Section 5.3 5.4: Modelling Data with a Line of Best Fit and a Curve of Best Fit 1
You will be expected to: Graph data, and determine the polynomial function that best approximates the data. Interpret the graph of a polynomial function that models a situation, and explain the reasoning. Solve, using technology, a contextual problem that involves data that is best represented by graphs of polynomial functions, and explain the reasoning. You are not expected to determine the equation of a regression line or curve without the use of technology. Graphing calculators, mobile devices, smartphones and tablet applications, computer programs or online tools, such as Geogebra, can be used to calculate and graph regression lines/curves. Quick Graph 2
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Terminology Independent Variable: the variable that is being manipulated. always placed on the x axis. Dependent Variable: the variable that is being observed. always placed on the y axis. Scatter Plot: A set of points on a grid, used to visualize a relationship or possible trend in the data. Line of Best Fit: A straight line that best approximates the trend in a scatter plot. Regression Function: A line or curve of best fit developed through a statistical analysis of data. Interpolation: the process used to estimate a value within the domain of a set of data, based on a trend. Extrapolation: is the process used to estimate a value outside the domain of a set of data, based on a trend. 4
Step 1. Enter the data. How to Create a Scatter Plot and a Regression Line or Curve using TI 83 Press STAT key Select EDIT Clear any numbers that are written in L1, L2 (Place cursor on L1, select CLEAR, then down arrow key. Repeat for L2) Under Column L1, enter the data (x values) Under Column L2, enter the data (y values) Step 2. Create scatter plot. Press 2nd Y= Select 1:Plot1 Select On Select Type (Select first one) Xlist:L1, Ylist: L2 Mark. 2nd Mode Press ZOOM 9:ZoomStat Step 3. Obtain the function. Press STAT key Select CALC Select #4 LinReg (or #5 QuadReg or #6 Cubic Reg) Enter L1, L2, VARS Select Y VARS Select #1 Function Y1 5
Linear Regressions The slope intercept form of the equation of a line is y = mx + b, where m is the slope of the line and b is the y intercept of the line. Examples: 1. The one hour record is the farthest distance travelled by bicycle in 1 h. The table below shows the world record distances and the dates they were accomplished. a. Use technology to create a scatter plot and to determine the equation of the line of best fit. Round to three decimal places. b. Interpolate, using your equation a possible world record distance for the year 2006, to the nearest hundredth of a kilometre. 6
2. The following scatter plot shows the change in temperature from 1 0pm 1 to 2 5 pm 3 4 in 5Port 6 Hope 7 8 Simpson. 9 10 11 12 13 14 15 00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 00 1 12 3 4 2 5 6 37 8 9 4 10 11 12 5 13 14 15 0 0 1 2 3 4 5 0 1 2 3 4 5 A) Draw the line of best fit for the scatterplot below. (Note: A Linear regression results in an equation that balances the points in the scatter plot on both sides of the line. B) Use Graphing technology to determine the actual linear regression equation. C) What is the slope and y intercept of the regression equation? What does each one represent in this situation? 7
3. Consider the data in the table. Use technology to create a scatter plot and to determine the equation of the line of best fit. a. Determine, to the nearest tenth, the value of y when x is 10.6. Use your calculator. b. Determine, to the nearest tenth, the value of x when y is 9.8. Use the regression equation. 8
4. Matt buys T shirts for a company that prints art on T shirts and then resells them. When buying the T shirts, the price Matt must pay is related to the size of the order. Five of Matt s past orders are listed in the table below. Matt has misplaced the information from his supplier about price discounts on bulk orders. He would like to get the price per shirt below $1.50 on his next order. a. Use technology to create a scatter plot and determine an equation for the linear regression function that models the data. Round to three decimal places. b. What do the slope and y intercept of the equation of the linear regression function represent in this context? 9
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