Scatter Plots and Line of Best Fit

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Presentation transcript:

Scatter Plots and Line of Best Fit Essential Question: How do we use scatter plots to find the line of best fit by hand and by using a calculator?

DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have a positive correlation, which means that the points can be approximated by a line with a positive slope.

DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have a negative correlation, which means that the points can be approximated by a line with a negative slope.

DETERMINING THE CORRELATION OF X AND Y In this scatter plot, x and y have relatively no correlation, which means that the points cannot be approximated by a line.

DETERMINING THE CORRELATION OF X AND Y TYPES OF CORRELATION Positive Correlation Negative Correlation No Correlation

Line of Best Fit line of best fit - A line of best fit (or "trend" line) is a straight line that best represents the data on a scatter plot. A graphing calculator computes the equation of a line of best fit using a method called linear regression. The graphing calculator gives you the correlation coefficient r. -1 0 1 negative correlation no correlation positive correlation

Finding the Line of Best Fit Using a Graphing Calculator

Finding the Line of Best Fit Using a Graphing Calculator

Finding the Equation of Line of Best Fit Example 1: Find an equation for the trend line and the correlation coefficient. Estimate the number of calories in a fast-food that has 14g of fat. Show a scatter plot for the given data. Calories and Fat in Selected Fast-Food Meals Fat(g) 6 7 10 19 20 27 36 Calories 276 260 220 388 430 550 633 Solution: Use your graphing calculator to find the line of best fit and the correlation coefficient.

Cont (example 1)... Step 1. Use the EDIT feature of the STAT screen on your graphing calculator. Enter the data for fat (L1) and the data for Calories (L2) Calories and Fat in Selected Fast-Food Meals Fat(g) 6 7 10 19 20 27 36 Calories 276 260 220 388 430 550 633

Cont (example 1)... Step 2. Use the CALC feature in the STAT screen. Find the equation for the line of best fit  LinReg (ax + b) LinReg y = ax + b a = 13.60730858 b = 150.8694896 r2 = .9438481593 r = .9715184812 Slope y-intercept Correlation coefficient The equation for the line of best fit is y = 13.61x + 150.87 and the correlation coefficient r is 0.9715184812

Cont (example 1)... Estimate the number of calories in a fast-food that has 14g of fat. Solution: Graph the Linear Regression Y= , VARS, Statistics, EQ, RegEQ *Highlight Plot 1 ZOOM , 9(Zoom Statistics) 2nd, Trace value, ____, enter y = 341.37 There is approximately 341g of calories when there is 14g of fat.

Finding the Equation of Line of Best Fit Recreation Expenditures Example 2. Use a graphing calculator to find the equation of the line of best fit for the data at the right. What is the correlation coefficient? Estimate the recreation expenditures in 2010. Year Dollars (Billions) 1993 340 1994 369 1995 402 1996 430 1997 457 1998 489 1999 527 2000 574 Answers: y = 32.33x - 2671.67 r = 0.9964509708 The expenditures in 2010 will be 885 billions Let 1993 = 93

Do these… Find the equation of the line of best fit. Let 91 correspond to 1991. What is the correlation coefficient? Show a scatter plot. If the trend continues,, how much will the gross be in 2009? Yearly Box Office Gross for Movies (Billions) 1991 1992 1993 1994 1995 1996 1997 1998 1999 $4.8 $4.9 $5.2 $5.4 $5.5 $6.0 $6.4 $7.0 $7.4 Answers: y = 0.33x – 25.35 r = 0.9751360069 $10.44 billions

Ave July Temperature (F) Ave Annual Precipitation (in.) Do these… Use a graphing calculator to find the equation of line of best fit for the data. Find the value of the correlation coefficient r. Show a scatter plot for each data. 2. The data below represents the average July temperature and the annual precipitation of the cities. Estimate the average rainfall for a city with average July temperature of 75F. Estimate the temperature if the ave. precipitation is 40 in. City Ave July Temperature (F) Ave Annual Precipitation (in.) New York 76.4 42.82 Baltimore 76.8 41.84 Atlanta 78.6 48.61 Jacksonville 81.3 52.76 Washington, DC 78.9 39.00 Boston 73.5 43.81 Miami 82.5 57.55 Answers: y = 1.58x – 76.74; r = 0.725197858 About 41.45 in; about 74.08 F

Do these… Use a graphing calculator to find the equation of line of best fit for the data. Find the value of the correlation coefficient r. 3. Use the equation to predict the time needed to travel 32 miles on a bicycle. How many miles will he travel for 125 mins. Speed on a Bicycle Trip Miles 5 10 14 18 22 Time (min) 27 46 71 78 107 Answers: y = 4.56x + 2.83 r. = 0.9881161783 about 148.85 min or 149 min 26.77 or 27 miles.