1. Correlation and Regression
Chapter 10
Correlation and Regression

2. 10.2 - Correlation Definitions: Correlation:
Linear correlation coefficient (r) :

3. Types of Correlation

4. Requirements for calculating r
The sample of paired (x,y) data is
Notation: ρ – “rho” is
Note: r and ρ are

5. Formulas for calculating r
1. 𝑟= 𝑛 𝑥𝑦 − 𝑥 𝑦 𝑛( 𝑥 2 )− 𝑥 𝑛 𝑦 2 − 𝑦 2
(can go to STAT > CALC > 2-Var Stats to find 𝑥𝑦 , 𝑥 , 𝑦 , 𝑥 2 , 𝑦 2 )
2. 𝑟= 𝑧 𝑥 𝑧 𝑦 𝑛−1 (much harder to do by hand)
zx is the z score for the sample value x
zy is the z score for the sample value y

6. Example 1: Calculation the Linear Correlation Coefficient r
Use the calculator to find the value of the linear correlation coefficient r for the following sample data:
Costs of a Slice of Pizza and Subway Fare (in dollars)
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Subway Fare
1.35
1.50
What does r tell us in this case?

7. Interpreting the Linear Correlation Coefficient r
Hypothesis Test for Correlation (using Test statistic r)
Testing Statistic:
Critical values:
Conclusion:
If 𝑟 > critical value, _____________. Conclude that there ________ _________evidence to support the claim of a linear correlation
If 𝑟 ≤ critical value, _____________. Conclude that there _______ _________ evidence to support the claim of a linear correlation

8. Interpreting the Linear Correlation Coefficient r
Hypothesis Test for Correlation (using p-value from a t Test)
Testing Statistic: , where df =
Conclusion:
If p-value ≤ significance level, _____________. Conclude that there _________ ______________ evidence to support the claim of a linear correlation
If p-value > significance level, ______________. Conclude that there ________ ______________ evidence to support the claim of a linear correlation

9. Heights of Presidents and Runners-Up
Theories have been developed about the heights of winning candidates for the U.S. presidency and the heights of candidates who were runners-up. Listed below are heights (in inches) from recent presidential elections. Is there a linear correlation between the heights of candidates who won and the heights of candidates who where runners-up? Use α=0.05 to find the linear correlation coefficient r, the critical values, and the p-value to find out.
Winner
69.5 73 74
74.5 71
Runner-Up 72 70 68 76

10. 10.3 - Regression Definitions: Regression equation: ŷ = Notation:
Population Parameter
Sample statistic
y-intercept of regression equation
Slope of regression equation
Equation of the regression line
Note:

11. Requirements and Formulas
The sample of paired (x,y) data is
Slope:
y-intercept:
On the Calculator: Use

12. Predicting the value of y
Check the following to determine if the regression equation is a good model:
If all of those are satisfied, then
Otherwise, if the equation is not a good model, then

13. Predicting Subway Fare (Example 4 page 519)
The table below includes the pizza/subway fare costs from 10-2 as well as the total number of runs scored in the baseball World Series for six different years. Suppose right now that the slice of pizza in New York City is $2.25, and 33 runs were scored in the last World Series.
Use the table to predict the cost of a subway fare given that a slice of pizza is $2.25.
Use the table to predict the cost of a subway fare in a year in which 33 runs were scored in the World Series
Year
1960
1973
1986
1995
2002
2003
Cost of Pizza
0.15
0.35
1.00
1.25
1.75
2.00
Runs Scored in the World Series 82 45 59 42 85 38
Subway Fare
1.35
1.50

14. Global Warming
Find the best predicted temperature for a recent year in which the concentration (in parts per million) of CO2 is Is the predicted temperature close to the actual temperature of 14.5⁰ C?
First, we need to find r to determine if the equation is a good model of our data
CO2
314
317
320
326
331
339
346
354
361
369
Temperature
13.9
14.0
14.1
14.3
14.5
14.4