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Section 2-7: Scatter Plots and Correlation

Section 2-7: Scatter Plots and Correlation. Goal : See correlation in a scatter plot and find a best-fitting line. (. 2 , 6. ). –. 1. ANSWER. 2. Write an equation of the line through and. –. (. 2 , 5. ). (. 4 , 8. ). 1. 6. y. x. +. ANSWER. =. 2. 2.

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Section 2-7: Scatter Plots and Correlation

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

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

  3. 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.

  4. 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

  5. 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.

  6. 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.

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

  8. Example 2 Year,x 0 1 2 3 4 5 1.24 1.29 1.26 1.34 1.39 1.48 Admissions,y Year,x 6 7 8 9 10 11 1.47 1.42 1.49 1.63 1.57 1.53 Admissions,y 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.

  9. Example 2 STEP 1 Draw a scatter plot of the data. Sketch the line that appears to best fit the data. A possibility is shown. STEP 2 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. Find a Best-Fitting Line SOLUTION

  10. Example 2 Write an equation of the line. First find the slope using the two points: STEP 4 – 1.6 1.34 0.26 m 0.0325 = = = – 11 3 8 Now use point-slope form to write an equation. Choose (x1,y1) (11, 1.6). = – – ( ) y y1 Point-slope form m x x1 = – – ( ) y Substitute for y1, m, and x1. 0.0325 x 1.6 11 = – – y Distributive property 0.0325x 1.6 0.3575 = Find a Best-Fitting Line

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

  12. Example 3 Speed(yd/sec) 0.8 0.85 0.9 1.3 1.4 1.6 1.75 1.9 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 Stride (yd) 2.15 2.5 2.8 3.0 3.1 3.3 3.35 3.4 Speed(yd/sec) 0.9 1.0 1.05 1.15 1.25 1.15 1.2 1.2 Stride (yd) 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.

  13. Example 3 SOLUTION a. Draw a scatter plot of the data. Use a Best-Fitting Line a. Approximate the best-fitting line for the data. • Predict the stride length for a class member walking • at 2 yards per second. 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).

  14. Example 3 – 1 0.6 0.4 m 0.25 = = = – 2.5 0.9 1.6 ANSWER An approximation of the best-fitting line is + y 0.25x 0.38. = Use a Best-Fitting Line Write an equation of the line. First find the slope using the two points: Use point-slope form as in Example 2 to write an equation.

  15. Example 3 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 Write the linear model. 0.25x 0.38 = + y Substitute 2 for x. 0.38 = y Simplify. 0.88 = ( ) 0.25 2 ANSWER A class member walking at 2 yards per second will have a stride length of about 0.88 yard. Use a Best-Fitting Line

  16. Checkpoint Years, t 0 5 10 15 20 25 30 35 3.81 3.35 3.01 2.80 2.72 2.36 2.10 1.91 Percent, p – + ANSWER Sample answer: p 0.05t 3.66; 1.41 = Find and Use a Best-Fitting Line 2. 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?

  17. Homework: p. 110 – 111 #7 – 21 all

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