1. STATISTICS ELEMENTARY C.M. Pascual
Chapter Correlation and Regression
C.M. Pascual

2. Chapter 9 Correlation and Regression
9-1 Overview
9-2 Correlation
9-3 Regression
9-4 Variation and Prediction Intervals
9-5 Multiple Regression
9-6 Modeling

3. 9-1 Overview Paired Data is there a relationship
if so, what is the equation
use the equation for prediction
page 506 of text

4. 9-2
Correlation

5. Definition Correlation
exists between two variables when one of them is related to the other in some way

6. Assumptions 1. The sample of paired data (x,y) is a random sample.
2. The pairs of (x,y) data have a bivariate normal distribution.
page 507 of text
Explain to students the difference between the ‘paired’ data of this chapter and the investigation of two groups of data in Chapter 8.

7. Definition Scatterplot (or scatter diagram)
is a graph in which the paired (x,y) sample data are plotted with a horizontal x axis and a vertical y axis. Each individual (x,y) pair is plotted as a single point.
Relate a scatter plot to the algebraic plotting of number pairs (x,y).

8. Scatter Diagram of Paired Data

9. Scatter Diagram of Paired Data
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10. Positive Linear Correlationy y y x x x
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(a) Positive
(b) Strong
positive
(c) Perfect
positive
Figure Scatter Plots

11. Negative Linear Correlationy y y x x x
(d) Negative
(e) Strong
negative
(f) Perfect
negative
Figure Scatter Plots

12. (h) Nonlinear Correlation
No Linear Correlation y y x x
Emphasize that graph (h) does have a correlation - just not linear.
Other types of correlation, such as (h), will be briefly discussed in Section 9-6.
(g) No Correlation
(h) Nonlinear Correlation
Figure Scatter Plots

13. Linear Correlation Coefficient r
Definition
Linear Correlation Coefficient r
measures strength of the linear relationship between paired x and y values in a sample
page 509 of text

14. Linear Correlation Coefficient r
Definition
Linear Correlation Coefficient r
measures strength of the linear relationship between paired x and y values in a sample
nxy - (x)(y)
r =
n(x2) - (x) n(y2) - (y)2
Formula 9-1

15. Linear Correlation Coefficient r
Definition
Linear Correlation Coefficient r
measures strength of the linear relationship between paired x and y values in a sample
nxy - (x)(y)
r =
n(x2) - (x) n(y2) - (y)2
Formula 9-1
Calculators can compute r
(rho) is the linear correlation coefficient for all paired data in the population.

16. Notation for the Linear Correlation Coefficient
n = number of pairs of data presented
 denotes the addition of the items indicated.
x denotes the sum of all x values.
x indicates that each x score should be squared and then those squares added.
(x)2 indicates that the x scores should be added and the total then squared.
xy indicates that each x score should be first multiplied by its corresponding y score. After obtaining all such products, find their sum.
r represents linear correlation coefficient for a sample
 represents linear correlation coefficient for a population

17. Rounding the Linear Correlation Coefficient r
Round to three decimal places so that    it can be compared to critical values    in Table A-6
Use calculator or computer if possible
Page 510 of text

18. Interpreting the Linear Correlation Coefficient
If the absolute value of r exceeds the value in Table A - 6, conclude that there is a significant linear correlation.
Otherwise, there is not sufficient evidence to support the conclusion of significant linear correlation.
page 511 of text
Discussion should be held regarding what value r needs to be in order to have a significant linear correlation.

19. TABLE A-6 Critical Values of the Pearson Correlation Coefficient r
= .05
= .01 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 70 80 90
100
.950
.878
.811
.754
.707
.666
.632
.602
.576
.553
.532
.514
.497
.482
.468
.456
.444
.396
.361
.335
.312
.294
.279
.254
.236
.220
.207
.196
.999
.959
.917
.875
.834
.798
.765
.735
.708
.684
.661
.641
.623
.606
.590
.575
.561
.505
.463
.430
.402
.378
.361
.330
.305
.286
.269
.256
Table A-6 is found on the Formula and Tables insert of the text as well as in the Appendix, page 778.

20. Example 1 Construct a scatter plot for the given data Age, x 43 48 5661 67 70
Pressure, y
128
120
135
143
141
152

21. Example 1
Solution: Draw and label the x and y axes. Plot each point on the graph below

22. Example 2
A Statistics professor at a state university wants to see how strong the relationship is between a student’s score on a test and his or her grade point average. The data obtained from the sample follow:
Test score, x 98
105
100
106 95
116
112
GPA, y
2.1
2.4
3.2
2.7
2.2
2.3
3.8
3.4

23. Subject
Test Score
GPA x y xy
x^2
y^2 1 98
2.1
205.8
9604
4.41 2
105
2.4
252
11025
5.76 3
100
3.2
320
10000
10.24 4
2.7
270
7.29 5
106
2.2
233.2
11236
4.84 6 95
2.3
218.5
9025
5.29 7
116
3.8
440.8
13456
14.44 8
112
3.4
380.8
12544
11.56
SUM
832
22.1
2321.1
86890
63.83

24. Example 2 Solve of SSxy, SSxx, and Ssyy; SSxy = ∑xy – [(∑x) (∑y )]/n
= – [(832)(22.1)]/8 = 22.7
SSxx = ∑x2 – (∑x)2/n = – [(832)2]/8
= 362
SSyy = ∑y2 – (∑y)2/n = – [(22.1)2]/8
= 2.78

25. Example 2 Substitute in the formula and solve for r;
r = SSxy/(SSxx * Ssyy)0.5
= 22.7/[(362)(2.78)]0.5 = 0.716
The correlation coefficient suggests a strong positive relationship between the test score and the grade point average.

26. Properties of the Linear Correlation Coefficient r
2. Value of r does not change if all values of either variable are converted to a different scale.
3. The r is not affected by the choice of x and y. Interchange x and y and the value of r will not       change.
4. r measures strength of a linear relationship.
page 512 of text
If using a graphics calculator for demonstration, it will be an easy exercise to switch the x and y values to show that the value of r will not change.

27. Common Errors Involving Correlation
1. Causation: It is wrong to conclude that correlation implies causality.
2. Averages: Averages suppress individual variation and may inflate the correlation coefficient.
3. Linearity: There may be some relationship between x and y even when there is no significant linear correlation.
page 513 of text

28. Common Errors Involving Correlation
FIGURE 9-2 50
100
150
200
250 1 2 3 4 5 6 7 8
Distance
(feet)
One example of data that does have a relationship but not a linear one.
Time (seconds)
Scatterplot of Distance above Ground and Time for Object Thrown Upward

29. Formal Hypothesis Test
To determine whether there is a significant linear correlation between two variables
Two methods
Both methods let H0: =
(no significant linear correlation)
H1: 
(significant linear correlation)
page 514 of text

30. Method 1: Test Statistic is t (follows format of earlier chapters)
n - 2
This method is preferred by some instructors because it follows the format presented in Chapter 7 for hypothesis testing.

31. Method 1: Test Statistic is t (follows format of earlier chapters)
n - 2
Critical values:
use Table A-3 with
degrees of freedom = n - 2
This is the first example in the text where the degrees of freedom for Table A-3 is different from n Special note should be made of this.

32. Method 1: Test Statistic is t (follows format of earlier chapters)
This is the drawing used to verify the position of the sample data t value in regard to the critical t values for the example which begins on page Drawing is at the bottom of page 516.
Figure 9-4

33. Method 2: Test Statistic is r
(uses fewer calculations)
Test statistic: r
Critical values: Refer to Table A-6
(no degrees of freedom)
This method is preferred by some instructors because the calculations are easier.

34. Method 2: Test Statistic is r
(uses fewer calculations)
Test statistic: r
Critical values: Refer to Table A-6
(no degrees of freedom)
Reject
= 0
Fail to reject
 = 0
Reject
= 0
This is the drawing used to verify the position of the sample data r value in regard to the critical r values for the example which begins on page Drawing is at the top of page 517. -1
r =
r = 1
Figure 9-5
Sample data:
r = 0.828

35. FIGURE 9-3 Testing for a Linear Correlation
Start
Let H0:  = 0
H1:   0
Select a
significance
level 
Calculate r using
Formula 9-1
METHOD 1
METHOD 2
The test statistic is
t =
1 - r 2
n -2 r
Critical values of t are from Table A-3 with n -2 degrees of freedom
The test statistic is
Critical values of t are from
Table A-6 r
page 515 of text
If the absolute value of the
test statistic exceeds the
critical values, reject H0:  = 0
Otherwise fail to reject H0
If H0 is rejected conclude that there
is a significant linear correlation.
If you fail to reject H0, then there is
not sufficient evidence to conclude
that there is linear correlation.

36. Is there a significant linear correlation?
0.27 2
1.41 3
2.19
2.83 6 4
1.81
0.85 1
3.05 5
Data from the Garbage Project
x Plastic (lb)
y Household
This is exercise #7 on page 521.

37. Is there a significant linear correlation?
0.27 2
1.41 3
2.19
2.83 6 4
1.81
0.85 1
3.05 5
Data from the Garbage Project
x Plastic (lb)
y Household
n =  = H0:  = 0
H1 :  0
Test statistic is r = 0.842
Using Method 2 to solve this problem.

38. Is there a significant linear correlation?4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 25 30 35 40 45 50 60 70 80 90
100 n
.999
.959
.917
.875
.834
.798
.765
.735
.708
.684
.661
.641
.623
.606
.590
.575
.561
.505
.463
.430
.402
.378
.361
.330
.305
.286
.269
.256
.950
.878
.811
.754
.707
.666
.632
.602
.576
.553
.532
.514
.497
.482
.468
.456
.444
.396
.335
.312
.294
.279
.254
.236
.220
.207
.196
= .05
= .01
n =  = H0:  = 0
H1 :  0
Test statistic is r = 0.842
Critical values are r = and 0.707
(Table A-6 with n = 8 and  = 0.05)
TABLE A-6 Critical Values of the Pearson Correlation Coefficient r

39. Is there a significant linear correlation?
Reject
= 0
Fail to reject
 = 0
Reject
= 0
- 1 1
r =
r =
Placement of the sample data r value in regard to the critical r values.
Sample data:
r = 0.842

40. Is there a significant linear correlation?
> 0.707, That is the test statistic does fall within the
critical region.
Reject
= 0
Fail to reject
 = 0
Reject
= 0
- 1 1
r =
r =
Sample data:
r = 0.842

41. Is there a significant linear correlation?
> 0.707, That is the test statistic does fall within the
critical region.
Therefore, we REJECT H0:  = 0 (no correlation) and conclude
there is a significant linear correlation between the weights of
discarded plastic and household size.
Reject
= 0
Fail to reject
 = 0
Reject
= 0
- 1 1
r =
r =
Sample data:
r = 0.842

42. Justification for r Formula
page 517 of text

43. Justification for r Formula
Formula 9-1 is developed from
 (x -x) (y -y)
r =
(n -1) Sx Sy

44. Justification for r Formula
Formula 9-1 is developed from
 (x -x) (y -y)
r =
(x, y) centroid of sample points
(n -1) Sx Sy
The mean of the x-values is x-bar, and the mean of the y-values is y-bar.

45. Justification for r Formula
Formula 9-1 is developed from
 (x -x) (y -y)
r =
(x, y) centroid of sample points
(n -1) Sx Sy
x = 3 y
x - x = = 4 24 •
(7, 23) 20
y - y = = 12
Quadrant 2
Quadrant 1 16 • 12
y = 11
For one point (7,23) the differences from the centroid are found. These values would be used in the r formula.
(x, y) 8 •
Quadrant 3 •
Quadrant 4 • 4
FIGURE 9-6 x 1 2 3 4 5 6 7