1. Correlation and Regression
Statistics
Correlation and Regression

2. Warm-up
Suppose that the average salary for a K-12 teacher in South Carolina is $40,000. Which of the following values would be a reasonable value for the standard deviation?
a) -1,000
b) 0
c) 1,000
d) 5,000
e) 23,000

3. Liquid Diet Prepared Meals Low Carb They are the same
Warm-up A personal trainer was interested in studying the effects of different types of diets (liquid diet, prepared meals, and low carb) on total weight loss in two months. Which boxplot has the biggest IQR?
Liquid Diet
Prepared Meals
Low Carb
They are the same
Cannot be determined

4. Objectives
Introduce linear correlation, independent and dependent variables, and the types of correlation
Find a correlation coefficient
Distinguish between correlation and causation

5. A relationship between two variables.
Correlation
Correlation
A relationship between two variables.
The data can be represented by ordered pairs (x, y)
x is the independent (or explanatory) variable
y is the dependent (or response) variable

6. Correlation
A scatter plot can be used to determine whether a linear (straight line) correlation exists between two variables. x 2 4 –2
– 4 y 6
Example: x 1 2 3 4 5 y
– 4
– 2
– 1

7. Types of Correlation As x increases, y tends to decrease.
As x increases, y tends to increase.
Negative Linear Correlation
Positive Linear Correlation x y x y
No Correlation
Nonlinear Correlation 7

8. Example: Constructing a Scatter Plot
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
A marketing manager conducted a study to determine whether there is a linear relationship between money spent on advertising and company sales. 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.

9. Solution: Constructing a Scatter Plotx y
Advertising expenses
(in thousands of dollars)
Company sales
Appears to be a positive linear correlation. As the advertising expenses increase, the sales tend to increase.

10. Example: 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.
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65

11. 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…
STATPLOT 1 5 50
100
From the scatter plot, it appears that the variables have a positive linear correlation.

12. Correlation Coefficient
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

13. Correlation Coefficient
The range of the correlation coefficient is -1 to 1. -1 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

14. Linear Correlation Strong negative correlationx y x y
r = 0.91
r = 0.88
Strong negative correlation
Strong positive correlation x y x y
r = 0.42
r = 0.07
Weak positive correlation
Nonlinear Correlation 14

15. 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-value by its corresponding y-value and find the sum. 15

16. Calculating a Correlation Coefficient
In Words In Symbols
Square each x-value and find the sum.
Square each y-value and find the sum.
Use these five sums to calculate the correlation coefficient. 16

17. Example: Finding the Correlation Coefficient
Calculate the correlation coefficient for the advertising expenditures and company sales data. What can you conclude?
Advertising expenses, ($1000), x
Company sales ($1000), y
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215

18. Solution: Finding the Correlation Coefficientx y xy x2 y2
2.4
225
1.6
184
2.0
220
2.6
240
1.4
180
186
2.2
215
540
5.76
50,625
294.4
2.56
33,856
440 4
48,400
624
6.76
57,600
252
1.96
32,400
294.4
2.56
33,856
372 4
34,596
473
4.84
46,225
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558

19. Solution: Finding the Correlation Coefficient
Σx = 15.8
Σy = 1634
Σxy =
Σx2 = 32.44
Σy2 = 337,558
r ≈ suggests a strong positive linear correlation. As the amount spent on advertising increases, the company sales also increase.

20. Example: Using Technology to Find a Correlation Coefficient
Use a technology tool to calculate the correlation coefficient for the Old Faithful data. What can you conclude?
Duration x
Time, y
1.8 56
3.78 79
1.82 58
3.83 85
1.9 62
3.88 80
1.93
4.1 89
1.98 57
4.27 90
2.05
4.3
2.13 60
4.43
2.3
4.47 86
2.37 61
4.53
2.82 73
4.55
3.13 76
4.6 92
3.27 77
4.63 91
3.65

21. 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 ≈ suggests a strong positive correlation.

22. Interpreting r
Statisticians use the following scale to provide a rough estimate of a relationship’s strength:
r value (+/-)
Strength
0 -.19
poor
fair
moderate
good
.8 – 1.0
excellent

23. Correlation and Causation
The fact that two variables are strongly correlated does not in itself imply a cause-and-effect relationship between the variables.
If there is a significant correlation between two variables, you should consider the following possibilities.
Is there a direct cause-and-effect relationship between the variables?
Does x cause y?

24. Correlation and Causation
Is there a reverse cause-and-effect relationship between the variables?
Does y cause x?
Is it possible that the relationship between the variables can be caused by a third variable or by a combination of several other variables?
Is it possible that the relationship between two variables may be a coincidence?

25. Example – McGwire-Sosa Data
Using the McGwire-Sosa Data, compute the correlation coefficient between batting average (x) and homeruns (y) for both players.
Assess the stregth of the relationship. Do you think batting average is a good predictor of homeruns?

26. Found a correlation coefficient
Summary
Introduced linear correlation, independent and dependent variables and the types of correlation
Found a correlation coefficient
Distinguished between correlation and causation

27. Homework
Pg , #2-28 even