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Linear Regression and Scatter Plots for Data Analysis

Learn how to fit scatter plot data using linear models and make predictions. Discover different types of correlations and find equations for lines of best fit.

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Linear Regression and Scatter Plots for Data Analysis

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  1. Bell Ringer Find each slope: 1. (5, –1), (0, –3) 2. (8, 5), (–8, 7)

  2. 1.4 Objectives Fit scatter plot data using linear models with and without technology. Use linear models to make predictions. A line of best fit may also be referred to as a trend line.

  3. Negative Correlation Positive Correlation No Correlation Constant Correlation Four Kinds of Correlations (you will learn about in Transition)

  4. Scatter Plots + Calculator • 1) STAT • #1  • L1 (x) , L2 (y) (enter data; use arrow keys to select column)  • STAT  • CALC  • 4enter LinReg(ax+b)  • 2nd  • 8) y=  • plot1  • on  • TYPE • X list L1 & Y list L2  • mark (select on)  • GRAPH

  5. Example 1 That’s to much work with paper & pencil Albany and Sydney are about the same distance from the equator. (a)Make a scatter plot with Albany’s temperature as the independent variable. (b)Name the type of correlation. (c)Then sketch a line of best fit and (d)find its equation. Tables: ACT How to: Calculator data entry Enter ______ in list L1 by pressing STAT and then 1. Enter _______ in list L2 by pressing  Make scatter plot in the following way: Press 2nd Y= PLOT 1 set up desired type when done, press GRAPH cont inued

  6. o o Does yours look like this ? example 1 continued Albany and Sydney are about the same distance from the equator. (a)Make a scatter plot with Albany’s temperature as the independent variable. (b)Name the type of correlation. (c)Then sketch a line of best fit and (d)find its equation. (hint: what is m? b?) • • • • • • • • • • •

  7. Example 2 (a)Make a scatter plot for this set of data. (b)Identify the correlation (c)sketch a line of best fit (d)find its equation.

  8. example 2 continued Step 1 Plot the data points. Step 2 Identify the correlation. Notice that the data set is positively correlated–as time increases, more points are scored • • • • • • • • • •

  9. example 2 continued Step 3 Sketch a line of best fit. Draw a line that splits the data evenly above and below. • • • • • Step 4 Identify the equation for the data. • • • • • end

  10. Example 3: Anthropology Application Anthropologists can use the femur, or thighbone, to estimate the height of a human being. The table shows the results of a randomly selected sample. (a)Make a scatter plot for this set of data. (b)Identify the correlation (c)sketch a line of best fit (d)find its equation.

  11. example 3 continued a. Make a scatter plot of the data with femur length as the independent variable. • • • • • • • •

  12. Example 3 Continued b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem. Enter the data into lists L1 and L2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4:LinReg. The equation of the line of best fit is h ≈ 2.91l+ 54.04.

  13. Example 3 Continued What does the slope indicate about problem?

  14. Example 3 Continued c. A man’s femur is 41 cm long. Predict the man’s height. The equation of the line of best fit is h ≈ 2.91l+ 54.04. Use the equation to predict the man’s height. For a 41-cm-long femur, Substitute 41 for l. h ≈ 2.91(41)+ 54.04 h ≈ 173.35 The height of a man with a 41-cm-long femur would be about 173 cm. end

  15. Example 4 The gas mileage for randomly selected cars based upon engine horsepower is given in the table. • • • • • • • • • • a. Make a scatter plot of the data with horsepower as the independent variable.

  16. Example 4 Continued b. Find the correlation coefficient r and the line of best fit. Interpret the slope of the line of best fit in the context of the problem. Enter the data into lists L1 and L2 on a graphing calculator. Use the linear regression feature by pressing STAT, choosing CALC, and selecting 4:LinReg. The equation of the line of best fit isy ≈ –0.15x + 47.5.

  17. Example 4 Continued What does the slope indicate ? The slope is about –0.15, so for each 1 unit increase in horsepower, gas mileage drops ≈ 0.15 mi/gal. c. Predict the gas mileage for a 210-horsepower engine. The equation of the line of best fit is y ≈ –0.15x+ 47.5. Use the equation to predict the gas mileage. For a 210-horsepower engine, y ≈ –0.15(210)+ 47.50. Substitute 210 for x. y ≈ 16 The mileage for a 210-horsepower engine would be about 16.0 mi/gal. end

  18. Exit Question: complete on graph paper attached to Exit Question sheet (a)Make a scatter plot for this set of data using your calculator (b)find its equation.

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