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Ronald G. ArmontroutThe Hotchkiss SchoolLakeville, CT 06039email: [email protected] http://armontrout.wikispaces.com/Algebra Data Exploration Lab: 2007 Life ExpectanciesCountry Life Expectancy Infant Mortalityper 1,000 birthsEthiopia 49.2 91.9Afghanistan 43.8 157.4Argentina 76.3 14.3Bolivia 66.2 50.4Mexico 75.6 19.6United States 78.0 6.4China 72.9 22.1India 68.6 34.6United Kingdom 78.8 5.0N Korea 71.9 22.6S Korea 77.2 6.1Botswana 50.6 44.0https://www.cia.gov/library/publications/the-world-factbook/geos/xx.htmlMake a scatter plot of the data (Infant mortality, Life Expectancy). Note: For those of you who may have forgotten how to enter the data in your calculator and produce the scatter plot on the TI-84: STAT EDIT X values in L1 and Y values in L22nd STAT PLOT PLOT1 ON ZOOM 99 ZoomStatOr, you can name the list by pushing your curser up to thelist name you used for infant mortality and using 2nd Insert type MORT. Similarly you can name you life expectancy list LIFEAssuming a linear model is appropriate for the data, with your partner use the procedure below to find a "toothpick" best fit line.Toothpick Best-Fit Lines: Work with a partner. One calculator will have the stat plot and the other will be used todo all calculations. Be very careful not to move the toothpick once you have determined your fit.Green Math: Investigations of Environmental & Social IssuesRon Armontrout 2Create a Stat Plot of the data. Lay a toothpick in the graphing window such that in your opinion you have the best line to fit the data. See Fig 1. (This is not your life expectancy data) figure 1Now using the free curser arrows and not Trace, determine two ordered pairs on your line. You can see oneof the ordered pairs in figure 2. It is perhaps best to choose one on the line near the other side of the window. figure 2Do not move your toothpick. Use your partner’s calculator to do thefollowing arithmetic. Use these two points to calculate an equation for the toothpick, and put the equation into Y1. Did your linear function hit the toothpick exactly? If not, determine where you went wrong and try it again.Now, let’s use this technique with your Life Expectancy and Infant Mortality Data:1. Place your toothpick on the statplot and calculate the slope of this toothpick line. Determine the real-world significance of this slope.2. Find an equation of the toothpick line.3. What is the Life expectancy intercept and what is its real-world significance? Explainhow you found it.4. What is the Infant mortality intercept and what is its real-world significance? Explain how you found it.5. Predict the life expectancy if the infant mortality rate for a country is 100 per 1000 births. 6. What infant mortality rate would you expect if the life expectancy stretched to 60 years?One of the problems with the tooth-pick best fit is that all pairs of students most likely had different ideas for what the best fit is. Use the least squares Sketchpad file to investigate linear regression.• To have the calculator determine an equation for the linear regression line, on the TI-84:STAT CALC44 LinReg (ax+b) L1, L2,Y1 or MORT, LIFE,Y1Green Math: Investigations of Environmental & Social IssuesRon Armontrout 3Residuals, (see the bi-variant data handout) are built into the TI-84. To look at the residuals, set up the STAT PLOT. The RESID list is found under LIST. (2nd STAT). Then, ZOOM#99 ZoomStat7. Use the TI-84 to calculate the linear regression equation for the Life Expectancy data. Write the function and the r value. Compare it to you toothpick line. How did you do?Algebra Data Exploration Lab: Burglary DataUtah Burglary Data 2006CITY BURGLARIES POPULATIONBountiful 156 42424Midvale 207 28055Ogden 820 80861Orem 269 92637Pleasant Grove 143 40327Provo 541 117156Salt Lake City 2244 183901Taylorsville City 344 59899Washington 61 14114Wendover 11 1673West Valley 877 116992http://www.fbi.gov/ucr/cius2006/data/table_08_ut.html1. Describe the stat plot. In your opinion is Salt Lake City a dangerous city for burglaries? Explain.2. For the Utah burglary data, calculate the linear regression model. Which variable is the independent variable? What is the r-value for the regression line? What does this number suggest about the data?3. What is the burglary intercept and what is its real-world significance? Explain how you found it.4. Recall your answer in #1. Do you feel the same about the danger of burglary in Salt Lake City as you did before? Explain.Green Math: Investigations of Environmental & Social IssuesRon Armontrout 4Salt Lake City can be thought of as an outlier point in the data set. There is another line we can fit to the data called the median-median line. Below is an algorithm for calculating theline by hand.PROCEDURE for FINDING the MEDIAN-MEDIAN LINE by Hand1. First sort the data in ascending order according to the independent variable. Separate the data into three groups, called summary groups, of equal size (or as close to equal as possible) according to values of the horizontal coordinate (the independent variable). If there is one extra piece of data, put it in the middle summary group. If there are two extra pieces of data, put one in each of the two outside summary groups.2. Find the summary point for each group by determining the median x-value of the group and the median y-value.3. Find the slope of the line through the summary points of the outer groups. Call this line L.4. Slide this line 1/3 of the way to the middle summary point. (NCSSM, Contemporary Precalculus, Janson Publications, page 12.)Now this is a very difficult thing to do algebraically. There is an easier way to do this.Or, using a simpler geometric method: (Use the Centroid Sketchpad file to investigate the properties of the centroid-the point of concurrency of the medians in a triangle.)From geometry you may have discovered that the centroid splits a median in a 2 to 1 ratio. That is 2/3 the distance from a vertex to the midpoint of the opposite side making the centroid the point that is 1/3 the distance to the middle summary point! If not, this would bea good time to explore the centroid with a computer and dynamical software such as the Geometer’s Sketchpad.The coordinates of the centroid are the averages of the three vertices.i.e. , 5. Find the equation of the line through the centroid of the triangle that is parallel to line L note: The TI-84 has a


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