Fitting a Line to Data

Transcription

1 Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Fitting a Line to Data Objective: Students will learn to draw a line of good fit and to write its equation. Connections to Previous Learning: Students should be able to draw scatterplots and write the equation of a line. Connections to AP*: AP Statistics Topic: Bivariate data Materials: Student Activity pages, rulers, graph paper. Teacher Notes: This lesson is designed to introduce constructing a trend line from data in order to make a prediction. This lesson can be introduced when students are learning how to write the equations of lines either in slope-intercept or point-slope form. While most linear regression equations are written in the form y = a + bx, it would not complicate matters to let students write their equation in y = mx + b form. This is a non-calculator lesson so students should use graph paper to construct their scatterplots and a ruler to draw their trend lines. The objective is to draw a line that fits the data so many teachers start to use the terminology best fit line or fitted line instead of trend line as this is more in keeping with statistical terminology. This line will be a line of good fit while it may not be the line of best fit. In order to determine a line of best fit students would have to use techniques beyond the realm of Algebra I. Note: When drawing the scatterplot, the variable that is being predicted goes on the vertical axis and the other variable (explanatory or independent) goes on the horizontal axis. Emphasize with students that the line they are going to draw needs to go through the middle of the data so they should think about this and experiment a little with their ruler before drawing their line. Another concept that is introduced is that of a residual. One objective in regression is to make a prediction for the y-value when you are given some x-value. The difference between the actual y- values and the predicted y-values determined from the fitted line are called residuals (residual = actual value predicted value). If the line is a good fit we would anticipate that most of the residuals will be small. The first example is a group instruction problem. The second can be a partner problem or an individual problem *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 1

2 Student Activity Fitting a Line to Data A large company is expanding its workforce and needs to hire some new administrative assistants. The company wants to know what the relationship is between the amount of experience that its current administrative assistants possessed when they were hired and their starting salary with the company. The data for ten randomly selected administrative assistants is given below. Experience (in months) Starting Salary (in $1000) a) What kind of the relationship would you expect between length of experience and starting salary for the randomly selected administrative assistants. b) Construct a scatterplot of the data. Which variable should be the independent variable and which is the dependent variable? Remember to include scales and labels for your axes. c) Does the scatterplot confirm your description in part (a)? Explain your answer. d) Suppose you wanted to hire an administrative assistant who had 17 months experience. Predict what the starting salary would be. Describe how you used the scatterplot to help you. How does your prediction compare with other student s predictions? Since there seems to be an almost linear relationship between starting salary and amount of job experience in months, a line can be drawn on the scatterplot to summarize this relationship. Such a line helps us to predict the value of the variable on the vertical axis (the dependent variable) from the value of the variable on the horizontal axis (the independent variable). On your scatterplot in part (b), draw a line that you think summarizes or fits the data. This line should go through the middle of the set of data points. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 2

3 Student Activity e) Use the line you drew to predict the approximate starting salary for an assistant with 12 months experience. f) According to the data, the actual salary of an assistant with 12 months experience was $33,000. How close was your prediction? Calculate this by subtracting the predicted value from the actual value. This difference in called a residual. Did you over-predict or under-predict your estimate? Justify your answer. g) Compare your prediction to those made by others in your class. Who was the closest to the actual value? What can we say about the residual of the person with the closest prediction? h) Look at the line drawn by the person who had the smallest residual value. Do you think that this person had the line that best fits the data? Explain your answer. i) Instead of using the graph of the fitted line to predict a starting salary based on experience, you can use an equation for the line to do it. Pick two ordered pairs on the line you drew on the scatterplot of length of experience and starting salary. Use your ordered pairs to write an equation of the line. Write your answer in the form of y = mx + b. j) What does the variable y represent in your equation? What does the variable x represent in your equation? k) What is the slope of this line? Interpret the slope in the context of this problem. l) Use your equation from part (i) to predict the starting salary for an assistant who has 17 months experience. m) Compare your prediction from part (l) to your estimate from the scatterplot in part (d). Are they reasonably close? Do you think having the equation for the fitted line makes it easier to predict a value? Explain your answer. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 3

4 Student Activity 2. Car dealers across North America use the Blue Book to help them determine the value of used cars that customers trade in when purchasing new vehicles. The book lists on a monthly basis the amount paid at recent used-car auctions and indicates the trade-in values according to condition and optional features. A study was completed to determine whether the odometer reading would serve as a useful predictor of trade-in value. Five-year-old cars of the same make, model, condition, and options have been randomly selected. The trade-in value and mileage are shown below. Odometer Reading Trade-in Value ($) 58, , , , , , , , , , , a) What kind of relationship would you expect between trade-in value and odometer reading for the randomly selected cars. Explain your answer. b) Code the data to make it easier to use. c) Construct a scatterplot of the coded data. Which variable should be the independent variable and which is the dependent variable? Do you think that odometer reading depends on trade-in value or trade-in value depends on the odometer reading? Remember to include scales and labels for your axes. d) Does the scatterplot confirm your description in part (a)? Explain your answer. e) On your scatterplot in part (b), draw a line that you think summarizes or fits the data. Pick two ordered pairs on the line you drew on the scatterplot of trade-in value and odometer reading. Use your ordered pairs to write an equation of the line. Write your answer in the form of y = mx + b. f) What does the variable y represent in your equation? What does the variable x represent in your equation? Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 4

5 Student Activity g) What is the slope of this line? Interpret the slope in the context of this problem. h) Use your equation from part (e) to predict the trade-in value of a vehicle with 52,000 miles. i) According to the data, the actual trade-in value of a car with 52,000 miles was $4,000. How close was your prediction? Calculate the residual. Did you over-predict or underpredict your estimate? Justify your answer. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 5

6 Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Fitting a Line to Data Answers: 1. a) I would expect that as the level of experience increases the starting salary for an administrative assistant will also increase. b) The independent variable should be experience and the dependent variable should be starting salary. 40 Salary ($1000) Experience (months) 20 c) The scatterplot confirms the description in part (a) since the data forms a positive relationship between experience and starting salary. d) There is no data point corresponding to an assistant with 17 months experience. If you were to lay a ruler on the scatterplot going through the data points around 17 months we can see that the salary range is in the neighborhood of $35,000 to $37,000, so let s estimate it as $36,000 Answers may vary depending on what students describe as their method. e) Answers will vary depending on the line that the student drew and how accurate the student can estimate. f) Suppose that the student estimates the salary to be $32,500. The residual would be $33,000 $32,5000 or $500. If the residual is positive, then the actual value is more than the predicted value so the line is under-predicting; if the residual is negative, then the predicted value is more than the actual value so the line would be over-predicting at that point. g) Answer will vary. h) It is possible that this is the line that fits the data best but it could just be that the line is off substantially for other points and just happens to be close for this particular value. Ideally the best line should consistently give residuals that are small for each of the data points in the problem. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 6

7 Answers i) Answers will vary depending on the chosen points but should be close to the equation y = 0.80x j) The variable y represents the starting salary in $1000 and the variable x represents the experience in months for an administrative assistant. It is helpful to write the equation representing the line in the context of the situation. This helps emphasize the relationship between the variables in the problem. Thus the equation would be written as Salary = 0.80 (months) k) The slope of the line is This means that for every 1 month increase in experience the starting salary increases by approximately 0.80(1000) = $800. l) Answers will vary depending on the student s equation but should be approximately y = 0.80(17) = The predicted starting salary for an administrative assistant who possesses 17 months experience would be approximately $35,400. m) Part (d) the predicted value from the scatterplot was $36,000 and from the fitted line the prediction was $35,400. They are reasonably close to each other but it is easier to use the fitted equation rather than eyeball an estimate form the scatterplot. 2. a) As odometer readings increases a car has been driven more so the trade-in value will decrease. b) Code the data by using odometer readings in thousands of miles and the trade-in value in hundreds of dollars. Odometer Reading (1000s) Trade-in Value ($100) Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 7

8 Answers c) 55 Trade-in Value ($100) Miles (1000s) d) Yes. There is a negative linear relationship between odometer reading and trade-in value. e) Student responses may vary considerably depending where they draw their line to account for the high point of (39,55). Ideally it should be close to y = -0.46x f) The variable y represents the trade-in value in hundreds of dollars and the variable x represents the odometer reading in thousands of miles. The equation could therefore be mileage written in the context of the situation as Value = ( )(100 dollars) 1000 g) Answers will vary depending on student s work. If the slope of the line is given by , then for every 1000 mile increase in odometer reading the trade-in value of the car will decrease by approximately (.46)(100) = $ Note: the units of the slope are units of y given by ; that is, hundreds of dollars per thousand miles. units of x h) Value = -0.46(52) = The predicted trade-in value for a car with 52,000 miles is approximately $4213. i) Residual = actual-predicted = = -$213. Since the residual has a negative value the fitted line is over-predicting the trade-in value. Copyright 2009 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: 8