Transcript Plasma television screens

Section 2-7: Scatter Plots and
Correlation
Goal: See correlation in a scatter plot
and find a best-fitting line.
Warm-Up Exercises
1.
Find the slope of the line through
ANSWER
2.
3.
y =
1
2
and ( – 2,. 5 )
x + 6
2
and contains the point
3
. Write an equation of the line.
( 6, 1 )
A line’s graph has slope
ANSWER
( 2, –6 )
–1
Write an equation of the line through
ANSWER
and ( – 5, 1 .)
y =
2
3
x – 3
( 4, 8 )
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.
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
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
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)
ANSWER
relatively no correlation.
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
Admissions, y
Year, x
Admissions, y
0
1
2
3
4
5
1.24
1.29
1.26
1.34
1.39
1.48
6
7
8
9
10
11
1.47
1.42
1.49
1.63
1.57
1.53
Example 2
Find a Best-Fitting Line
SOLUTION
STEP 1
Draw 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.
Example 2
STEP 4
Find a Best-Fitting Line
Write an equation of the line. First find
the slope using the two points:
m =
1.6
– 1.34
11 – 3
=
0.26
8
= 0.0325
Now use point-slope form to write an equation. Choose
1.6).
=
y –
y1 = m ( x –
x1)
y –
1.6 = 0.0325 ( x – 11)
y –
1.6 = 0.0325x
– 0.3575
(x1, y1)
(11,
Point-slope form
Substitute for y1, m, and x1.
Distributive property
Example 2
Find a Best-Fitting Line
y = 0.0325x
+ 1.2425
ANSWER
An approximation of the best-fitting line is
y = 0.0325x
+ 1.24.
Solve for y.
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
0.85
0.9
1.3
1.4
1.6
1.75
1.9
Stride (yd)
0.5
0.6
0.6
0.7
0.7
0.8
0.8
0.9
Speed (yd/sec)
2.15
2.5
2.8
3.0
3.1
3.3
3.35
3.4
Stride (yd)
0.9
1.0
1.05
1.15
1.25
1.15
1.2
1.2
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:
m =
1 – 0.6
2.5 – 0.9
=
0.4
1.6
= 0.25
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
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 =
0.25( 2 ) + 0.38
Substitute 2 for x.
y =
0.88
Simplify.
ANSWER
A class member walking at 2 yards per second will have a stride length of
about 0.88 yard.
Checkpoint
2.
Find and Use a Best-Fitting Line
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?
Years, t
Percent, p
ANSWER
0
5
10
15
20
25
30
35
3.81
3.35
3.01
2.80
2.72
2.36
2.10
1.91
Sample answer:
p = – 0.05t
+ 3.66; 1.41
Homework:
p. 110 – 111
#7 – 21 all