $130,000, are on the market for an average of 70 days before selling. Assume that the distribution of days on the market in Chicago for homes in this price range is approximately normal with a standard deviation of 25 days. Homes in Fargo, North Dakota, in the same price range are on the market for an average of 150 days. Also assume that the distribution of days on the market in Fargo for homes in this price range is approximately normal with a standard deviation of 50 days. (a.) Sketch the distribution of days on the market for both homes in Fargo and in Chicago (in this price range) on the same axis. (b.) Indicate on your sketch what area will correspond to the proportion of homes in Chicago (in this price range) that sell in the first month (that is, in less than 30 days); then find the proportion

12 Summer 2006 I2T2 Probability & Statistics Page 132 FISH IN THE FINGER LAKES Fishermen in the Finger Lakes Region have been recording the dead fish they encounter while fishing in the region. The DEC monitors the pollution index for the Finger Lakes Region. The data table below shows this information for the past ten years. Year Pollution Index Deaths ) a) Enter the data and produce a scatter plot. Use the Index as the x-variable and the deaths as the y-variable. b) What is the meaning of slope in this model? c) Determine the equation of the line of best fit. d) Predict the number of dead fish for a pollution index of 15. e) What pollution index would result in the death of 150 fish? * using the STAT, EDIT feature of the TI84+, the Index will be placed in L1 and the Deaths in L2. * using the window shown STAT PLOT 1 (2nd Y=) is turned on and following the directions in (a) L1 contains the Index (x) and L2 contains the number of deaths (y). * using the WINDOW, set the xmin/max and ymin/max appropriately. * using the GRAPH, show the scatter plot. 2) Discussing which mathematical model best fits this data can be a very rich classroom conversation. When working with paper and pencil, this can be demonstrated using a piece of "linguine." In this model the slope of the line of best fit is:

13 Summer 2006 I2T2 Probability & Statistics Page 133 Deat hs Pollution deaths per unit of pollution. * To determine the line of best fit using the calculator the STAT CALC feature is used and #4 LinReg is selected. By entering the arguments L1, L2, (2nd #1, 2nd #2) we are assured of the correct data sets being used in the calculation. Then we have to tell the calculator where to place the regression line VARS ; Y- VARS 1 (Function), Y 1 ENTER. * In order to graph the Linear Regression Line just found we select the Y= menu and place our cursor at the function location of our choice. * using the GRAPH, the scatter plot gives us a mathematical model with which to make predictions. Using this model you can answer parts (d) and (e). FISH IN THE FINGER LAKES 1. Enter Data 2. Stat Plot 3. Stat Plot 4. Zoom Stat 5. Graph 6. Guess 7. Graph 8. Linear Reg 9. ax + b 10. Vars 11. Line Place 12. Y2 placed 13. Y2 Enter 14. Lin Reg 15. Y1 & Y2 On 16. Graph 17. Y2 only 18. Answer d 19. 2nd Calc. 20. x = Delete outlier 22. Guess 23. Graph nd D

14 Summer 2006 I2T2 Probability & Statistics Page Diag. On 27. TI84+ Line 28. ax + b 29. place line 30. Y4 place 31. Correlate 32. Y3 &Y4 33. Graph 34. Y4 only 35. Zoom out 36. x = 15 Final Answer x = 15 y =

15 Summer 2006 I2T2 Probability & Statistics Page 135 DESCRIBING ONE-VARIABLE DATA *Topic 1 Histograms and Frequency Tables from Raw Data (Explorations: pgs ) a) set up the plot b) set up the window 1. Send & Receive 2. Phill 3. Transmit 4. Receive 5. Stat Edit 6. Stat Plot 7. Zoom 9 8. Trace 9. Window 10. Nice Limits 11. X-scl= X-scl= Class Limits 14. (L1+L2)/2 15. Frequency 16. Relative Freq. 17. Mod. Box Plot 18. Outliers 19. Mod. vs. Box 20. Side by Side c) Why does City Hall appear to be taller after you change the Xscl = 20?

16 Summer 2006 I2T2 Probability & Statistics Page 136 *Topic 6 Box Plots and Five-Number Summary (Explorations: pg. 21) a) What is the Five-Number Summary? min# Q1 med Q3 max#

17 Summer 2006 I2T2 Probability & Statistics Page 137 S7 - MODELING WITH THE TI-84 Investigating Elasticity Objective: To have students test their understanding of the equation of a line to fit data from an experiment. Concepts/Skills: Students perform an experiment, collect data, graph the data and fit an equation of a line to the data. Needs: Paper cup String to hang cup Rubber band or slinky Stand or yardstick supported across two chairs About 50 pennies, either pre-1982 or post Ruler or measuring tape Getting started: 1. Prepare a cup by punching holes and running string across the top. Attach either a rubber band or a slinky to the string. If using a rubber band, attach a paper clip to the top. 2. Hang the cup from a ring stand or from a yardstick so that it is hanging freely. The textbook suggests taping the paper clip to the edge of a table so that it hangs over the edge. This is fine as long as neither the cup nor the rubber band is hitting the table. 3. If using the edge of the table, you can tape a ruler or measuring tape to the edge near the cup. Have 0 at the top of the rubber band. Otherwise, use a ruler, yardstick or measuring tape to measure the stretch of the rubber band or slinky. 4. Record the distance from the top of the rubber band or slinky to the bottom of the cup. This is the initial distance. 5. Add 10 pennies to the cup. Record the number of pennies in the cup and the distance from the top of the rubber band or slinky to the bottom of the cup. 6. Repeat Step 5 several more times until you have 50 pennies in the cup. Each time, record the number of pennies in the cup and the distance from the top of the rubber band or slinky to the bottom of the cup. pennies distance

18 Summer 2006 I2T2 Probability & Statistics Page 138 Investigating the data: 1. Make a scatter plot of the data you have collected. Describe any patterns you see. 2. Find an equation for a best-fitting line. You may want to use a graphing calculator or computer. 3. Explain what the y-intercept and the slope mean in terms of your data. 4. Describe a reasonable domain and range of your equation. Write inequalities to represent both. Making conjectures: 5. Make a conjecture about what the distance from the top of the rubber band or slinky to the bottom of the cup would be if you placed 100 pennies in the cup. 6. Test your conjecture. What relationship do you see between the number of pennies used and the length of the stretch? (Actually this is not a linear relationship as suggested by the book. After a certain point, the rubber band will not continue to stretch at the same rate.) 7. Make a conjecture about how the length of the rubber band or slinky might affect the total distances. Would the equations be the same or different? 8. How would your graph be different if you measured from the floor up to the bottom of the rubber band or slinky? (Using the CBR, you are measuring from the floor to the bottom of the cup. Note: Using Hooke s Law to Explore Linear Functions by Chris McGlone and Gary Nieberle, Mathematics Teacher, May 2000, is a similar activity. Using the Graphing Calculator: 1. Make a scatter plot of the data you have collected. a. Using the graphing calculator, press STAT EDIT, enter. Clear the lists L1 and L2. Enter the values for the number of pennies in L1. Enter the corresponding distances in L2. b. Go to STAT PLOT by pressing 2nd Y= Press ENTER to go to Plot 1. Press ENTER to turn Plot 1 on. Select the scatter plot (first choice). For Xlist enter L1, for Ylist enter L2. Choose one of the first two marks. c. Press ZOOM then choose 9:ZoomStat. d. Describe any patterns you see. 2. Find an equation for a best-fitting line. a. Go to CATALOG (2nd 0). Press D (ALPHA x 1 ). Arrow down to DiagnosticOn. Press ENTER twice, the word Done should be on the screen. b. Go to STAT CALC. Arrow down to 4:LinReg(ax+b). Press ENTER. After LinReg(ax+b) enter L1 (2nd 1), then comma (above 7), then L2 (2nd 2). Now press VARS Y-VARS, choose 1:Function, then 1:Y1. On the main screen press ENTER. You should see values for a and b. The values for r and r 2 indicate how closely the equation fits the data. The closer r and r 2 are to 1 or 1, the better the fit. c. The equation has been pasted into Y1. Press graph to see a graph of the line. Compare the line to the plotted data.

19 Summer 2006 I2T2 Probability & Statistics Page 139 Area and Perimeter: Building a Garden Fence Adapted from Graphing Calculator Activities for Enriching Middle School Mathematics, Browning & Channell, Texas Instruments, Concepts/Skills: Geometry, measurement, dimensions, area, maximum area given a fixed perimeter. Needs: Graphing calculators, toothpicks or raffle tickets, grid paper or dot paper, student worksheet, calculator instruction sheet. Getting started: Tell students the following story: You and a friend are visiting her grandparents on their small farm. They have asked the two of you to design a small, rectangular-shaped vegetable garden along an existing wall in the backyard. They wish to surround the garden with a small fence to protect their plants from small animals. W W L To enclose the garden, you have 24 sections of 1 meter long rigid border fencing. In order to grow as many vegetables as possible, your task is to design the fence to enclose the maximum possible area. How many sections of fencing should you use along the width and the length of the garden? There are many rectangular shapes that can be formed using the 24 fencing sections and, before the digging begins, you should do some calculations. Procedure: Have students use their manipulatives (toothpicks or raffle tickets) to represent the sections of border fencing. They should fill out the first three questions on the student worksheets. Once they have answered the initial questions and filled out the table, work with students to enter the data into the lists on the calculator, graph the results, and answer the remaining questions on the Question sheet.

20 Summer 2006 I2T2 Probability & Statistics Page 140 Additional Explorations: The following explorations use the list capabilities of the graphing calculator to investigate the situations. In each case, students should produce a scatterplot of the widths and areas. Students will continue to use the 24 sections of fencing in forming the border. 1. A friend suggests that you plant your grandparent's garden at the back corner of the yard so that the existing fence can border two of the four sides of your garden. What are the dimensions of the garden with the largest possible area? Is this configuration an improvement over the original plan? Explain your reasoning. W L 2. Suppose the garden were placed at the corner of a barn so that it was positioned as shown below. The garden has an L-shaped configuration as it wraps around the corner of the bard. What dimensions would give the largest garden area? barn W W L L

21 Summer 2006 I2T2 Probability & Statistics Page 141 Student Questions for Garden Fence 1. If you were to use three sections of fencing along each width of the garden, how many sections would remain to form the length? What will be the area of this garden? Copy these values into the Table below, and then enter three more possible garden sizes into the table. Try to guess the width and length of the garden with the largest possible area. Compare your results with others in your class. Possible Dimensions of Garden Fence Width (m) Length (m) Area (m 2 ) 3 2. If you know what the width is, how can you find the length? Write an equation that shows this relationship between width and length. 3. The smallest number of fencing pieces you can use along the garden width is one. What is the largest number of pieces that you can use along the width of the garden? Explain how you know this. Use calculator instruction sheet to enter L1 and L2 in your calculator. 4. How can the values for L3 (the areas) be determined from L1 and L2? Remember that L1 stores the possible widths and L2 stores possible lengths. Enter L3 into your calculator. 5. Scroll through the values in L3. Are the values you computed earlier contained in this list? Describe any patterns you see in the data values contained in L3.

22 Summer 2006 I2T2 Probability & Statistics Page Examine the third list to find the dimensions of the rectangular garden that has the largest possible area. Complete the following sentence to provide a solution to the original question: A rectangle with a width of meters and a length of meters gives the largest possible garden of square meters. Return to instructions on Displaying a Graph. 7. When creating the scatterplot of the areas, you entered the settings shown at right for the display window. Why do you think these values were used? 8. You used TRACE to move through the data points in the scatterplot. Which point corresponds to the maximum area? What sets it apart from the other points on the plot? 9. How do any patterns that you observed in the lists show up in the scatterplot of the data?

23 Summer 2006 I2T2 Probability & Statistics Page 143 Instructions for using the Calculator for Garden Fence 1. To clear the first three lists in the calculator, press STAT 4:ClrList ENTER, and then press 2 nd [L1], then a comma, 2 nd [L2], then a comma, 2 nd [L3] ENTER. You will use L1 to store the possible widths and then calculate values for the corresponding lengths and areas. Once you calculate the values, you will store the lengths in L2 and the areas in L3. 2. Press STAT 1:Edit and press ENTER. Note: If L1, L2, and L3 are not visible, then press STAT 5:SetUpEditor ENTER. Then repeat step Enter the whole numbers from 1 to 11 (the largest possible width) into L1 on your calculator by typing each number and pressing ENTER until all widths have been entered. (See the example below.) You should have noticed in your earlier computations that the number of fencing pieces remaining for the length of the garden can be found by subtracting twice the number used for a width from the 24 fencing pieces available. We want to store in L2 the lengths that correspond to the widths in L1. 4. Press to move to the second list. Press to move to the top so that L2 is highlighted. Press Enter. 5. Enter [L1] as the definition for L2. Your display should look like the example at the right. 6. Press ENTER and the column of possible lengths should appear in L2. Answer #4 on the Question sheet. This will help you find an expression to enter for L3. 7. Press to move to the third list and then press to move to the top so that L3 is highlighted. Press Enter. 8. Enter 2 nd [L1] x 2 nd [L2] ENTER to enter the expression for L3 that you found in question 4. Answer #5 and #6 on the Question sheet.

24 Summer 2006 I2T2 Probability & Statistics Page 144 Displaying a Graph for Garden Fence A scatterplot is often used to present a visual display of the relationship between two sets of paired data like the width and area measurements. Your calculator can produce a scatterplot display of the numbers in its lists. 1. Press 2 nd [STAT PLOT] ENTER. Press Enter to turn on Plot 1. Edit the window so that yours looks like the one at the right. To make a selection, press to move the blinking cursor on top of the desired location and then press ENTER. 2. Press WINDOW and edit the numbers so that your window looks like the one shown at the right. 3. Press Y= and clear any equations in any of the lines. Press GRAPH to view a scatterplot. The horizontal axis represents the garden widths and the vertical axis shows the corresponding areas. 4. You can view the coordinates of each plotted point by pressing TRACE followed by the left and right arrow keys. Answer #7, #8, and #9 on the Question sheet.

25 Summer 2006 I2T2 Probability & Statistics Page 145 A Fence for Fido: Area & Perimeter Adapted from T3 MSM Summer Institute, Texas Instruments, Concepts/Skills: Geometry, measurement, dimensions, area, maximum area given a fixed perimeter. Needs: Graphing calculators, toothpicks or raffle tickets, grid paper or dot paper, Student question sheet. Getting started: Suppose we want to make a rectangle with a perimeter of 12. What are the possible rectangles that could be made? Procedure: Divide the students into small groups. Each group will need 12 toothpicks or raffle tickets, to represent the perimeter of a rectangle, and a sheet of grid paper. As a group, students are to arrange the toothpicks/tickets into every possible rectangular shape to represent the given perimeter of 12. Sketch each figure on the grid paper. Record the dimensions of each rectangle. Compile the data from every group into one chart (similar to the one below). Figure the area of each figure. Length Width Perimeter Area Present the problem: Tell students the following story: Fido, our mutt, keeps running away and we need to fence him in. The fencing costs $3 a foot and we have only $126 to spend. How much fencing material can we buy? Fido is a very hyperactive dog. We need to build the largest possible rectangular pen so he has plenty of room to run. What do the dimensions of the pen need to be? a. Sketch the possible rectangles on the grid paper b. In the cooperative group, compile all of the data on a chart c. As a class compile all of the data into one chart. 1. Look for patterns, compare width and length, compare dimensions with the area. 2. Discuss the maximum area of the dog pen. Are there solutions that are not whole numbers that would give us a bigger area? Have students enter the class data in a graphing calculator using lists. Instructions are basically the same as for the Garden Fence. First have them clear L1, L2, L3, L4.

26 Summer 2006 I2T2 Probability & Statistics Page 146 Discuss how they can write an expression for the length, width, perimeter, and area in terms of the length. Listed are examples and there may be other equivalent expressions that are valid. The amount of fence we can afford would be $126/$3 = 42, which would be the perimeter. Then half the perimeter is 21. They can calculate L2 using 21 length, or use (42 2*length)/2. a. L1 = length b. L2 = width (L2 = 21 c. L3 = perimeter (L3 = 2L d. L4 = area (L4=L1* Using the lists, what is the maximum area of Fido's pen? What are the dimensions of the pen? Students can view a graphical solution to the problem with a scatterplot. Instructions are basically the same as for the Garden Fence. Ask students which two lists they should use to provide a graphical solution to the problem. Have them explain why. a. x-axis (length) = L1 b. y-axis (area) = L4 Have them predict what the graph will look like based on the information in the lists. Selection of the window values also provide good information on the students' understanding of the constraints of the problem. Using TRACE compare the ordered pairs to the corresponding values in the lists. (See questions on Student Question sheet.) If the length = x, then how can you represent the area as a function of x? area = x(21-x) (See questions on Student question sheet. ) Students can also view the relationship between the length and area by using the table feature of a graphing calculator. By changing the ΔTbl, they can zoom in on the desired value for x and y.

27 Summer 2006 I2T2 Probability & Statistics Page 147 See Writing Activities, questions 11 and 12 on Student Question sheet. One possible answer to #11 is that a long rectangular pen would give Fido more length to run. See Related Extension Activities, questions 13, 14, 15 on Student Question sheet.

28 Summer 2006 I2T2 Probability & Statistics Page 148 Student Questions for Fido's Fence Fido, our mutt, keeps running away and we need to fence him in. The fencing costs $3 a foot and we have only $126 to spend. How much fencing material can we buy? Fido is a very hyperactive dog. We need to build the largest possible rectangular pen so he has plenty of room to run. 1. How much fencing can we buy? How can we calculate this? 2. What do the dimensions of the pen need to be? a. Sketch the possible rectangles on the grid paper. b. In the cooperative group, compile all of the data on a chart. c. As a class compile all of the data into one chart. i. Look for patterns, compare width and length, compare dimensions with the area. ii. Discuss the maximum area of the dog pen. Are there solutions that are not whole numbers that would give us a bigger area? 2. How can you write an expression for the length, width, perimeter, and area in terms of the length? a. length = x b. width = c. perimeter = d. area = If L1 = length, then what would you enter for the formulas for the following? a. L2 = width = b. L3 = perimeter = c. L4 = area = 3. Using the lists, what is the maximum area of Fido's pen? What are the dimensions of the pen? 4. To see a graphical solution for this problem, which two lists should you use? Explain why. 5. What should the window settings be? Explain why. 6. Predict what the graph will look like based on the information in the lists.

29 Summer 2006 I2T2 Probability & Statistics Page Using TRACE compare the ordered pairs to the corresponding values in the lists. a. Discuss how well they predicted the shape of the graph. b. Why are there two points with the same area? c. Estimate the maximum area. What are the length and width for this area? d. Estimate the dimensions of the figure if the area is 75. (Use the arrow keys.) 8. If the length = x, then how can you represent the area as a function of x? Enter the area function in Y1. Graph this function along with the scatter plot. a. Press the TRACE key. Press or to change to the graph instead of the plot. b. What do the x and y represent in the display? c. Verify the maximum area from your previous estimation in 7c. d. Verify the dimensions of a figure with an area of 75 (from 7d). e. What happens to the area as the length approaches zero? f. What does the graph of the continuous function show us that the scatter plot cannot? 9. Use the table feature to view the relationship between the length and area. Set up the table and estimate the dimensions when the area is 75. a. Using the TBLSET allows us to get closer to the area approximation by changing the size of ΔTbl. Start with TblStart=1 and ΔTbl=1 and scroll to about 75. Keep changing these values until you get closer to Y1=75. b. Use [TABLE] to visualize the maximum area. 10. What will be the dimensions of Fido's dog pen? 11. What are some reasons for not building a pen that has a maximum area? 12. Your friend has asked you how to find the maximum area of a rectangle with the perimeter of 50. Write the directions for finding this.

30 Summer 2006 I2T2 Probability & Statistics Page After all of your work, you now found out that you can use the side of the house for one of the sides of the pen. What is the maximum area you can fence? What are the dimensions? 14. You have found fencing materials at You-Save-A-Lot Building Supply for $2.25 a foot. What is the maximum area of a rectangular pen you could build while still spending the same amount of money ($126)? What are the dimensions? 15. Fido still needs a pen and we have only 42 feet of fence. But now consider that the fence can bend. Does the rectangular pen still provide the maximum area given the fixed perimeter? (Should we consider a different shape?)

31 Summer 2006 I2T2 Probability & Statistics Page 151 Box It Up: Volume Adapted from Graphing Calculator Activities for Enriching Middle School Mathematics, Browning & Channell, Texas Instruments, Concepts/Skills: Maximum volume of an open box, dependent and independent variables. Needs: Graphing calculators, sheet of graph paper for each student or group of students, material to fill boxes to measure volume (puffed rice, popcorn, etc.), scissors, tape, rulers Getting started: Tell students the following story: Ms. Hawkins, the physical sciences teacher at Buffalo High School, needs several opentopped boxes for storing laboratory materials. She has given the industrial technologies class several pieces of metal sheeting to make the boxes. Each of the metal pieces is a rectangle measuring 30 cm. by 50 cm. The class plans to make the boxes by cutting equal-sized squares from each corner of a metal sheet, bending up the sides, and welding the edges. Procedure: To better understand this problem, have the class start by making paper models of the kind of box Ms. Hawkins has requested. a. Using graph paper, where each square represents 1 cm 2, mark off a rectangle 30 squares x 50 squares. b. Mark off equal-sized squares at each of the corners and then cut out the squares. c. Fold the side tabs up and tape each edge together to form an open-topped box. d. Determine whose box has the greatest volume. Fill each box with puffed rice, popcorn, or some other material. Pour into a large container and mark the volume of each box. Which box has the greatest volume? e. Compare the general size and shape of your box with the boxes made by others in your classroom. Note that the size of the square cut from each corner of the paper determines the dimensions of the final product. What happens to the height, length, and width of the box as the size of the cut squares gets larger? You can see that the industrial technologies class could make several differently shaped boxes. Students should answer questions 1-5 on the Student worksheet. See Student notes for directions for setting up the table. Answers to worksheet questions follow below in italics. 1. Height:5 cm, Length: 40 cm. Width: 20 cm, Volume: 4000 cm 3 2. Height: 2 cm, Length: 46 cm, Width: 26 cm, Volume: 2392 cm 3 3. Does cutting out and throwing away less sheet metal necessarily result in a box with a larger volume? Explain. from previous examples 4. What is the largest possible value for the length of the square? Explain why no larger value could be used. 14 cm

32 Summer 2006 I2T2 Probability & Statistics Page 152 Height (cm) H Length (cm) L Width (cm) W Volume (cm 3 ) V h 50 2h 30 2h (50 2h)(30 2h)h

33 Summer 2006 I2T2 Probability & Statistics Page Write a sentence that would describe the meaning of the two numbers in the first row of the table. When a square of 1 cm is cut out from each corner of the rectangular piece of metal, a box with a height of 1 cm and a volume of 1344 cm 3 can be produced. 8. Describe any patterns you see in the sequence of volumes as you continue to scroll to the value x = 14. For example, as the size of the cut squares increases, what happens to the volume of the box? The volume increases to a maximum value of 4104 cm 3 at x=6 cm. After that the volume decreases to a low of 616 cm 3 at x = 14 cm. If you continue to scroll past 14 cm, you get zero and negative volumes. 9. What size of square cut (value of x) gives a box with the largest volume (value of Y1)? Height: 6 cm, Length: 38 cm, Width: 18 cm, Volume: 4104 cm What are the dimensions and volume of the box that the industrial technologies class should make for Ms. Hawkins? Height: 6 cm, Length: 38 cm, Width: 18 cm, Volume: 4104 cm You should have found that the box, of largest volume, is made by cutting squares with sides of length 7 cm. However, this result assumes the height x is restricted to whole number values. What if we could now make cuts to tenths of a centimeter in the sheet metal? Between what two whole number heights do you believe the maximum volume would be found? (Do not have to be sequential numbers.) Write down those two values along with their corresponding volumes. X = 5, Y = 4000 and X = 7, Y = Discuss these choices in your group and provide an argument to show why the maximum volume must occur for some x between your two chosen values. The volume has been gradually increasing from X=1 to 6, then decreasing from X=6 to 14. So the maximum must be between 5 and Describe any surprises as you scrolled through the table of values. Students often anticipate that is will be halfway between 5 and 6 or 6 and What is the new best value for x? 6.1cm 15. What is the volume of the best box? cm Is the volume of this box significantly larger than the one with a height of 6 cm? no 17. In summary for this numerical look at volume, write three key points or ideas your group has found while determining the maximum volume for the boxes Ms. Hawkins needs for her physical sciences class. Sample ideas: Volume of box increases and decreases as you increase the height; we were able to find the volume knowing only one dimension; we were able to write an equation relating height and volume; relationship between height and volume is not linear.

34 Summer 2006 I2T2 Probability & Statistics Page 154 Extension A graphical look at Box It Up entails using the same equation, setting the window, then analyzing the graph, zooming in on the maximum point of the graph. See student notes for directions. Answers are in italics below. 1. Why must x have a value between 0 and 15? Because 1 cm is the smallest cut and at 15 the box has no volume. 4. What are the values of x and y at the highest point on the graph? x= , y= Record these x-coordinates , Record the coordinates from the screen. Sample values: x= , y= By observing the x-coordinates of the points on either side of this point, you can again determine an interval that contains the desired solution. What x-interval contains the value associated with the maximum height of the graph? Sample interval (to nearest hundredth) is 6.04, Write the new interval for the maximum volume. Determine whether you can provide an x-value accurate to the nearest tenth. Remember, you want all numbers in your interval to round to the same tenth s value. Sample interval is 6.064, Based on the results from your trace of the graph, what size squares should the class cut from the pieces of sheet metal? What are the dimensions and volume of the box with the largest volume? Side of square:6.1. Height: 6.1. Length: Width: 17.8, Volume: You have now investigated the box volume problem using two methods: tabular and graphical. Discuss in your groups the advantages of both methods. Write those advantages below. Some prefer the tabular approach with the ΔTbl feature making zooming in fairly easy. Others prefer a picture and would rather zoom in on the graph for a maximum value. Sometimes when viewing a table of values, it is difficult to get a sense of how the function behaves over a larger interval; the graph provides a bigger picture. One can find a maximum value fairly quickly on the table without having much sense about how the function behaves. When creating a graph you have the additional challenge of defining the window, which requires some knowledge of the behavior of the function.

35 Summer 2006 I2T2 Probability & Statistics Page 155 Student Worksheet for Box It Up Ms. Hawkins, the physical sciences teacher at Buffalo High School, needs several opentopped boxes for storing laboratory materials. She has given the industrial technologies class several pieces of metal sheeting to make the boxes. Each of the metal pieces is a rectangle measuring 30 cm by 50 cm. The class plans to make the boxes by cutting equal-sized squares from each corner of a metal sheet, bending up the sides, and welding the edges. To better understand this problem, start by making paper models of the kind of box Ms. Hawkins has requested. a. Using graph paper, where each square represents 1 cm 2, mark off a rectangle 30 squares x 50 squares. b. Mark off equal-sized squares at each of the corners and then cut out the squares. c. Fold the side tabs up and tape each edge together to form an opentopped box. d. Determine whose box has the greatest volume. Fill each box with puffed rice, popcorn, or some other material. Pour into a large container and mark the volume of your box. Which box has the greatest volume? e. Compare the general size and shape of your box with the boxes made by others in your classroom. Note that the size of the square cut from each corner of the paper determines the dimensions of the final product. What happens to the height, length, and width of the box as the size of the cut squares gets larger? You can see that the industrial technologies class could make several differently shaped boxes.

36 Summer 2006 I2T2 Probability & Statistics Page 156 Questions 1. If the class decided to cut 10-cm squares from the corners of the metal sheets, determine the dimensions and volume of the box that could be formed from the remaining sheet metal. Size of square: 5 cm Height: cm Length: cm Width: cm Volume: cm 3 2. What would be the dimensions and volume of the box if 5-cm squares were cut from the corners of the metal sheets? Size of square: 2 cm Height: cm Length: cm Width: cm Volume: cm 3 3. Does cutting out and throwing away less sheet metal necessarily result in a box with a larger volume? Explain. 4. Squares of many different sizes can be removed from the corners of the metal sheets. If cuts were made in whole centimeter lengths, the smallest square would have a side of length 1 cm. What is the largest possible value for the length of the square? Explain why no larger value could be used. 5. Ms. Hawkins decides she wants to have boxes made with the largest possible volume. How could the class determine which size square to cut out so that a box with the largest volume is produced? If you enter the information you found earlier in a table, it may help you find a relationship between the dimensions of the box. Notice that the size of the cut-out square determines the height of the box. Complete all but the last line of the table.

37 Summer 2006 I2T2 Probability & Statistics Page 157 Height (cm) H Length (cm) L Width (cm) W h Volume (cm 3 ) V Since you can choose the height of the box, H, height is called the independent variable. The other dimensions depend upon your choice of H, so they are dependent variables. When you pick H, the resulting length, L, is 50 2H, for example 50 cm 2(5 cm) = 40 cm. The width, W, is found by using the expression 30 2H. Finally, the volume, V, is the product of all three dimensions, of L*W*H. Substitute in the other expressions for L and W in the product for finding volume if the height = h: V = L*W*H = ( )( )h. Complete the last row of the table using the expressions you have found for L, W, and V in terms of h. 6. How would you calculate the volume of a storage box if you knew only the height of the box?

38 Summer 2006 I2T2 Probability & Statistics Page 158 Using the Calculator You can use the graphing calculator to generate a table of values. Then the box with the largest volume can be found by searching through these values. In order to generate a table of values, you must provide the calculator with three pieces of information: An expression written using only one variable, x. The value of x to begin the table calculations. The size of the step to be used in moving from one value of x to the next. For this situation, the expression we have written is (50 2h)(30 2h)h. We need only to refer to the variable height as x instead of h. So, V = (50 2x)(30 2x)x. We decided that the smallest cut-out square would have a side length of 1 cm, which means that the table should start at x = 1. The value of x should increase by 1 each time the next table entry is calculated, so the table will contain whole number values for the cuts. a. Press Y=. If there are any expressions listed, press the arrows to each line and press CLEAR. If any Plots are highlighted, arrow up to them and press ENTER to turn them off. Enter the expression (50 2x)(30 2x)x for the volume of the box. b. Press 2 nd [TblSet] to access the Table Setup window. Enter the TblStart = 1, the smallest cut. Enter the ΔTbl = 1, the increase in the x value. Make sure that Auto is highlighted on the last two lines. c. Press 2 nd [TABLE] to see the first few rows of the table. 7. Write a sentence that would describe the meaning of the two numbers in the first row of the table.

39 Summer 2006 I2T2 Probability & Statistics Page 159 Even though the window display shows only 7 lines of the table, the up-down arrows can be used to move up and down the display. Use the down arrow to move past the last line in the display. 8. Describe any patterns you see in the sequence of volumes as you continue to scroll to the value x = 14. For example, as the size of the cut squares increases, what happens to the volume of the box? 9. What size of square cut (value of x) gives a box with the largest volume (value of Y1)? Height: cm Length: cm Width: cm Volume: cm What are the dimensions and volume of the box that the industrial technologies class should make for Ms. Hawkins? Height: cm Volume: cm 3 Length: cmwidth: cm 11. You should have found that the box, of largest volume, is made by cutting squares with sides of length 7 cm. However, this result assumes the height x is restricted to whole number values. What if we could now make cuts to tenths of a centimeter in the sheet metal? Between what two whole number heights do you believe the maximum volume would be found? (Do not have to be sequential numbers.) Write down those two values along with their corresponding volumes. X = X = Y = Y = 12. Discuss these choices in your group and provide an argument to show why the maximum volume must occur for some x between your two chosen values. Set TblStart to the smaller of the two X numbers you argued for above. Set ΔTbl to.1 so that the change between x values will not be tenths of a centimeter. Construct a new table. 13. Describe any surprises as you scrolled through the table of values. 14. What is the new best value for x?

40 Summer 2006 I2T2 Probability & Statistics Page What is the volume of the best box? 16. Is the volume of this box significantly larger than the one with a height of 6 cm? 17. In summary for this numerical look at volume, write three key points or ideas your group has found while determining the maximum volume for the boxes Ms. Hawkins needs for her physical sciences class. a. b. c. A Graphical Look at Box It Up You have already built a table showing the volumes that resulted when squares of certain sizes were removed. Another approach often used to solve such problems is to generate a graph of the relationship between box height and volume on a coordinate grid. To graph a relationship using a graphing calculator, it is necessary to carry out two tasks: Define the relationship in terms of two variables, x and y. Define the portion of the coordinate plane over which you wish to view the graph. If you have not cleared your Y= screen, you should still have Y1 defined as (50 2x)(30 2x)x. That is the first task. The second task is to define the portion of the coordinate plane that you wish to view, referred to as the window on the calculator. Press the WINDOW key. The values for Xmin and Xmax are the left and right endpoints of the viewing window. The values for Ymin and Ymax are the lower and upper limits of the viewing window. Xscl and Yscl are used to define the distance between the tick marks on the two axes; they do not affect the viewing window or appearance of the graph.

41 Summer 2006 I2T2 Probability & Statistics Page 161 Now define the limiting values for the viewing window considering the equation (50 2x)(30 2x)x and think about the values used to answer the problem using the table. Given the conditions of the problem, x must take on the values between 0 and Why must x have a value between 0 and 15? If tick marks are desired, you can set Xscl to 1 and have tick marks at every unit along the x-axis. Defining the limits on y is a little more challenging. A given value of Y1 represents the volume of a box for some height x. Volume cannot be negative, so you can define Ymin to be slightly less than 0. You are looking for the x that would produce the largest value of y possible, but this doesn t indicate how large y can get. However, your previous work with tables showed that all volumes were less than 4200 cubic centimeters, so a reasonable value for Ymax is 5000 (to give room to display the function at the top of the screen). Since the height of this view is very large, the Yscl should be large. Approximately 20 tick marks per axis is about right, and 5000/20 = 250, so enter 250 for Yscl so that tick marks on the y-axis are placed at 250, 500, 750,, When entering negative values, be certain to use the (-) key and not the minus key. Enter the following values into WINDOW. Xmin = 1, Xmax = 17, Xscl = 1, Ymin = 10, Ymax = 5000, Yscl = 250 Press the GRAPH key. A graph similar to the one on the right should appear. 2. In your group, write a brief explanation of why the graph should look the way it does based on your previous work with the volume table and the patterns you observed there.

42 Summer 2006 I2T2 Probability & Statistics Page 162 Press the TRACE key once to read coordinates of points on this graph. Note that the blinking crosshair and the coordinates are displayed at the bottom of the screen. This informs you that the point where x = 8 and y = 3808 lies on this view of the graph. 3. Do the coordinates of points on the graph agree with the values you computed earlier in this activity? Explain what these two values mean with respect to the box volume problem. Use the left and right arrows to move the blinking crosshair. The crosshair will move only along the graph, as long as you are in TRACE, and not just anywhere in the coordinate plane. Notice that the coordinates at the bottom of the screen change as you press the arrow keys. The x-coordinates are determined by the calculator based upon the current values of Xmin and Xmax. The y-coordinates are computed using the expression (50 2x)(30 2x)x. You will probably see that the x-values are not always nice numbers, unlike those you saw when you generated the tables. Move the cursor until you locate the point on the graph that gives the largest y-coordinate. 4. What are the values of x and y at the highest point on the graph? x = y = Not all of the decimal places shown in the coordinates of this point are meaningful. In fact, this point may not be the one you are looking for. The screen is restricted by the number of pixels that can be shown, so this point is the closest point on this view of the graph. By examining the x-coordinates of the points on each side of this point, you can find an interval that contains the best value of x. Use the arrow keys to determine the x-coordinates of the points immediately to the left and to the right of the point found above. 5. Record these x-coordinates. Point to the left: Point to the right: The solution you are looking for lies somewhere between these two x- values. Check the table of values to see that the tabular solutions fall somewhere between the two values above. (Press 2 nd [TABLE] to access the table of numbers.)

43 Summer 2006 I2T2 Probability & Statistics Page 163 Just as we did when we used the tables to investigate this problem, it is easy to examine the volume function over a smaller interval in order to obtain a more precise approximation to the desired solution. This can be done in a number of ways, but the fastest is to use ZOOM. Press the ZOOM key and consider the two options Zoom In and Zoom Out. Zooming in is similar to bringing a portion of your current view closer for a finer examination. Zooming out is equivalent to enlarging your field of view so that you see more of the graph, but less detail. Since you want a closer, more detailed look at the points near the apparent maximum if the graph, you must zoom to the region containing that maximum. A quick way to do this is to use the option ZBox. ZBox lets you use the cursor to select opposite corners of a box to define the portion of the current view that you wish to enlarge. Press ZBox. You will see a different crosshair in the center of the screen. Use the arrow keys to move the cursor somewhere above and to the left of the highest point of the graph. Once you are satisfied with the cursor s location, press ENTER. Now move the crosshair to the right and down until you reach the opposite corner of the box you are forming. As you do this, you will see the box forming on the screen. Try to center the point of interest in the box. One possible zoom box is shown on the right. When you are satisfied with the location of the second corner, press ENTER again. The calculator should redraw the graph in the window you have just defined. Use TRACE and the arrow keys again to locate where you believe the maximum value to be. 6. Record the coordinates from the screen. x = y = 7. By observing the x-coordinates of the points on either side of this point, you can again determine an interval that contains the desired solution. What x-interval contains the value associated with the maximum height of the graph? If the tenths position in the x-interval is not the same for both values, or if both x-values when rounded to the nearest tenth are not the same, then the best choice for x we can make has to be a whole number. If you zoom one more time on the place where you believe the maximum volume will be, you will be able to provide a value for x that is accurate to the nearest tenth.

44 Summer 2006 I2T2 Probability & Statistics Page 164 Press ZOOM and select ZBox again. Make another box around the area on the graph where you think the maximum volume lies. Once you ve made the box, press ENTER. A new graph appears looking something like the graph displayed. (Remember, we haven t necessarily chosen identical zoom boxes.) Press TRACE to find new x-values for an interval containing the maximum volume as you have done previously. 8. Write the new interval for the maximum volume. Determine whether you can provide an x-value accurate to the nearest tenth. Remember, you want all numbers in your interval to round to the same tenth s value. 9. Based on the results from your trace of the graph, what size squares should the class cut from the pieces of sheet metal? What are the dimensions and volume of the box with the largest volume? Side of square: Length: Volume: Height: Width: 10. You have now investigated the box volume problem using two methods: tabular and graphical. Discuss in your groups the advantages of both methods. Write those advantages below. 11. Which method do you prefer and why?

45 Summer 2006 I2T2 Probability & Statistics Page 165 S9 - Statistics with Graphing Calculators Exploring the different graphs possible: NY Standards: 7.S.4, 5, 6; A.S.4, 5, 6, 7, 8, 14; A2.S.6 1. Enter data: Go to STAT Edit. To enter the column labels, to L1, press 2nd INS to insert a new column. Enter ST for Student, then Enter. Enter the numbers 1, 2, 3, 4, 5, 6, 7, 8 for the students. Student Verbal Score Math Score Angella Bob Carol David Jean Lisa Peter Matthew Similarly insert and label the column for Verbal Scores VS and the column for Math Scores MS. Enter the SAT scores from the chart above. Press 2nd QUIT to return to the home screen. 3. Statistical computations: Go to STAT, CALC, select 1-Var Stats, press Enter to the home screen.. Then 2nd LIST to VS, Enter, Enter. These same computations are available individually under 2nd STAT MATH. then to mean(, stddev(, median(, min(, max(.

46 Summer 2006 I2T2 Probability & Statistics Page 166 Record the statistics in the table below. Repeat for the Math Score. Record the Mean: Record the Standard Deviation: Record the Median: Record the Minimum: Record the Maximum: Calculate the Range: Verbal Score Math Score 4. Box Plots: First press Y= and clear out any functions there. Then go to 2nd STAT PLOT, select Plot 1, Enter. Press Enter to turn Plot 1 On, to Type, then to the box plot icon (also called box and whiskers), Enter. Then to Xlist. Go to 2nd LIST to VS, Enter. Press ZOOM, then to ZoomStat. You now see the box plot for the verbal scores. Repeat the above steps, using Plot 2, for the Math Scores. Go to ZoomStat again to graph the two box plots together. (It has to resize the window for the second plot.) What statistics are used to create the box plots? Use TRACE to help determine this. Use the arrows to go between plots. The "whiskers" are the long lines extending from each box. A point that is at an extreme, where the distance between that point and the median is greater than 1.5 times the distance between Q1 and Q3, is called an "outlier." Looking at the data, do you think that one of the whiskers is exceptionally long due to an outlier? You might have noticed in the STAT PLOT menu there were two box plots. The one with a dot at the end of one whisker does not connect outliers, but shows them as separate points.

47 Summer 2006 I2T2 Probability & Statistics Page 167 Shows no outliers. 5. Calculations with the Data: Return to the statistics editor. (STAT EDIT). Make a new list called CS for combined scores. Position the cursor on the new label and press ENTER. You should have a blinking cursor at the bottom after the name. Enter the following: Go to 2nd LIST to MS, Enter. Then type +, then go to 2nd LIST to VS, Enter, Enter. The CS column now has a list of the sums of each score. If you arrow up to the name of the list, you will see {946, 1140, 1047, }. If you change anyone's score in MS or VS, CS will have to be recalculated because it contains a list, not the formula. To attach a formula to CS, after CS= type a " quotation mark, then follow the above instructions to enter the formula. When you press Enter, the same values appear in the column, but when you arrow up to the name, you will see the formula instead of the list. Now if you change anyone's score in MS or VS, the value in CS changes automatically. What is Angella's (1) combined score? Who has the highest combined score? 6. Histograms: Turn all the plots off by going to the STAT PLOT menu and choosing 4: PlotsOff. Press ENTER until you see Done on the home screen. Now go back to the STAT PLOT (or PLOT) menu and choose Plot 1 again. Turn Plot 1 on, choose the histogram. First graph the Verbal Scores, then the Math Scores, then the Combined Scores. Remember to use ZoomStat to get a good window. (Note that you cannot graph more than one histogram at a time with the graphing calculator.)

48 Summer 2006 I2T2 Probability & Statistics Page 168 To increase the number of columns, go back to WINDOW and change the Xscl to 10. Trace to see the range of each column and the number of students in that range. How might a school guidance counselor use this graph? 7. Sorting: To display the combined scores in ascending order, remember that we have to include the other three lists too to keep the correct scores with the correct person. Go to STAT, or go to 2nd LIST, OPS. Choose SortA(, then go to 2nd LIST to CS, Enter, comma, then go to 2nd LIST to ST, Enter, comma, then go to 2nd LIST to VS, Enter, comma, then go to 2nd LIST to MS, Enter, ) Enter. Now go back to STAT Edit. What are the benefits of sorting? How is this useful? Who has the highest combined score? 8. Scatter Plot: A scatter plot can give information on whether the two scores are related. For example, if the points seem to go from lower left to upper right, we consider the scores to have a positive correlation. In other words, there is some relationship between getting high math scores and high verbal scores. On the other hand, if the general trend is from upper left to lower right, there is a negative correlation. This would suggest that students, who score high in math, score low in verbal, and vice versa. Do you believe there should be a positive, negative, or no correlation between math and verbal SAT? We can test this conjecture with our small sample.

49 Summer 2006 I2T2 Probability & Statistics Page 169 Go to STAT PLOT, turn off all other plots except for Plot 1. Type should be scatter plot (dots), Xlist: VS, Ylist: MS, any Mark you prefer. Press ZOOM, ZoomStat and look at the graph. Is there a correlation? Does it make a difference if Xlist is MS and Ylist is VS? Switch them and regraph. 9. Determine the regression line: Go to STAT CALC, choose LinReg(ax+b), Enter. Now whatever is currently your Xlist (2nd LIST to MS), Enter, comma, whatever is currently your Ylist (2nd LIST to VS), Enter, comma, VARS, Y-VARS, Function, Y1, Enter, Enter. On the home screen, you should see the calculated values for a and b, and the correlation coefficient r. If you do not see a value for r and r 2, then do the following: Press 2nd CATALOG (above 0 on TI-84.) Arrow down to DiagnosticOn, press Enter. Then press 2nd ENTRY twice to get back to the LinReg command, and press Enter. The closer r is to +1 or 1, the better the fit between the function and the data. The r-value for this function tells you that a line is not a particularly good match for this data. Now press GRAPH to see the regression line graphed with your data. Because the regression line has a positive slope, you can see that your data has a positive correlation.

50 Summer 2006 I2T2 Probability & Statistics Page 170 Problem #9 The World's Fastest Men & Women NY Standards: 7.S.4, 6, 8, 12; A.S.7, 8, 17; A2.S.6, 7 Is there a pattern to the times recorded for the gold medallists in the men's 100-meter run from past Olympics? If there is a pattern, can you determine what the winning time in Barcelona, 1992 might have been? Don't look up the winning time yet! What about Atlanta in 1996, Australia in 2000, and Greece in 2004? A scatterplot is a good way to see if there is a pattern in the relationship between the year of the Olympics and the gold medal winning time. This scatterplot is easily created by hand, with computer software, or with graphing calculators. Before getting started with the graphing calculator, check the MODE settings and put them all on the default settings (the ones on the left). Entering data: First clear all lists by pressing 2nd MEM ClrAllLists. Now enter the data (STAT EDIT). Under L 1, enter the year of each entry and press ENTER after each one. Then go to the top of the List L 2 and enter the time for each year in the same order. When finished press 2nd QUIT to return to the home screen. Men s 100 Meter Run Gold Medallists for the Modern Olympic Games Year Name Time (Sec.) 1896 Thomas Burke, United States Francis W. Jarvis, United States Archie Hahn, United States Reginald Walker, South Africa Ralph Craig, United States Charles Paddock, United States Harold Abrahams, Great Britain Percy Williams, Canada Eddie Tolan, United States Jessie Owens, United States Harrison Dillard, United States Lindy Remigino, United States Bobby Morrow, United States Armin Hary, Germany Bob Hayes, United States Jim Hines, United States Valeri Borzov, USSR Hasely Crawford, Trinidad Allan Wells, Great Britain Carl Lewis, United States Carl Lewis, United States 9 99 Setting the window: Before drawing the scatterplot, set the WINDOW because the scatterplot is drawn on the same graphics screen you have used to graph functions. Our X data range from 1896 to 1988, so set Xmin to 1890 and Xmax to 2000 (or similar figures) and Xscl to 20. Because the winning times range from 12 down to 9.99, set Ymin to 9 and Ymax to 13 and Yscl to 0.5. Drawing the scatterplot: First clear all functions out of the Y= screen. Next go to STATPLOT by pressing 2nd, then Y=. At Plot 1, press ENTER. Under Plot 1, choose ON, then for TYPE choose the first icon for scatterplot. Next choose L 1 for Xlist and L 2 for Ylist (unless you named them something else). 2nd 1 for L1, 2nd 2 for L2.

51 Summer 2006 I2T2 Probability & Statistics Page 171 In order to see your points better, choose the first or second icon under Mark. Then press the GRAPH key. Finding the line of best fit: The TI-84 can determine very quickly the regression line for these data. From the Home screen (2nd QUIT), press STAT. Then under CALC, choose LinReg(ax +b). L1 and L2 are the default lists, so you do not have to enter anything unless you named the lists something else. To graph the line of best fit, we need to enter it into Y1. To do this, while still on the home screen, before pressing Enter after LinReg(ax+b), do the following: Press VARS, then arrow to Y-VARS, choose Function, then choose Y1. Press ENTER and you will see values for a and b. Record these values below. a = b = The value of b represents the y-intercept, a represents the slope of the regression line. Now press GRAPH and you will have the scatterplot and the line of best fit. What part of the data does b represent? How can we interpret the value of a with respect to the data? Is a increasing or decreasing? What does that mean with respect to running times? With this mathematical model you can now make a number of predictions about Olympic times. 1. What might have been a good winning time in Barcelona? There are several ways to answer that question. One is to trace the function, get X as close to 1992 as you can, and read Y. Another is to store 1992 into X, go to VARS, Y- VARS, choose Y1, and press Enter. A third method is to use the TABLE menu. 2. Now go to a reference book or the Internet. Who won in Barcelona? What was the actual winning time? 3. What time might have been expected for the winner at the Atlanta Games in 1996? 4. What was the actual winning time? 5. Notice from the table on the previous page that there is no time listed for some years. Why not? Use your mathematical model to predict what the winning times might have been, had there been Olympics in those years. 6. Using the model, what should we have expected for Australia in 2000? What was the actual winning time? 7. What should we expect for 2004 in Greece? What was the actual winning time?

52 Summer 2006 I2T2 Probability & Statistics Page 172 Fat Versus Calories NY Standards: 7.S.4, 6, 8; A.S.7, 8, 13, 14, 17; A2.S.6, 7 The following table contains the number of calories and grams of fat for selected fast foods. Does the scatterplot of these variables for each item look linear? Is there a positive correlation between fat and calories the more fat, the higher the number of calories? Fast food item Grams of fat Calories Burger King Whopper McDonald's Big Mac Wendy's Big Classic Arby's Roast Beef Hardee's Roast Beef Roy Roger's Roast Beef Burger King Whaler McDonald's Filet-O-Fish Arby's Chicken Breast Sandwich Burger King Chicken Tenders Church's Fried Chicken (2 pc.) Hardee's Chicken Filet Sandwich Kentuckv Fried Chicken (2 Pc.) Kentucky Fried Chicken Nuggets McDonald's Chicken Nuggets Roy Roger's Chicken (2 pc.) Wendy's Chicken Fillet Sandwich To Graph, remember to do the following: Clear all graphs in the Y= screen and turn off all other STAT PLOTS. Enter the data as sets of ordered pairs with x as the number of grams of fat and y as the number of calories. Select an appropriate viewing window, such as [10,40] by [200,600]. Choose Xscl and Yscl so you don't have more than 20 hash marks on each. Choose LinReg(ax +b) and put your equation in Y 1 or Y2. 1. The standard form of the regression line uses a as slope and b as the y-intercept, so an approximate equation of a line through the data is: y = 2. What kind of correlation is there between fat and calories positive, negative, or neither? 3. What does b, the y-intercept, mean in regards to fat and calories? What does the value of a tell us regarding fat vs calories?

53 Summer 2006 I2T2 Probability & Statistics Page If you found out that a particular meal at Perkins had 575 calories, what would you expect the grams of fat to be? 5. If you found out that a Steak & Shake hamburger contained 15 grams of fat, what would you expect the calories to be? Scrabble statistics. NY Standards: 7.S.4, 6, 8, 12; A.S.7, 8, 17; A2.S.6, 7 This problem is based on an activity, A-B-C, from the Mathematics Teacher, April 2004, by Mary Richardson and John Gabrosek. You can read about the history of the game on the website or In the activity, students are to emulate how Alfred Butts determined the number of each letter tile by tallying the number of times each letter of the alphabet appears in an article. The purpose of the activity is to examine the relationship between a letter s relative frequency in an article, the percent of Scrabble tiles per letter, and the point value of each tile. Our variation will just look at the relationship between the frequency of each letter in a Scrabble game and its point value. There are 100 tiles in a Standard Edition designed for American English. In a classroom, you could have students count and record the frequency of each tile and its point value. I was able to locate the following information on the Internet. Notice that no point values were given for N and C. Letter Value Distribution Letter Value Distribution Letter Value Distribution A 1 9 J 8 1 S 1 4 B 3 2 K 5 1 T 1 6 C? 2 L 1 4 U 1 4 D 2 4 M 3 2 V 4 2 E 1 12 N? 6 W 4 2 F 4 2 O 1 8 X 8 1 G 2 3 P 3 2 Y 4 2 H 4 2 Q 10 1 Z 10 1 I 1 9 R 1 6 blank Do you think the association between the Scrabble tile value and the letter s relative frequency will be positive or negative? Why? 2. Using lists and Plot 1, make a scatterplot with each letter s Scrabble value on the vertical axis and each letter s distribution on the horizontal axis. (Do not use the numbers for the blank.) 3. Using the scatterplot, describe the form, strength, and direction of the association between a letter s distribution and its value.

54 Summer 2006 I2T2 Probability & Statistics Page Use the graphing calculator to fit a straight-line model to the data. (The regression equations model average point values and may predict values that are not possible for actual data.) 5. Turn off Plot 1 and the line equation. Use your calculator to compute the residuals for the straight-line model. Using Plot 2, plot the residuals (on the vertical axis) versus the distribution of each letter (on the horizontal axis). Change your window to accommodate both positive and negative y-values. Give an intuitive explanation for the pattern of this plot. Whenever the graph of the residuals follow a pattern, are not randomly scattered, then this is an indication that the model chosen is not a good model. Based on your observation of the residual graph, do you think a straight-line model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 6. Turn off Plot 2. Use your calculator to fit the quadratic model y = a! x 2 + b! x + c to the data. Turn on Plot 1 to graph your fitted quadratic equation on the scatterplot produced in question Turn off Plot 1 and the quadratic graph. Use your calculator to compute the residuals for the quadratic model. Turn on Plot 2 to graph the residuals (on the vertical axis) versus the distribution of each letter (on the horizontal axis). Change your window to accommodate both positive and negative y-values. Give an intuitive explanation for the pattern of this plot. Whenever the graph of the residuals follow a pattern, are not randomly scattered, then this is an indication that the model chosen is not a good model. Based on your observation of the residual graph, do you think a quadratic model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 8. Turn off Plot 2. Use your calculator to fit the cubic model y = a! x 3 + b! x 2 + c! x + d to the data. Turn on Plot 1 to graph your fitted cubic equation on the scatterplot produced in question Repeat Step 7 with the residuals. Based on your observation of the residual graph, do you think a cubic model is a good model to use to describe the relationship between a letter s distribution and its Scrabble tile point value? 10. Use your cubic regression equation to predict the Scrabble-tile point value for the letters C and N. (Actual values are C = 3, N = 1)

55 Summer 2006 I2T2 Probability & Statistics Page 175 Simulating an epidemic: NY Standards: 7.S.1, 4, 6; A.S.7, 8; A2.S.6, 7 To simulate this in the classroom, assign a number to each person and have everyone stand up. For demonstration purposes, we shall use n = 10 people. The larger the number, the better the statistics generated. We want to generate random integers from 1 to n, count how many people are infected after each time period (depending on the disease, this could be 1 day, 1 week, 1 month, etc.), and record this number in a table (see below). For our demonstration, we will use randint(1,10). To start press Enter once to get the first random integer. result 8 Person #8 is now infected and should sit down. In row #1 of the table, enter 1. Press Enter once. result 2 Person #2 is now infected and should sit down. In row #2 of the table, enter 2 for number infected. Now we will modify the function to read randint(1,10,2) because each of these infected people could infect two additional people. result 4 1 Persons #4 and #1 are now infected and should sit down. In row #3 of the table, enter 4 for number infected. Press 2nd ENTRY to modify out function to randint(1,10,4) because we have 4 infected people. result These people should now sit down. In row #4 of the table, enter 8 for number infected. Modify the function to read randint(1,10,8) because we have 8 infected people. result Person #6 is now infected and should sit down.. Persons #1, #3, and #2 were already infected. In row #5 of the table, enter 9 for number infected. Our function is now randint(1,10,9) because we have 9 infected people. result Person #9 is now infected and should sit down. All the rest were already infected. In row #6 of the table, enter 10 for number infected. time random # number infected , , 7, 3, With a different size group, you might need more or fewer time periods. Having students sit down makes it easier to count how many are infected. You could also list all the number and cross them off as people are infected. Typically the number newly infected starts slowly, then increases rapidly, and then slows down again as most of the population is infected.

56 Summer 2006 I2T2 Probability & Statistics Page 176 Go to STAT and Edit. Clear L1 and L2. To analyze this data, enter the time in L1 in the calculator. Enter the number infected in L2. Go to 2nd STAT PLOT, turn off all plots except for Plot 1. Turn Plot 1 On. Arrow down to Type and choose the first plot, a scatter plot, press Enter. For Xlist, press 2nd 1 to choose L1, Enter. For Ylist, enter L2. Choose the + mark. Now press Y= to clear out any equations there. Press ZOOM, then ZoomStat to set the window. For our small sample, the data looks sort of linear. You can experiment with various regression equations to see what fits best. We do want the equation pasted into Y1, so on go to VARS, Y-VARS, Function, Y1. The r-value is.98 for our limited data, which indicates a good fit. Press GRAPH to see both the data and the equation. We know that an epidemic is not linear. The appropriate regression equation is the logistic equation. Go to STAT CALC, and to Logistic, Enter. L1 and L2 are the defaults, so go to VARS, Y-VARS, Function, Y2, and Enter. Press Enter to calculate this will take a while. The logistic formula is c "bx, which is not something you would find in your typical textbook. The 1 + a! e curve is S shaped. Go to Y= and turn off Y1 by positioning the cursor over the = sign and pressing Enter. Now press GRAPH. Even with our small sample we can see that the logistic curve is a better model for this type of data.

57 Summer 2006 I2T2 Probability & Statistics Page 177 Final exam scores for 50 students in General Chemistry The Five-Number Summary group of data consists of the following five numbers: 1. L, the smallest value in the data set 2. Q 1, the first quartile (The first quartile is the 25 th percentile.) 3. Q 2, the median 4. Q 3, the third quartile (The third quartile is the 75 th percentile.) 5. H, the largest value in the data set The difference between the first and third quartiles is called the interquartile range. It is the range of the middle 50% of the data. A box-and-whiskers display is a graphic representation of the 5-number summary. The box represents the middle half of the data that lies between the two quartiles. The whiskers are line segments used to represent the other half of the data. One line segment represents the quarter of the data that is smaller than the 1 st quartile. The other line segment represents the quarter of the data that is larger than the 3 rd quartile. A vertical line is placed in the box at the location of the median.

58 Summer 2006 I2T2 Probability & Statistics Page 178 Box and Whisker Plots To create a box and whisker plot for data in List 1, press [2 nd ][Y=] and choose Plot1. Then set the calculator for a boxplot. Press [ZOOM] and choose 9: ZoomStat to let the calculator choose an appropriate viewing window. Press [GRAPH] to see the box and whisker plot. Suppose the boxplot below represents the final exam grades for 50 chemistry students. Write a paragraph discussing how the students did on the exam.

59 Summer 2006 I2T2 Probability & Statistics Page 179 What s My Data? 1. For each of the box and whisker plots, create a 7-value data set that would create it. Warning: You may need to change the automatic window settings.

60 Summer 2006 I2T2 Probability & Statistics Page 180 See Handouts in session S11 Tinker Plots

61 Summer 2006 I2T2 Probability & Statistics Page 181 S12 - HS Regression Labs Workshop Regression Types: Needed for Math B Linear Quadratic (not required) Logarithmic Exponential Power You can calculate the least-square fit to altered set of data for five types of curves. They are the line (LinReg), the quadratic (QuadReg), the exponential curve (ExpReg), and the power curve (PwrReg). y = a + bx y = ax 2 + bx + c y = a + b ln x y = ab x y = ab x Warning! These curve regressions work by fitting a line to logarithm of the x-data set, the y-data set, or both. If you have zero or negative values in the data, you may get an error.

62 Summer 2006 I2T2 Probability & Statistics Page 182 Statistics Steps Directions: (a) Statistics Menu: press STAT key. (1) Press 1: Edit (You will see columns labeled, etc. ) (2) Type in some paired data point(s) followed by ENTER key. For example For example enter 2, 2.25, 2.60 etc. enter 1.06, 2, 1.80, etc. (b) Graphing data using Stat Plot Menu (1) Press 2nd Y = keys. (2) Choose 1: Plot 1 (a) In Plot 1, choose ON (b) Type: SCATTERPLOT (choice 1) (c) Xlist: (d) Ylist (e) Mark: (3) Zoom: Press Zoom key and Choose 9: ZoomStat (4) Press Graph key. (5) Start with step b again and choose this time Type: SCATTERPLOT (choice 2 ) and Make a scatter plot of the data. GRAPH (c) Finding a Linear regression, Quadratic regression, etc. (Look at the sheet for what type reg.) The TI 83 OR TI 84 calculator has a built-in feature that allows it to compute best-fitting line through a set of data. This procedure is called a linear regression. To perform a linear regression on the data you have collected, (1) Press key and move the cursor to Calc STAT (2) Select Linear regression: now you end up at your home screen.

63 Summer 2006 I2T2 Probability & Statistics Page 183 (3) On you home screen???? Type in (4) Press enter (5) r^2 and r give us the information how good of fit the equation is to the data. The closer r is to + 1 and 1 the better fit. r 2 is also determines the best fit. Correlation Coefficient/Coefficient of Determination The TI 83 or TI 84 will compute both the correlation coefficient and the coefficient of determination. However, these are not shown on the screen unless the Diagnostic is turned on. With Diagnostic On, the values of r and r 2 will be displayed when???reg is executed. One way to turn Diagnostic On: 2 nd CATALOG ALPHA D and mouse down to Diagnostic On Another way to turn Diagnostic On: VARS 5: Statistics EQ 7: r and 8: r 2 The coefficient of determination (r 2 ) gives a measure of the strength of the linear association between x and y. The square root of the coefficient of determination, r, is called the correlation coefficient. It is a measure of linear association, or the way the data points cluster around the least squares regression line. The least square line is the line that minimizes the sum of squared residuals. The r 2 formula uses the sum of squares residuals from the regression line. Therefore, it is a measure of how strongly the data points follow a linear relationship. If all data points fall on a line, the x-variable predicts the y-variable perfectly. In that case, r 2 = 1 and r = ±1. The value of r 2 must always satisfy and the value of r must satisfy.