1. Lesson 1.7 Linear Models and Scatter Plots
Essential Question: How do you write equations to model real-world data?

2. Before we start… Is the following statement always true?
The taller you are, the more you weigh.

3. Scatter Plots and Correlation
Many real-life situations involve finding relationships between two variables, such as the year and the number of employees in the cellular telecommunications industry.
In a typical situation, data are collected and written as a set of ordered pairs. The graph of such a set is called a scatter plot.

4. What is a scatter plot?
A graph used to determine whether there is a relationship between paired data.
Scatter plots can show trends in the data. These trends are described by correlation.

5. The data in the table show the numbers E (in thousands) of employees in the cellular telecommunications industry in the United States from 2002 through Construct a scatter plot of the data.

6. Construct a scatter plot of the following data.
Height
(in inches
Shoe size
(U.S.) 68 10 61 8 57 6 60 7 58 66 9 65

7. What is correlation? Positive Negative None
The relationship between the x and y values.
Positive
Negative
None

8. Positive Correlation
y tends to increase as x increases.

9. Negative Correlation
y tends to decrease as x increases

10. No Correlation
The points show no obvious pattern.

11. Scatter Plots and Correlation
Positive correlation
Negative correlation
No correlation

12. On a Friday, 22 students in a class were asked to record the numbers of hours they spent studying for a test on Monday and the numbers of hours they spent watching television. The results are shown below. (The first coordinate is the number of hours and the second coordinate is the score obtained on the test.)
Study Hours:
(0, 40), (1, 41), (2, 51), (3, 58),
(3, 49), (4, 48), (4, 64), (5, 55),
(5, 69), (5, 58), (5, 75), (6, 68),
(6, 63), (6, 93), (7, 84), (7, 67),
(8, 90), (8, 76), (9, 95), (9, 72),
(9, 85), (10, 98)
TV Hours:
(0, 98), (1, 85), (2, 72), (2, 90),
(3, 67), (3, 93), (3, 95), (4, 68),
(4, 84), (5, 76), (7, 75), (7, 58),
(9, 63), (9, 69), (11, 55), (12, 58), (14, 64), (16, 48), (17, 51), (18, 41), (19, 49), (20, 40)
Construct a scatter plot for each set of data.
Determine whether the points are positively correlated, are negatively correlated, or have no discernible correlation. What can you conclude?

13. What is a best-fitting line?
The line that lies as close as possible to all the data points.
If the correlation coefficient for a set of data is near -1 or 1, the data can be modeled by this line.
You can approximate a best fitting line by graphing.

14. What is the correlation coefficient?
Denoted by r, this is a number from -1 to 1 that measures how well a line fits a set of data pairs (x, y)
If r is near 1, the points lie close to a line with positive slope.
If r is near -1, the points lie close to a line with negative slope.
If r is 0, the points do not lie close to any line.

16. How do you write equations to model real-world data?
Using a graphing calculator, create a scatter plot.
Find the line of best fit either by hand or using linear regression (the process of finding the best fitting line).
This line models your data.

17. Find a linear model that relates the year to the number of employees in the cellular telecommunications industry in the United States.

18. The data in the table show the estimated numbers v (in thousands) of alternative-fueled vehicles in use in the United States from 2001 through (Source: Energy Information Administration). Find a linear model for this data.

19. Find a linear model that relates the amount of rainfall to the depth of a stream.
(inches)
Depth, D 8
27.1 7
26.8 6
26.5 5 26 4
25.8 3
25.3 2
24.9 1
24.4 24

20. How closely does the model represent the data?
The median purchase prices (in thousands of dollars) for new one-family homes in the United States from 2000 to 2008 are given by the following ordered pairs.
Use the regression feature of a graphing utility to find a linear model for the data. Let t represent the year, with t = 0 corresponding to 2000.
How closely does the model represent the data?
(2000, $169.0)
(2001, $175.2)
(2002, $187.6)
(2003, $195.0)
(2004, $221.0)
(2005, $240.9)
(2006, $246.5)
(2007, $247.9)
(2008, $232.1)

21. The number of women W (in millions) in the civilian labor force in the United States from 2001 to 2008 are shown in the table.
Find the least squares regression line that fits this data, where t = 1 represents 2001.
Use a graphing utility to plot the data and graph the model in the same viewing window.
Year W
2001
66.8
2002
67.4
2003
68.3
2004
68.4
2005
69.3
2006
70.2
2007
71.0
2008
71.8

22. How do you write equations to model real-world data?

23. Ticket Out the Door
Grapefruit The table shows the price (in dollars) for one pound of grapefruit for the years 1997 through Write an equation that models the price of grapefruit since 1997.
Years since 1997
Price (dollars)
0.53 1
0.55 2
0.58 3 4
0.60 5
0.62