Linear Models and Scatter Plots

2015

Objectives

• Interpret correlation

• Use a graphing calculator to find linear models and make predictions about data.

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HWQ 8/11/15

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3 13 2, find f x x f x

Extra practice:

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13 3 10, find f x x f x

Interesting result:

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2

1

16 , 0 4

find

f x x x

f x

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x2 4

–2

– 4

y

2

4

6

A scatter plot represents data graphically using points plotted on a rectangular coordinate system.

Example: (x, y)

(1, – 4)

(2, – 2)

(3, – 1)

(4, 0)

(5, 2)

(6, 4)

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In a collection of ordered pairs (x, y), if y tends to increase as x increases, the collection has a positive correlation.

x

y

x

y

x

y

If y tends to decrease as x increases, the collection has a negative correlation.

positive correlation no correlationnegative correlation

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Example: The average salary S (in millions of dollars) for professional baseball players from 1996 through 2002 is shown in the table. Let t = 6 represent the year 1996. Draw the scatter plot.

Year Salary, S1996 1.11997 1.31998 1.41999 1.62000 1.82001 2.12002 2.3

S

4t

8

1

2

Year (6 1996)

Sal

ary

(in

mil

lion

s of

dol

lars

)

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Year Salary, S1996 1.11997 1.31998 1.41999 1.62000 1.82001 2.12002 2.3

Graphing Utility: Draw the scatter plot that describes the data.

Stat Plot Menu:

Zoom Menu:

Stat Edit Menu:

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Finding a linear model to represent the relationship described by a scatter pot is called fitting the line to data.

Year Salary, S1996 1.11997 1.31998 1.41999 1.62000 1.82001 2.12002 2.3

4 t8

S

1

2

Example: The table and scatter plot for the average salary S (in millions of dollars) for professional baseball players from 1996 through 2002 is shown. Let t = 6 represent the year 1996.

Find the equation of the line.

Example continues.

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Year Salary, S1996 1.11997 1.31998 1.41999 1.62000 1.82001 2.12002 2.3

4 t8

S

1

2

(6, 1.1)

(12, 2.3)

The equation of this line is S = 0.2t – 0.1.

This line approximates the data.

Example continued:

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Year Salary, S1996 1.11997 1.31998 1.41999 1.62000 1.82001 2.12002 2.3

Graphing Utility: Find a linear model that describes the data.

Stat Menu:

A linear model for this data is S = 0.2t – 0.14. This equation is very close to the equation found using two data points. What does the model predict Baseball players salaries to be in 2015?

Stat Edit:

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The correlation coefficient (or r-value) of the data gives the measure of how well the model fits the data.

The closer |r| is to 1, the better the points can be described by a line.

r = 0.99 strong positive

correlation

r = – 0.93 negative

correlation

0 13

2

0

52.6

52.30 13

r = 0.66 weak correlation

509050

100

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Year Radio

Stations, R1970 6,7601975 7,7441980 8,5661985 10,3591990 10,788

1995 11,8342000 13,058

A linear model for this data is R = 209.257t +6733.857.

15,104

You Try: The number R of U.S. radio stations for selected years is shown in the table. Use a graphing utility to find a linear model that describes the data. Let t=0 correspond to 1970.For the year 2010, how many radio stations does the model predict?

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Homework

Pg. 773-6 all, 7-15 odds, 27-37 odds