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Prediction
Intro: Let the Bones Speak!
Read the article provided as an introduction to this unit. It will provide you with some clues to completing Activity 1.
Activity 1: Using Your Head
In this unit you will be asked to think like a forensic scientist. After studying the data in Table 1 of the preparation reading, you will begin to sort out clues about the deceased from their bones. For the final project at the end of this unit, you will write a report detailing their story.
1.Study the data in Table 1 of the preparation reading.
a)A forensic scientist would tell you that these bones belonged to at least two people. How would the scientist know this for sure?
b)Which bones do you think belonged to the same person? On what assumptions did you base your answer? How sure are you of your answer? (Name the dead people Bones 1, Bones 2, and so on, to make it easier to classify their bones.) You will need to refer to your answers to this item later.
c)Do you think the deceased were male or female? On what evidence did you base your answer?
d)Do you think the deceased were young children or adults? Defend your answer.Guess the heights of the deceased. How accurate do you think your guesses are?
2.One place to look for some help in estimating heights is artists’ guidebooks for sketching human figures. Artists have found that the rule of thumb, “draw a 14-year-old 7 head-lengths tall,” helps them draw teenagers with heads correctly proportioned to their bodies.
- How closely do you think the dimensions of real students, such as those in your class, match the ideal relationship suggested byartists?
- Write an equation to represent the relationship between the head length and the height of a 14-year old based on the information given above.
- Suppose the following data represented the actual measurements of 8 14-year olds in a class. Using this data, determine a linear model that best fits the data. You may use your calculator or an Excel spreadsheet to do this. How closely does this equation match the one you wrote in part b?
A / B / C
1 / Head Length (cm) / Actual Height (cm)
2 / Student 1 / 19 / 134
3 / Student 2 / 20 / 145
4 / Student 3 / 21 / 150
5 / Student 4 / 21.5 / 153
6 / Student 5 / 22 / 160
7 / Student 6 / 24 / 170
8 / Student 7 / 23 / 160
9 / Student 8 / 20.5 / 148
Calculator instructions:
Enter the head lengths in list 1 and the heights in list 2
2nd Y= to set up a scatterplot of the data
Zoom Stat (Zoom 9) will give a reasonable window for the graph.
To find the linear regression model, STAT CALC LinReg Enter VARS YVars Enter Enter [this will place the equation in Y1]
Look at the graph to determine if this line is a good fit for the data.
Excel instructions:
Enter the data into a spreadsheet just as shown above.
Highlight columns B and C and select Insert (Graph) Scatterplot
Edit scatterplot to label the axes and adjust the scales as needed.
Click on the data points, right click to select “add trendline”.
Choose Linear and check add equation to graph, then close.
The line and its equation will appear on the scatterplot.
3. Now collect some data on your own. Be sure to collect at least 10 sets of data if possible.
a)Within your group, measure each person’s head length (from chin to the top of the head). Record your data in column Bof a spreadsheet similar to the one shown below. Be sure to specify your units of measurement in the column headers. (It may be easiest to record measurements in cm.)
b)Use the relationship “Height = 7 head lengths” to predict each person’s height. Record your results in column Cof your spreadsheet.
c)Next, measure each person’s actual height and record your results in column D. In almost every situation in which predictions are made from data, it is useful to examine the residual errors. [ Residual errors are defined as the difference between the actual value and the predicted value for each point in your data.]
d)Calculate the residual errors corresponding to the people represented in your table by subtracting the predicted heights in column Cfrom the actual heights in column D. Record the results in column E. So that you have sufficient data to detect patterns, collect the data from another group and add it to the bottom of your table.
e)If a residual error is positive, what does that tell you about your prediction? What if an error is negative? What if an error is zero?
f)Are the residual errors fairly evenly divided between positive and negative values? How well did the relationship “Height = 7 head lengths” do in predicting the actual heights of members of your group?
g)Would a multiplier different from 7 do a better job? If so, what multiplier would you choose? What process did you use to determine this multiplier? Why do you think it does a better job than the multiplier 7? Test this out by changing the formula in your spreadsheet.
A / B / C / D / E1 / Head Length (cm) / Predicted Height (cm) / Actual Height (cm) / Residuals
(Error)
2 / Person 1 / (Person 1) / =B2*7 / (Person 1) / =C2-D2
3 / Person 2 / (Person 2) / =B3*7 / (Person 2) / =C3-D3
Etc….
h)Next create a scatterplot of your data, plotting the Head Length vs Actual Ht. To do this, highlight columns B and D and then select Data Graph Scatterplot from the menu bar. Be sure to label your axes appropriately and adjust your axes as needed. To determine the equation of a linear equation that fits the data, right click on the data points, then select Add Trendline and choose Linear along with display equation. Discuss how the slope of this trendline relates to the multiplier you determined to be the best “fit” in part g.
4.The relationship between height and head length changes with age. Therefore, artists adjust their guideline based on the age of the person they are drawing.
a)When drawing sketches of adults (ages 18–50) artists follow this guideline: Draw the figure of an adult approximately seven and one-half head-lengths tall. Write a formula that describes the relationship between height, H, and head length, L, according to the artists’ guideline for drawing an adult.
b)How does this formula compare with the one you calculated on Excel?
c)Look at the equations you have found for 14-yr olds and for adults. How are the graphs of these two equations the same, and how are they different? What effect does changing the value of the multiplier have on the graph?
d)Using the artists’ guidelines for adults, predict the height of a person whose head length measures 23.0 cm. Without doing further calculations, would your estimate be higher or lower if you knew the person was only 13 years old? Explain how you could use your graphs to answer the preceding question.
5.Juan decides to draw a picture of his mother standing by a window. He follows the artists’ guidelines for drawing adults. He makes a preliminary sketch, but then decides that the figure is too small. So, for his final sketch, he draws the head of his figure 1 cm longer than in his preliminary sketch and continues to follow the artists’ guidelines. How much taller than his preliminary sketch is Juan’s final sketch. Justify your answer.
6.Think about how you might use one of the artists’ guidelines or the relationship that your group determined between height and head length to make a rough prediction of the height of the person whose skull length was recorded in Table 1, 230 mm in length.
a)What assumptions might you make in order to make your prediction?
b)Predict the height of the person in cm. Describe the process you used in making your prediction.
c)Does your prediction result in a height that is reasonable for a person? Explain.
d)Do you think your prediction is likely to be close to the actual height of the person? Why?
7.What information do you think might be helpful in determining better estimates of the heights of the deceased whose bone lengths are recorded in Table 1 of the introductory article? How or where might you obtain this information?
Activity 2: Leg Work
Dr. Mildred Trotter (1899–1991), a physical anthropologist, was well known for her work in the area of height prediction based on the length of the long bones in the arms and legs. Here is one of the relationships proposed by Dr. Trotter.
First formula: H = 2.38F + 61.41
where H is the person’s height (in cm) and F is the length of the femur (in cm).
Figure 2. The femur (thighbone)
1. Suppose, for most adults, femurs range in size from about 38 cmto 55 cm. According to Dr. Trotter’s formula, how tall is aperson with a 38-cm femur? How tall is a person with a 55-cm
femur?
2. Use the axes provided below to sketch a graph of this situation.
Figure 3. Axes for height and femur length
Notice that the horizontal axis is scaled from around 35 cm to 60 cm (a slightly wider range than the minimum and maximum femur lengths) with tick marks every 5 units. A zigzag has been added to indicate that there is a break in this scale between 0 and 35.
a)Draw a scale on the vertical axis that would be appropriate for data on adult heights (in cm).
b)Sketch a graph of Dr. Trotter’s relationship on the set of axes provided. (You may want to plot several points before drawing the graph.)
3. Jason’s femur measures 40 cm. His brother’s measures 41 cm. Based on Dr. Trotter’s first formula, predict the difference in thetwo brothers’ heights.
4. The femurs of two men differ by one centimeter. Predict thedifference in their heights. Explain how you were able todetermine your answer even though the lengths of the two men’s
femurs were not given. In addition, tell how you could read offyour answer from Dr. Trotter’s first formula.
5. Suppose that a woman is 172.7 cm (about 5 ft 8 in.) tall. Explainhow you could use your graph to estimate the length of her femur.What is your estimate?
a)Draw a horizontal line at approximately H = 172.7 cm. Findthe F-coordinate that corresponds to the point where the horizontal line and the graph of Dr. Trotter’s equationintersect.
b)Write an equation (based on Dr. Trotter’s first formula) thatdescribes how you could predict the length of the femur froma person’s height.
c)Use your equation in b) to predict the length of a woman’sfemur if the woman is 172.7 cm tall. Compare your answer tothe one from a).
6. Another of Dr. Trotter’s equations predicts height from theperson’s tibia:
Second formula: H = 2.52T + 78.62, where H and T are measured in cm.
a)The length of the tibia described in Table 1 was 416 mm. Using Dr. Trotter’s second formula, predict the person’sheight. Is your answer a reasonable height for a person?(Recall that 2.54 cm 1 in.)
b)Write a set of algebraic steps to solve the second formula, H = 2.52T + 78.62, for T. (A doctor might use such an equation to check that the lengthof a person’s tibia is normal for a person of that height.)
c)If a person is 172.7 cm tall, use your equation from b) topredict the length of his or her tibia.
7.In the third formula, Dr. Trotter used both the tibia and the femurto predict height:
H = 1.30(F + T) + 63.29. (All measurements are in cm.)
a)Suppose that students measure the femur and tibia of askeleton and determine that the femur is 42 cm long and thetibia is 43 cm long. Predict the height of the person using Dr.Trotter’s third formula:
H = 1.30(F + T) + 63.29.
b)Compare the prediction in a) with the predicted height usingDr. Trotter’s equation H = 2.38F + 61.41.
c)Compare the predictions in a) and b) with the predicted heightusing Dr. Trotter’s second formula, H = 2.52T + 78.62.
d)You should have found a fairly large discrepancy betweenyour predictions in a) - c). One possibility is that you did notget precise measurements of the bone lengths. Suppose that aman is 175 cm tall (about 5 ft 9 in.). Based on Dr. Trotter’sequations in b) and c), would you expect his tibia or his femurto be longer and by how much?
e)Repeat part d) for a person whose height is 160 cm.
f)Based on Dr. Trotter’s equations, is there any evidence thatindicates that you may have made faulty measurements?Explain.
8.Use one or more of Dr. Trotter’s equations to estimate the heightsof two of the people whose bones are described in Table 1 of thepreparation reading. Using her equations, do you think thesebones might have belonged to at least three people? Do yourcalculations give you cause to change any of the assumptions thatyou made in Item 1b), Activity 1? If so, which assumption(s)?
In Activity 1 and in Activity 2, you examined and interpretedequations established by artists and by a scientist. You used some ofDr. Trotter’s models to estimate the heights of Bones 1 and Bones 2(described in the preparation reading). Dr. Trotter’s formulas mayhave challenged some of the assumptions that you made in Item 1b),Activity 1. However, for the equations given in Activity 2, sheassumed that the deceased were adult white males. If this assumptionis not valid, your estimates based on Dr. Trotter’s equations may not beaccurate.
Activity 2 Extension: Under Investigation
Unlike the artists’ guidelines for drawing figures, Dr. Trotter’sequation, H = 2.38F + 61.41 (where height, H, and femur length, F, are in cm),is not a member of the y = mx family, but instead belongs to the larger y= mx + b family. You indicate members of this family by choosing
values for m and b. (What were Dr. Trotter’s choices for m and b?)
Recall that Dr. Trotter’s equation H = 2.38F + 61.41 was designed to work well for a particular population, adult whitemales. She later modified her formula by modifying the values of m
and b to adjust for age, ethnic background, and gender. To make suchadjustments, you will need to know how changes in m and b affect thegraph. Complete the following investigation to find out what happenswhen you make changes to m and b.Because there are two quantities to change, m and b, it may help todivide the investigation into two parts, as described below.
PART I: KEEP m THE SAME AND CHANGE b.
(1) Choose a value for m and one for b. What is your equation?
(2) Graph your equation.
(3) Choose several other values for b. What equations correspond tothese choices?
(4) Graph several of the equations from (3) and the equation from (2)in the same window.
PART II: KEEP b THE SAME AND CHANGE m.
Repeat Part I, reversing the roles of m and b.
1.Use your graphing calculator to investigate how changing the values of m and b affects the graph of a member of the y = mx + b family.
a)How does changing the value of b affect the graph of a member of the y = mx + b family? Illustrate using several examples. Continue experimenting with choices for b until you know what b controls on the graph.
b)How does changing the value of m affect the graph of a member of the y = mx + b family? Illustrate using several examples. Continue experimenting with choices for m until you know what m controls on the graph.
c)The numbers m and b are called the slope and y-intercept, respectively. Do you think slope and y-intercept are descriptive names for m and b? Why?
Open the Excel spreadsheet called “GraphingLines” and open the sheet called Slope Intercept. Play with the sliders to watch what happens to the line and to the equation.
If you want to create your own interactive spreadsheet that will do this, open the Excel file called SlopeIntercept, which has been created with instructions on how to create “sliders” or scroll bars.
By changing your window settings, you can affect the appearance of a line described by a member of the y = mx + b family without changing the values of m or b. At times, you may want to adjust your window settings to display your graph more effectively. However, you should
also be aware that some people, driven by an interest in distorting the truth, will tinker with their window settings until they achieve a graph that satisfies their purpose. Your understanding of how scale change affects the appearance of the line will help you interpret graphs correctly and avoid being misled by their distortions. The next investigation will help you learn the effects on a graph of changing the maximum settings for the horizontal or vertical axis.
2.In Activity 2, you drew a graph of Dr. Trotter’s equation by hand. Now you will reproduce your hand-drawn graph using a graphing calculator.
a)Set the viewing window on your calculator to match the scalings on the axes of your hand-drawn graph from Item 2, Activity 2. (For example, set xmin = 35, xmax = 60, xscl = 5. The y-settings will depend on your choice of scale for the vertical axis.) Enter Dr. Trotter’s equation into your calculator and then graph the equation. How does your calculator-produced graph compare with your hand-drawn graph?
b)Experiment with changing the scale on the vertical axis by first increasing the value of ymax and then decreasing the value of ymax. How would you change the value of ymax to make the graph of Dr. Trotter’s equation appear very steep? How would you change the value of ymax to make the graph appear much flatter?
c)Without actually changing the scaling on the horizontal axis, predict what would happen to the appearance of the graph if you changed the value of xmax from 60 to 120. Why do you think your graph will change as you predicted? Finally, check your prediction by changing the xmax setting from 60 to 120.