Investigation 1.3: Life Expectancy

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2 Investigation 1.3: Life Expectancy In July 2010, the former Soviet republic of Georgia claims a woman from a remote mountain village turned 130 years old. Her name is Antisa Khvichava, and she retired from her job as a corn and tea picker at the age of 85. This is yet to be verified. Prior to this, the oldest documented person was 114-year- old Eugenie Blanchard of Saint Barthelemy, France. Reference: 1. REPRESENTING & INTERPRETING DATA 2. PATTERNS AND 3. PROPORTIONAL REASONING How long people are expected to live, on average, varies from country to country. Also, the life expectancy for females is different from that of males. In this investigation, you will explore the life expectancy of people from many different countries around the world. As you explore this problem, keep in mind that this list does not include all countries so be careful of any generalizations you make. 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL 1 8. QUADRATIC

3 1. REPRESENTING & INTERPRETING DATA life expectancy (continued) CASIO put va lue back I n the equat IOn 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL country overall life expectancy Male life expectancy female life expectancy Japan Iceland Australia France Canada Singapore Greece United Kingdom Costa Rica South Korea Cuba United States Mexico Argentina Vietnam Syria Venezuela China Brazil Turkey Iran India Ghana a Consider the following graph types: bar graph, histogram, pie graph, scatterplot, and xy-line graph. Given the data presented here, which of these graphs are appropriate? B Construct each of the graphs you believe will appropriately represent the data. c Discuss each of the graphs you constructed and explain what information it provides. Be sure to discuss the strengths and weaknesses of each graph. 8. QUADRATIC 2

4 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA d Construct three median-box graphs (box plots) on the same axes--one for overall life expectancy, one for male life expectancy, and one for female life expectancy. Discuss what you find in these graphs. e Determine the median, mean, and mode for each of the three numerical lists. Which of these statistics do you think best summarizes the data? Explain your reasoning. f Determine the range, the interquartile range, and the standard deviation for the three lists. Which of these statistics do you think best summarizes the spread of the data? Explain your reasoning. G Write a paragraph summarizing the trends you identified in the life expectancy data for these countries. 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL 3 8. QUADRATIC

5 1. REPRESENTING & INTERPRETING DATA 2. PATTERNS AND Sample Solution: Life Expectancy CASIO put va lue back I n the equat IOn a Consider the following graph types: bar graph, histogram, pie graph, scatterplot, and xy-line graph. Given the data presented here, which of these graphs are appropriate? 3. PROPORTIONAL REASONING 4. LINEAR The data are numerical, so students should not use pie graphs or bar charts. Histograms can provide meaningful information so they would be appropriate for these data. An xy-line graph best shows trends over time, so we do not find it appropriate. A scatterplot may not describe the entire data set, but it may help us determine the relationship between men s and women s life expectancies. B Construct each of the graphs that you believe will appropriately represent the data. Before beginning, enter the Statistics menu and check the SET UP (by pressing Lp) to ensure that the window will be set automatically. Then, before we construct the graphs, we must enter the data into the calculator. 5. SYSTEMS OF LINEAR EQUATIONS To enter subtitles in, make sure the cursor is in the SuB row, press La for A-LOCK, Alpha Lock. This allows you to access all letters in red. When finished inputting the subtitle, press l. Enter the overall life expectancy in list 1, male life expectancy in list 2, and female life expectancy in list 3; see below for the beginning of the data: 6. LINEAR PROGRAMMING 7. EXPONENTIAL 8. QUADRATIC 4

6 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA HiStoGRaM: to create a histogram using the calculator: Press q(graph1) then arrow down to Graph type, press u for more options, and select q(hist); this indicates you want to create a histogram in Graph 1. Arrow down to xlist: and press q(list)1l. Arrow down to frequency and press q; this tells the calculator that each value in the list should be used once. If desired, you can also set up histograms for the men and women s life expectancies at the same time. If you do, scroll back to the top and press w to set up StatGraph2 as a Histogram using list 2. Once again, scroll back to the top and press e to set up StatGraph3 as a Histogram using list 3. Below shows the set ups for graphs 1 and 3, respectively: 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS Press d to back up; if you go too far, press q(graph). To view GRapH 1, press q. The calculator will suggest a starting value (60 is fine) and a width; we have chosen 5 for the width. 6. LINEAR PROGRAMMING Press l to draw the graph. 7. EXPONENTIAL 5 8. QUADRATIC

7 1. REPRESENTING & INTERPRETING DATA life expectancy (continued) CASIO put va lue back I n the equat IOn 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR Note that while there is a bar for 60 to 65, there is no country with an overall life expectancy between 65 and 70. Most of the countries seem to have life expectancies more than 70, with the highest frequency in the range. We can repeat this process and change the width of the bars to see if different patterns emerge. Below is the histogram using the calculator s suggested value of 2.52 years for the width of each bar. We ve used the Trace Feature (Lq) to find that the highest bar begins at years. 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING We can also view the histograms for men and women separately if we desire. 7. EXPONENTIAL 8. QUADRATIC 6

8 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA ScatteRplot: To create a scatterplot comparing the male and female life expectancies, set up a scatterplot as shown below. If your screen does not look like the one pictured, press q(graph1) then arrow down to graph type and press q(scatter); this indicates that you want to create a scatterplot in Graph 1. Arrow down to xlist: and press q(list)2l. Arrow down to ylist: and press q(list)3l; this tells the calculator the x-values for your scatterplot can be found in list 2, and the y-values can be found in list 3. Be sure the frequency is 1; press l and q(graph1) to graph the points. 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR the data appear to have a linear trend. to determine a linear equation that models these data using linear regression: Press q(calc) w(x). The variable in a linear equation is of degree 1, and for this reason we press x to indicate a linear regression. 5. SYSTEMS OF LINEAR EQUATIONS The calculator allows you to choose which form of a linear equation you prefer, y = ax+b or y = a+bx. We chose the standard slope-intercept form y = ax+b by pressing q(ax+b). To draw the best-fit line on top of the scatterplot, press u while viewing the Linear Regression screen. 6. LINEAR PROGRAMMING 7. EXPONENTIAL 7 8. QUADRATIC

9 1. REPRESENTING & INTERPRETING DATA life expectancy (continued) CASIO put va lue back I n the equat IOn 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL The slope tells us that, on average, for every year of increase in a man s life expectancy, there is approximately a 1.11 year increase in a female s life expectancy. The y-intercept literally says that if male s life expectancy is 0, then female s life expectancy is approximately years. Clearly this doesn t make sense; 0 is not in the domain of the data set. c Discuss each of the graphs you constructed and explain what information it provides. Be sure to discuss the strengths and weaknesses of each graph. Histogram: The histogram allows us to identify trends in the overall life expectancies of people in these countries. The data are grouped into age intervals and the frequency of each interval is displayed in the histogram. The histogram allows us to identify that very few countries have life expectancies in the 60 s, and most countries have life expectancies in the 70 s and low 80 s. The histogram allows us to see general trends in the data, but does not give us very much specific information about the entire data set. Scatterplot: Scatterplots allow us to identify trends as we compare the female and male life expectancy data for the selected countries. The scatterplot indicates that the data appear to have a positive linear trend. From the scatterplot, we can use linear regression to identify a linear equation that models the data. The correlation coefficient, r, then indicates the strength of the correlation, and the regression equation can be used to predict the life expectancy of a female given the life expectancy of a male, and vice versa. In many cases, the scatterplot and the regression equation together can provide us a wealth of information about a data set. For this reason, many of the investigations in this book rely heavily on scatterplots and regression equations to model data sets. d Construct three median-box graphs (box plots) on the same axes one for overall life expectancy, one for males life expectancy, and one for female life expectancy. Discuss what you find in these graphs. to create Med-Box graphs using the calculator: Press p to return to the Main Menu, then press 2 to access the Statistics mode. Graph the data as ordered pairs by pressing q(graph), and press u(set) to set up Graph QUADRATIC 8

10 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA If your screen does not look like the one pictured here, press q (GRAPH1) and arrow down to Graph type, press u for more options, then press w(medbox); this indicates that you want to create a Median- Box graph in Graph 1. Next, arrow down to xlist: and press q(list)1l. Arrow down to frequency and press q(1)l; this tells the calculator that the overall life expectancies can be found in list 1 and the frequency of each data point is 1. Arrow to outliers and press q(on)l. 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR use similar steps to set up Graph 2: Press u(set) to set up Graph 2. If your screen does not look like the one pictured here, press w(graph2) and arrow down to Graph type, press u for more options, and then select w(medbox). Arrow down to XList: and press q(list)2l. Arrow down to frequency and press q(1) and l; this tells the calculator that the male life expectancies can be found in list 2 and the frequency of each data point is 1. Arrow to outliers and press q(on)l. 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL 9 8. QUADRATIC

11 1. REPRESENTING & INTERPRETING DATA life expectancy (continued) CASIO put va lue back I n the equat IOn 2. PATTERNS AND 3. PROPORTIONAL REASONING also, use similar steps to set up Graph 3: Press u(set) to set up Graph 3. If your screen does not look like the one pictured here, press e(graph3) and then arrow down to Graph type, press u for more options, and then select w(medbox). Arrow down to xlist: and press q(list)3l. Arrow down to frequency and press q(1)l; this tells the calculator that the female life expectancies can be found in list 3 and the frequency of each data point is 1. Now, arrow to outliers and press q(on)l. 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS to graph all three Median-Box graphs on the same axes: Press r(select). 6. LINEAR PROGRAMMING If your screen does not look like the one shown here, turn all StatGraphs to drawon. To do so, highlight each StatGraph and press q(on). 7. EXPONENTIAL When finished, press u(draw) to draw all three median-box graphs simultaneously. 8. QUADRATIC 10

12 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA Here we can clearly see that women, represented in the bottom graph, live longer, on average, than men. There are outliers on each graph, indicating one country overall, one country for men, and two for women whose life expectancy is significantly less than that of other countries. We also note that there is more spread in the bottom 50% than in the top 50% as Q1 (the left whisker) is larger than Q4 (the right whisker ) and Q2 (the left side of the box) is larger than Q3 (the right side of the box). Thus, the countries with shorter life spans have a greater range than the countries with longer life spans. e Determine the median, mean, and mode for the three numerical lists. Which of these statistics do you think best summarizes the data? Explain your reasoning. to determine the median, mean, and the mode using the calculator: Press q(1-var); the cursor will be on StatGraph1 was shown here: 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR Press l to indicate that you want the statistics calculated for the data in StatGraph1. Notice you must arrow down to view the median and the mode(s). 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL QUADRATIC

13 1. REPRESENTING & INTERPRETING DATA life expectancy (continued) CASIO put va lue back I n the equat IOn 2. PATTERNS AND 3. PROPORTIONAL REASONING to determine the median, mean, and the mode for StatGraph2: Press q(1-var); the cursor will be on StatGraph1, so arrow down until the cursor is on StatGraph2 as shown below. Press l to indicate that you want the statistics calculated for the data in StatGraph2. Notice you must arrow down to view the median and the mode(s). 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS Repeat the process to determine the median, mean, and the mode for StatGraph3: 6. LINEAR PROGRAMMING 7. EXPONENTIAL 8. QUADRATIC 12

14 life expectancy (continued) 1. REPRESENTING & INTERPRETING DATA Note that the mean and median for the women are significantly higher than those for men. Students may argue that any of these measures of center is best to use. They should be able to provide support for their results. f Determine the range, interquartile range, and standard deviation for the three lists. Which of these statistics do you think best summarizes the spread of the data? Explain your reasoning. Each of these tells us something a little different. The standard deviation, represented by s x tells us that there is, on average, a greater difference among women s life spans than there is among men s (5.93 years compared to 5.14 years). The range of values, determined by calculating the difference between the maximum and minimum values, does give us a sense of how much difference there is. Overall, there is a 22.6 year difference (from 60 up to 82.6 years), for men a 20.6 year difference (from 59.6 to 80.2 years), and for women a 25.6 year difference (from 60.5 to 86.1 years) in life expectancies for these countries. The interquartile range is the difference in the middle 50% of the countries and is easily obtained by subtracting Q1 from Q3. We again find a larger range for women than we find for men. G Write a paragraph summarizing the trends you identified in the life expectancy data for these countries. Students, of course, will identify different trends; however, all should notice that the life expectancy for women exceeds the life expectancy of men. They may want to speculate why this is the case. They also may look at the locations of the countries, but again we caution that this list is incomplete, so trends about continents, for example, may not hold true if we looked at data from all countries. 2. PATTERNS AND 3. PROPORTIONAL REASONING 4. LINEAR 5. SYSTEMS OF LINEAR EQUATIONS 6. LINEAR PROGRAMMING 7. EXPONENTIAL QUADRATIC

15 1. POLYNOMIAL Investigation 6.5: Earth s Revolution Before the 1530s, many people believed the Earth was the center of the universe and that every celestial body revolved around it, including the sun! Copernicus, a Polish astronomer, presented a cosmological theory stating that the Earth and other heavenly bodies actually revolved around the sun. Although today his heliocentric system of the universe is widely accepted, in his day Copernicus s theories were considered extremely controversial. Reference: Kepler s First Law of Planetary Motion states that the orbit of a planet is elliptical, with the Sun at one of the foci. A complete revolution of the Earth about the Sun takes approximately days. During its revolution around the sun, the Earth winter solstice and summer solstice occur at the ends of the major axis. On January 3, the Earth is at its perihelion, approximately 147 million km from the sun. On July 4, it is at its aphelion (its farthest point), approximately 152 million km from the sun. (We re actually closer to the sun in the Northern Hemisphere s winter than we are in summer.) These two values are at the endpoints of the major axis of its elliptical path. a. Set up an axis system that can describe the Earth s revolution about the Sun using the center of the major axis of the ellipse as the origin. What are the coordinates of the Earth at its perihelion and aphelion? b. What are the coordinates of the Sun? c. What are the coordinates of the Earth when it is an equal distance from both foci? d. Graph this ellipse and select an appropriate viewing window. Which view window best represents the eccentricity of the ellipse? e. Use the calculator to verify the location of the foci determined in part a. Why might these answers vary slightly? f. What is the eccentricity of the ellipse? Discuss what that this means in general terms. Reference: RATIONAL 3. INVERSE RELATIONS AND 4. EXPONENTIAL AND LOGARITHMIC 5. TRIGONOMETRIC 6. CONIC SECTIONS 7. MATRICES 8. TIME, VALUE, AND MONEY

16 1. POLYNOMIAL CASIO put va lue back I n the equat IOn Sample Solution: Earth s Revolution 2. RATIONAL 3. INVERSE RELATIONS AND a. Set up an axis system that can describe the Earth s revolution about the Sun using the center of the major axis of the ellipse as the origin. What are the coordinates of the Earth at its perihelion and aphelion? The total distance along the major axis is 2A. We ll find the sum of the distance from the earth to the Sun at both the perihelion and aphelion. This will give us the length of the entire major axis. By dividing by 2, we ll have the value of A. In RUN: Enter the values, using the c key as shown. This can help us avoid potential mistakes with inputting the correct number of 0 s. 4. EXPONENTIAL AND LOGARITHMIC 5. TRIGONOMETRIC Consequently, A is x 10 8 km. At the perihelion, the Earth would be at x 10 8 km, and at the aphelion, it would be at x 10 8 km. b. What are the coordinates of the Sun? We will assume we are orienting the ellipse with the sun at the left focus. The Sun is located 1.47 x 10 8 km to the right of the perihelion and 1.52 x 10 8 km to the left of the aphelion. Though we need do only one of the calculations, for confirmation, we do both calculations as shown. 6. CONIC SECTIONS 7. MATRICES 8. TIME, VALUE, AND MONEY From this, we find that the sun is located at (-2.5 x 10 6, 0). c. What are the coordinates of the Earth when it is an equal distance from both foci? 16

17 1. POLYNOMIAL earth S ReVolution The minor vertices are equal distances from each focus. Letting (0, B) and (0, -B) represent these vertices; the Earth will be at these points when it is equidistant from the two foci. We note that either of these points, we ll use (0, B), the Sun, and the origin form a right triangle. The hypotenuse of the triangle is the distance from (0, B) to the Sun, which, as we found earlier, is at (-2.5 x 10 6, 0). We know, however, that the sum of the distances from the Sun to (0, B) and from (0, B) to the other focus is 2A, so the distance from the Sun to (0, B) is precisely A, which we already know is x 10 8 km. The leg from the origin to our point of interest has length B. The leg from the origin to the sun has length 2.5 x 10 6 km. Using the Pythagorean Theorem, we find that (2.5 x 10 6 ) 2 + B 2 = (1.495 x 10 8 ) 2. To find B, we used the RUN mode on the calculator. 2. RATIONAL 3. INVERSE RELATIONS AND 4. EXPONENTIAL AND LOGARITHMIC Thus we find that B is approximately x 10 8 km, which we note is almost identical to A. d. Graph this ellipse and select an appropriate viewing window. Which view window best represents the eccentricity of the ellipse? from the conic Graphs mode: Scroll down until the form of the ellipse you want is highlighted and press l. 5. TRIGONOMETRIC 6. CONIC SECTIONS Enter the values we have determined. Press u(draw) to draw the graph. At this point, you will likely want to adjust the viewing window. Press Le(V-Window) and make the following changes: xmin: -2.0c8l; xmax: 2.0c8l; ymin: -2.0c8l; ymax: 2.0c8l; and scales of 2.0c7l on both axes MATRICES 8. TIME, VALUE, AND MONEY

18 1. POLYNOMIAL CASIO put va lue back I n the equat IOn earth S ReVolution 2. RATIONAL Press du(draw) to view the graph with the new view window settings. 3. INVERSE RELATIONS AND 4. EXPONENTIAL AND LOGARITHMIC The window is not yet squared, however, so we cannot tell how far removed from a circle this elliptical path really is. Here the ellipse appears elongated. Press Le(V-Window). Select y(square) for a square view window, followed by q(y-base) to use the y-base. The calculator will then select the appropriate x-range for the square view. The screen shot shows the calculator s adjusted values for the xmin and xmax. Redraw the graph. 5. TRIGONOMETRIC 6. CONIC SECTIONS Note: how circular the path seems to be now. This is a much better visualization of how close the Earth s orbit is to a circle. e. Use the calculator to verify the location of the foci determined in part a. Why might these answers vary slightly? With the graph displayed, press Ly(G-Solv)q(FOCUS) to find the focus. Use! to find the other focus. 7. MATRICES 8. TIME, VALUE, AND MONEY 18

19 1. POLYNOMIAL earth S ReVolution Though we note slight rounding error, relative to the distances we are discussing, these are minor. What is noteworthy here is how close to the center the two foci appear to be. f. What is the eccentricity of the ellipse? Discuss what that means in general terms. With the graph displayed, press Ly(G-Solv)uq(e) to find the eccentricity value. 2. RATIONAL 3. INVERSE RELATIONS AND The eccentricity of an ellipse is a number between 0 and 1. The closer the eccentricity is to 0, the closer the ellipse is to a circle. The closer it is to 1, the more the ellipse collapses on itself, making a very narrow ellipse. Here we see the eccentricity is approximately ; the Earth s trip around the sun is very close to circular. 4. EXPONENTIAL AND LOGARITHMIC 5. TRIGONOMETRIC 6. CONIC SECTIONS MATRICES 8. TIME, VALUE, AND MONEY

20 Investigation 2.3: Colored Pencils: Rotation Problem There are groups devoted entirely to the love of the colored pencil. One such group is the Colored Pencil Society of America (CPSA). They have over 1,900 members and their mission is to educate about the fine art of using colored pencils. The World Record for the longest colored pencil drawing was set by Jainthan Francis in Sayreville, New Jersey in 2009 with a drawing that was twenty inches wide and five hundred yards long. It took three months to complete and had two hundred different colors. Reference: 1. PROPERTIES OF 2-DIMENSIONAL SHAPES 2. REFLECTIONS AND ROATATIONS 3. TRANSLATIONS AND DILATIONS 4. AREA AND PERIMETER In this investigation we will explore the rotation of an image about a point. To do so, we will use the picture of colored pencils stored in the PRIZM. a Draw a line segment connecting the tips of the two identical green colored pencils. b Draw at least four more line segments connecting tips of identical colored pencils. c Where is the center of rotation the point the image of the colored pencils are being rotated about? Explain your thinking. d Construct what you believe to be the center of rotation. How many degrees do you think the image was rotated about this point? Explain your thinking. e Use your observations to complete the following statements. A point and its image are the same from the center of rotation. If an image is rotated 180 about the center of rotation, segments joining every point and its image at a point. The point of intersection is the. 5. VOLUMNE AND SURFACE AREA 6. PYTHAGOREAN THEOREM 7. RIGHT ANGLE TRIGONOMETRY REASONING AND PROOF

21 1. PROPERTIES OF 2-DIMENSIONAL SHAPES 2. REFLECTIONS AND ROATATIONS Sample Solutions: Colored Pencils CASIO put va lue back I n the equat IOn a Draw a line segment connecting the tips of the two identical green colored pencils. First, to open the picture of the colored pencils use the following steps: Press px to access the Geometry mode of the calculator. 3. TRANSLATIONS AND DILATIONS Press q(file) and 2(Open) to access the pictures. 4. AREA AND PERIMETER Press u(strgmem), scroll to highlight Casio, press q(open), scroll down to the folder g3p, and press q(open). 5. VOLUMNE AND SURFACE AREA 6. PYTHAGOREAN THEOREM Scroll down to Colored_Pencils and press l. The picture of the colored pencils should appear on the screen. Press q(yes) to set initial window value. 7. RIGHT ANGLE TRIGONOMETRY 8. REASONING AND PROOF 22

22 investigating RotationS 1. PROPERTIES OF 2-DIMENSIONAL SHAPES 2. REFLECTIONS AND ROATATIONS 3. TRANSLATIONS AND DILATIONS Now draw a line segment connecting the tips of the two identical green colored pencils. To do so: 4. AREA AND PERIMETER Press e(draw) and 2(Line Segment). Scroll to the tip of one green colored pencil and press l. Scroll to the tip of the other green colored pencil and press l. Segment will be constructed. Press d to exit Draw mode. 5. VOLUMNE AND SURFACE AREA 6. PYTHAGOREAN THEOREM If you choose, you can change the formatting using these steps. Highlight the segment using the arrow tool and press l to select the segment. Press L5(format), 2(Line Style), and 2(Thick), Press 3(Line Color), and 7(Yellow). 7. RIGHT ANGLE TRIGONOMETRY Press d to exit Format mode REASONING AND PROOF

23 1. PROPERTIES OF 2-DIMENSIONAL SHAPES investigating RotationS CASIO put va lue back I n the equat IOn 2. REFLECTIONS AND ROATATIONS 3. TRANSLATIONS AND DILATIONS b Draw at least four more line segments connecting tips of identical colored pencils. to draw a line segment: Press e(draw) and 2(Line Segment). 4. AREA AND PERIMETER Scroll to the tip of a colored pencil and press l. Scroll to the other identical colored pencil and press l. A line segment will be constructed. Press d to exit Draw mode. 5. VOLUMNE AND SURFACE AREA Use this process to create at least three more segments. A screenshot that may result is shown here. 6. PYTHAGOREAN THEOREM 7. RIGHT ANGLE TRIGONOMETRY c Where is the center of rotation the point the image of the colored pencils are being rotated about? Explain your thinking. The image of the colored pencils appears to be rotated about the point where all of these line segments intersect, so that point appears to be the center of rotation. d Construct what you believe to be the center of rotation. How many degrees do you think the image was rotated about this point? Explain your thinking. 8. REASONING AND PROOF 24

24 investigating RotationS 1. PROPERTIES OF 2-DIMENSIONAL SHAPES to construct a point: Press e(draw) and 1(Point). Scroll to a point you believe to be the center of rotation and press l. if you choose, you can change the formatting using these steps. Highlight the point using the arrow tool and press l to select the point. Press L5(format), 2(Line Style), and 2(Thick), Press 3(Line Color), 3(Red), and d. 2. REFLECTIONS AND ROATATIONS 3. TRANSLATIONS AND DILATIONS 4. AREA AND PERIMETER The point and its image are equidistant from the center of rotation. The angle of rotation appears to be counterclockwise. e Use your observations to complete the following statements. A point and its image are the same distance from the center of rotation. If an image is rotated 180 about the center of rotation, segments joining every point and its image intersect at a point. The point of intersection is the center of rotation. 5. VOLUMNE AND SURFACE AREA 6. PYTHAGOREAN THEOREM 7. RIGHT ANGLE TRIGONOMETRY REASONING AND PROOF

25 INTRODUCTION DATA ENTRY Investigation 4.2: A Warming World Carbon Dioxide and Methane are the main greenhouse gases contributing to global warming. Over the last century the earth has warmed approximately 1 degree Fahrenheit. The multinational Arctic Climate Impact Assessment (ACIA) report concludes that in Alaska, western Canada, and eastern Russia, average temperatures have increased as much as 4 to 7 degrees Fahrenheit (3 to 4 degrees Celsius) in the past 50 years. The decade ending in 2009 was the warmest on record. Reference: You are curious about the proportion of Americans who believe that global warming is due to human activity. According to one study, the proportion is approximately 45%. a. If this is the case, how many people must be included in a study to ensure that, with 95% confidence, you have no more than a 3% error? B. For what value of π (proportion of the population) is the standard error the greatest? Explain. Use this value to answer Part A above. How many more people would you need in your sample size in this worst case scenario? c. Assuming you want 95% confidence in your results, explore the relationship between the total error and the sample size. Assume the worst case. Discuss your findings. Reference: environment_energy/energy_update 1. DESCRIPTIVE TECHNIQUES 2. COUNTING AND THE BINOMIAL DISTRIBUTION 3. NORMAL DISTRIBUTION 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 GROUPS 7. ANOVA AND CHI-SQUARE EXTENDING STATISTICAL THINKING

26 INTRODUCTION DATA ENTRY CASIO put va lue back I n the equat IOn 1. DESCRIPTIVE TECHNIQUES 2. COUNTING AND THE BINOMIAL DISTRIBUTION 3. NORMAL DISTRIBUTION 4. UNIVARIATE INFERENCES One Solution: A Warming World a. If this is the case, how many people must be included in a study to ensure that, with 95% confidence, you have no more than a 3% error? The total error is given by the product of the standard error and the confidence coefficient. The standard error of the sampling distribution for a confidence interval of a categorical variable is given by the formula, where π represents the proportion of the population who believe this (here it is not the value approximated by 3.14). Since we are to assume that 0.45 is the correct value for π, we will use it in the formula. Our confidence coefficient is our z-value. To determine the z-value for 95% confidence, we will use the inverse of the Standard Normal Curve, which has a mean of 0 and a standard deviation of 1. From the home screen in the STATISTICS menu, Press y(dist)q(norm)e(invn). Complete the screen as shown. Scroll down to execute; press q(calc). 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 GROUPS We will now do some algebra on the calculator to determine the sample size we need to limit our total error to Using that the total error in a point estimate is expressed in the formula π(1 π) TE = z * n, we need to solve the equation for n when we substitute 0.03 for TE (Total Error), 1.96 for z, and 0.45 for π. From the Main Menu, choose EQUATION. Then, 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING 28

27 INTRODUCTION DATA ENTRY a WaRMinG WoRld Press e(solver). Enter the equation as shown below left; press l. Press u(solve). 1. DESCRIPTIVE TECHNIQUES 2. COUNTING AND THE BINOMIAL DISTRIBUTION We find that we need at least 1,057 people in the study to ensure that our error is no more than 3%. B. For what value of π (proportion of the population) is the standard error the greatest? Explain. Use this value to answer Part A above. How many more people would you need in your sample size in this worst case scenario? In determining the standard error, the value of π appears twice, both times in the numerator under the square root. The standard error will be largest whenever the product of π and 1 π is the largest. The table below shows some quick calculations. π 1 π π (1 π ) This tells us that the standard error is greatest when the percent of people in the category is 50%. Note that if we increased instead of decreasing it, then the products remain the same. In this worst case scenario, we recalculate the sample size needed to ensure that, with 95% confidence, our error will be no more than 3%. 3. NORMAL DISTRIBUTION 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 GROUPS 7. ANOVA AND CHI-SQUARE EXTENDING STATISTICAL THINKING

28 INTRODUCTION DATA ENTRY CASIO put va lue back I n the equat IOn 1. DESCRIPTIVE TECHNIQUES 2. COUNTING AND THE BINOMIAL DISTRIBUTION a WaRMinG WoRld From the screen shown above right, Press q(repeat). Use the scroll bar and change the product of 0.45 x 0.55 to 0.50 x Press lu(solve). 3. NORMAL DISTRIBUTION 4. UNIVARIATE INFERENCES 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 GROUPS We find that we need at least 1,068 people in the study to ensure that, with 95% confidence, our error will not exceed 3%. This is a modest increase of only 11 people in the study. c. Assuming you want 95% confidence in your results, explore the relationship between the total error and the sample size. Assume the worst case. Discuss your findings. We will explore this graphically, treating the total error as a function of the sample size. Thus x represents the number of people in the same and y represents the total error. We know the worst case occurs when π is 50%. We have already determined that the confidence coefficient is From the Main Menu, choose GRAPH. First we will set a reasonable window and then enter the function. Press Le(V-Window). We have chosen to set xmin at 0, xmax at 3000, and xscale at 500. Our ymin is 0, ymax is 0.10, and yscale is Press d to return. We have entered the total error into function y1, using x, not n, as our independent variable. So we can see fairly readily when the total error is 1%, 3%, and 5%, we have entered these functions as y2, y3, and y4 respectively. 7. ANOVA AND CHI-SQUARE 8. EXTENDING STATISTICAL THINKING 30

29 INTRODUCTION DATA ENTRY a WaRMinG WoRld Press u(draw) to see the graphs. The blue graph, which represents the total error, shows that as we move to the right (adding more people to the sample), at first the total error drops dramatically. It then levels off; we reach a point when adding more people has only a small effect upon the total error. Even at the far right end of the graph, when the sample size is 3000, the total error is still greater than 1% (note the blue curve never gets as low as the red line). It appears that if we want the total error limited to 2%, we would need approximately 2,500 people in the study. Earlier we found that with 1,068 people in the study, our total error is 3%. Thus we would have to more than double the sample size to reduce the error from 3% to 2%. Perhaps this will help students understand why many major studies often use a little more than 1,000 people in their surveys. To see how many people we would need if we would accept an error of 5%, Press Ly(G-Solv)y(INTSECT). 1. DESCRIPTIVE TECHNIQUES 2. COUNTING AND THE BINOMIAL DISTRIBUTION 3. NORMAL DISTRIBUTION Press l when y1 and y4 are highlighted. 4. UNIVARIATE INFERENCES We see that we need only 385 people in the study if we are willing to accept an error of 5%. 5. REGRESSION MODELS AND ANALYSIS 6. DIFFERENCES BETWEEN 2 GROUPS 7. ANOVA AND CHI-SQUARE EXTENDING STATISTICAL THINKING