2 7 Curve Fitting with Linear Models Warm
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2 -7 Curve Fitting with Linear Models 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. 2 x + 4. Find x for each given value of y. 3. y = 7 x = – 15 Holt Algebra 2 4. y = 3. 5 x = 2. 5
Curve. Fittingwith. Linear. Models 2 -7 Curve Holt Algebra 22
2 -7 Curve Fitting with Linear Models Objectives Can you fit scatter plot data using linear models with and without technology? Can you use linear models to make predictions? Holt Algebra 2
2 -7 Curve Fitting with Linear Models Vocabulary regression correlation line of best fit correlation coefficient Holt Algebra 2
2 -7 Curve Fitting with Linear Models Researchers, such as anthropologists, are often interested in how two measurements are related. The statistical study of the relationship between variables is called regression. Holt Algebra 2
2 -7 Curve Fitting with Linear Models 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. Holt Algebra 2
2 -7 Curve Fitting with Linear Models 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. Helpful Hint Try to have about the same number of points above and below the line of best fit. Holt Algebra 2
2 -7 Curve Fitting with Linear Models 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. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 1 Continued Step 1 Plot the data points. Step 2 Identify the correlation. o Notice that the data set is negatively correlated–as the temperature rises in Albany, it falls in Sydney. • • • • o Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 1 Continued Step 3 Sketch a line of best fit. o Draw a line that splits the data evenly above and below. • • • • o Holt Algebra 2
2 -7 Curve Fitting with Linear Models 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. Point-slope form. y – y 1= m(x – x 1) y – 64 = – 0. 46(x – 35) y = – 0. 46 x + 80. 1 Substitute. Simplify. An equation that models the data is y = – 0. 46 x + 80. 1. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 2 Make a scatter plot for this set of data. Identify the correlation, sketch a line of best fit, and find its equation. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 2 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 • • • Holt Algebra 2 • •
2 -7 Curve Fitting with Linear Models Example 2 cont. Step 3 Sketch a line of best fit. Draw a line that splits the data evenly above and below. • • • Holt Algebra 2 • •
2 -7 Curve Fitting with Linear Models The correlation coefficient r is a measure of how well the data set is fit by a model. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 3 The gas mileage for randomly selected cars based upon engine horsepower is given in the table. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 3 cont. a. Make a scatter plot of the data with horsepower as the independent variable. The scatter plot is shown on the right. Holt Algebra 2 • • • •
2 -7 Curve Fitting with Linear Models Example 3 cont. The slope is about – 0. 15, so for each 1 unit increase in horsepower, gas mileage drops ≈ 0. 15 mi/gal. The correlation coefficient is r ≈ – 0. 916, which indicates a strong negative correlation. Holt Algebra 2
2 -7 Curve Fitting with Linear Models Example 3 cont. c. Predict the gas mileage for a 210 -horsepower engine. The equation of the line of best fit is y ≈ – 0. 15 x + 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. Holt Algebra 2
2 -7 Curve Fitting with Linear Models BELLWORK 2. 7 Use the table for Problems 1– 3. 1. Make a scatter plot with mass as the independent variable. Holt Algebra 2
2 -7 Curve Fitting with Linear Models BELLWORK 2. 7 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. 07 x – 5. 24 Holt Algebra 2
2 -7 Curve Fitting with Linear Models BELLWORK 2. 7 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. Holt Algebra 2
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