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Section 2-7: Scatter Plots and Correlation Goal: See correlation in a scatter plot and find a best-fitting line.

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Warm-Up Exercises Find the slope of the line through and. 2, 6() – 5, 1() – 1. )2, 5 Write an equation of the line through and. 4, 8()( – 2. A line’s graph has slope and contains the point. Write an equation of the line. 3. 3 2 6, 1 () ANSWER 2 1 yx = + 6 1 – 3 2 y x = – 3

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Scatter Plot Graph of a set of data pairs (x, y). A scatter plot can help you identify the type of relationship, or correlation, between two variables.

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Correlations Positive Correlation: as x increases, y tends to increase Negative Correlation: as x increases, y tends to decrease Relatively No Correlation: there is no obvious pattern between x and y

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Example 1 Identify Correlation Televisions The scatter plots compare unit sales of plasma television sets with those of LCD television sets and with those of analog direct-view color television sets (older-style “picture-tube” sets). Describe the correlation shown by each plot.

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Example 1 Identify Correlation SOLUTION The first scatter plot shows a positive correlation: as sales of plasma sets increased, sales of LCD sets increased. The second plot shows a negative correlation: as sales of plasma sets increased, sales of analog direct-view color sets decreased.

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Checkpoint Draw a scatter plot of the data. Then tell whether the data show a positive correlation, a negative correlation, or relatively no correlation. ANSWER relatively no correlation. Identify Correlation (1, 7), (1, 5), (2, 3), (3, 2), (3, 6), (5, 5), (6, 4), (6, 8), (7, 6), (8, 2)

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Example 2 Find a Best-Fitting Line Movies The table gives the total number y (in billions) of U.S. movie admissions x years after 1993. Approximate the best-fitting line for the data. Year, x0 Admissions, y 1.24 1 1.29 2 1.26 3 1.34 4 1.39 5 1.48 Year, x6 Admissions, y 1.47 7 1.42 8 1.49 9 1.63 10 1.57 11 1.53

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Example 2 Find a Best-Fitting Line SOLUTION STEP 1Draw a scatter plot of the data. STEP 2 Sketch the line that appears to best fit the data. A possibility is shown. STEP 3 Choose two points. The line shown appears to pass through the data point (3, 1.34) and through (11, 1.6), which is not a data point.

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Example 2 Find a Best-Fitting Line STEP 4 Write an equation of the line. First find the slope using the two points: – 1.6 – 11 = 1.34 38 = 0.26 = 0.0325 m Now use point-slope form to write an equation. Choose (x 1, y 1 ) (11, 1.6). = yy1y1 – = () xm x1x1 – Point-slope form y 1.6 – = () x0.0325 11 – Substitute for y 1, m, and x 1. y 1.6 – = 0.0325x 0.3575 – Distributive property

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Example 2 Find a Best-Fitting Line y = 0.0325x 1.2425 + Solve for y. ANSWER An approximation of the best-fitting line is y = 0.0325x 1.24. +

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Example 3 Use a Best-Fitting Line Walking In a class experiment, students walked a given distance at various paces, from normal to as fast as possible (“race walking”). By measuring the time required and the number of steps, the class calculated the speed and the stride, or step length, for each trial. The table shows the data recorded. Speed ( yd/sec ) 0.8 Stride ( yd ) 0.5 0.85 0.6 0.9 0.6 1.3 0.7 1.4 0.7 1.6 0.8 2.15 0.9 2.5 1.0 2.8 1.05 3.0 1.15 3.1 1.25 3.3 1.15 1.75 0.8 3.35 1.2 1.9 0.9 3.4 1.2 Speed ( yd/sec ) Stride ( yd )

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Example 3 Use a Best-Fitting Line a. Approximate the best-fitting line for the data. b.Predict the stride length for a class member walking at 2 yards per second. SOLUTION a.Draw a scatter plot of the data. Sketch the line that appears to best fit the data. A possibility is shown. Choose two points on the line. It appears to pass through (0.9, 0.6) and (2.5, 1).

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Example 3 Use a Best-Fitting Line Write an equation of the line. First find the slope using the two points: – 1 – 2.5 = 0.6 0.91.6 = 0.4 = 0.25 m Use point-slope form as in Example 2 to write an equation. ANSWER An approximation of the best-fitting line is y = 0.25x 0.38. +

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Example 3 Use a Best-Fitting Line ANSWER A class member walking at 2 yards per second will have a stride length of about 0.88 yard. b. To predict the stride length for a class member walking at 2 yards per second, use the equation from part (a), substituting 2 for x. y = 0.25x 0.38 + Write the linear model. y = 0.38 + Substitute 2 for x. () 20.25 y = Simplify. 0.88

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Checkpoint Employment The table shows the percent p of the U.S. work force made up of civilian federal government employees t years after 1970. Approximate the best-fitting line for the data. What does your model predict for the percent of the work force made up of civilian federal government employees in 2015 ? 2. 0 Percent, p 3.81 5 3.35 10 3.01 15 2.80 20 2.72 25 2.36 30 2.10 35 1.91 Years, t Find and Use a Best-Fitting Line ANSWER p = 0.05t 3.66 ; 1.41 + – Sample answer:

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Homework: p. 110 – 111 #7 – 21 all

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