Section 2-7: Scatter Plots and Correlation

Goal: See correlation in a scatter plot and find a best-fitting line.

Warm-Up Exercises

Find the slope of the line through and . 2, 6( )–5, 1( )–1.

)2, 5Write 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( )

ANSWER2

1y x= + 6

ANSWER 1–

ANSWER3

2y x= – 3

Scatter PlotGraph of a set of data pairs (x,

y). A scatter plot can help you identify the type of relationship, or correlation, between two variables.

CorrelationsPositive 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

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.

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.

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)

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, x 0

Admissions, y 1.24

1

1.29

2

1.26

3

1.34

4

1.39

5

1.48

Year, x 6

Admissions, y 1.47

7

1.42

8

1.49

9

1.63

10

1.57

11

1.53

Example 2 Find a Best-Fitting Line

SOLUTION

STEP 1 Draw a scatter plot of thedata.

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.

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

3 8=

0.26= 0.0325m

Now use point-slope form to write an equation. Choose (x1, y1) (11, 1.6). =

y y1– = ( )xm x1– Point-slope form

y 1.6– = ( )x0.0325 11– Substitute for y1, m, and x1.

y 1.6– = 0.0325x 0.3575– Distributive property

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.+

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 timerequired 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)

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).

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

0.9 1.6=

0.4= 0.25m

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.+

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

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:

Homework:p. 110 – 111#7 – 21 all