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2-7 Curve Fitting with Linear Models Warm Up Lesson Presentation

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1 2-7 Curve Fitting with Linear Models Warm Up Lesson Presentation
Lesson Quiz Holt Algebra 2

2 Warm Up Write the equation of the line passing through each pair of passing points in slope-intercept form. 1. (5, –1), (0, –3) 2. (8, 5), (–8, 7) Use the equation y = –0.2x + 4. Find x for each given value of y. 3. y = y = 3.5 x = –15 x = 2.5

3 Objectives Fit scatter plot data using linear models with and without technology. Use linear models to make predictions.

4 Vocabulary regression correlation line of best fit
correlation coefficient

5 Researchers, such as anthropologists, are often interested in how two measurements are related. The statistical study of the relationship between variables is called regression.

6 A scatter plot is helpful in understanding the form, direction, and strength of the relationship between two variables. Correlation is the strength and direction of the linear relationship between the two variables.

7 If there is a strong linear relationship between two variables, a line of best fit, or a line that best fits the data, can be used to make predictions. Try to have about the same number of points above and below the line of best fit. Helpful Hint

8 Example 1: Meteorology Application
Albany and Sydney are about the same distance from the equator. Make a scatter plot with Albany’s temperature as the independent variable. Name the type of correlation. Then sketch a line of best fit and find its equation.

9 • • • • • • • • • • • Example 1 Continued Step 1 Plot the data points.
Step 2 Identify the correlation. Notice that the data set is negatively correlated–as the temperature rises in Albany, it falls in Sydney.

10 • • • • • • • • • • • Example 1 Continued
Step 3 Sketch a line of best fit. o Draw a line that splits the data evenly above and below.

11 Step 4 Identify two points on the line.
Example 1 Continued Step 4 Identify two points on the line. For this data, you might select (35, 64) and (85, 41). Step 5 Find the slope of the line that models the data. Use the point-slope form. y – y1= m(x – x1) Point-slope form. y – 64 = –0.46(x – 35) Substitute. y = –0.46x Simplify. An equation that models the data is y = –0.46x

12 The correlation coefficient r is a measure of how well the data set is fit by a model.

13 You can use a graphing calculator to perform a linear regression and find the correlation coefficient r. To display the correlation coefficient r, you may have to turn on the diagnostic mode. To do this, press and choose the DiagnosticOn mode.

14 Example 2: 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.

15 • • • • • • • • Example 2 Continued
a. Make a scatter plot of the data with femur length as the independent variable. The scatter plot is shown at right.

16 Example 2 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

17 Example 2 Continued The slope is about 2.91, so for each 1 cm increase in femur length, the predicted increase in a human being’s height is 2.91 cm. The correlation coefficient is r ≈ which indicates a strong positive correlation.

18 A line of best fit may also be referred to as a trend line.
Reading Math

19 Lesson Quiz: Part I Use the table for Problems 1–3. 1. Make a scatter plot with mass as the independent variable.

20 Lesson Quiz: Part II 2. Find the correlation coefficient and the equation of the line of best fit on your scatter plot. Draw the line of best fit on your scatter plot. r ≈ 0.67 ; y = 0.07x – 5.24

21 Lesson Quiz: Part III 3. Predict the weight of a $40 tire. How accurate do you think your prediction is? ≈646 g; the scatter plot and value of r show that price is not a good predictor of weight.

22 Homework P , 17, 23, 27, 31, 33 Due Friday


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