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Embed code for: 9.1 correlation day 1 april 17 2017
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Monday April 17, 2017
Thursday April 20, 2017
© 2012 Pearson Education, Inc.
All rights reserved.
Section 9.1 Objectives
© 2012 Pearson Education, Inc. All rights reserved.
Introduce linear correlation, independent and dependent variables, and the types of correlation
Find a correlation coefficient
Test a population correlation coefficient ρ using a table
Perform a hypothesis test for a population correlation coefficient ρ
Distinguish between correlation and causation
A linear relationship between two variables.
The data can be represented by ordered pairs:
Types of Correlation
As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation
Example 1: Constructing a Scatter Plot
An economist wants to determine whether there is a linear relationship between a country’s gross domestic product (GDP) and carbon dioxide (CO2) emissions.
The data are shown in the table.
Display the data in a scatter plot and determine whether there appears to be a positive or negative linear correlation or no linear correlation.
(Source: World Bank and U.S. Energy Information Administration)
(trillions of $), x
CO2 emission (millions of metric tons), y
Example 1 workspace Proper labeling for credit!
Solution: Constructing a Scatter Plot
Appears to be a positive linear correlation. As the gross domestic products increase, the carbon dioxide emissions tend to increase.
Example 3: Constructing a Scatter Plot Using Technology
Old Faithful, located in Yellowstone National Park, is the world’s most famous geyser. The duration (in minutes) of several of Old Faithful’s eruptions and the times (in minutes) until the next eruption are shown in the table. Using a TI-83/84, display the data in a scatter plot. Determine the type of correlation.
Solution: Constructing a Scatter Plot Using Technology
Enter the x-values into list L1 and the y-values into list L2.
Use Stat Plot to construct the scatter plot.
STAT > Edit…
From the scatter plot, it appears that the variables have a positive linear correlation.
The range of the correlation coefficient is –1 to 1.
If r = –1 there is a perfect negative correlation
If r is close to 0 there is no linear correlation
If r = 1 there is a perfect positive correlation
r = –0.91
r = 0.88
Strong negative correlation
Strong positive correlation
r = 0.42
r = 0.07
Weak positive correlation
A measure of the strength and the direction of a linear relationship between two variables.
The symbol r represents the sample correlation coefficient.
A formula for r is
The population correlation coefficient is represented by ρ (rho).
n is the number of data pairs
Calculating a Correlation Coefficient
In Words In Symbols
Find the sum of the x-values.
Find the sum of the y-values.
Multiply each x and y. Find the sum.
Square each x. Find the sum.
Square each y. Find the sum.
Use these five sums to calculate the correlation coefficient.
Example 4:Finding the Correlation Coefficient by hand…. Skip?
Calculate the correlation coefficient for the gross domestic products and carbon dioxide emissions data.
What can you conclude?
Solution: Finding the Correlation Coefficient
Σ x = 23.1
Σ y = 5554
Σ xy = 15,573.71
Σ x 2 = 67.35
Σ y 2 = 3,775,842.76
Σ x 2 = 32.44
r ≈ 0.882 suggests a strong positive linear correlation. As the gross domestic product increases, the carbon dioxide emissions also increase.
Example 5: Using Technology to Find a Correlation Coefficient
Use a technology tool to calculate the correlation coefficient for the Old Faithful data. (this should already be in your calculator)
Solution: Using Technology to Find a Correlation Coefficient
To calculate r, you must first enter the DiagnosticOn command found in the Catalog menu
STAT > Calc
r ≈ 0.979 suggests a strong positive correlation.
9.1 Day 1 Assignment:
Larson/Farber 5th ed.
Vocabulary #3, 4
Problems: #23, 24
Scatter Plot Using Technology
© 2012 Pearson Ed