1 Chapter 10 Correlation and Regression 10.2 Correlation 10.3 Regression.
Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.
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Transcript of Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.
![Page 1: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/1.jpg)
Chapter 10
DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION
![Page 2: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/2.jpg)
Going ForwardYour goals in this chapter are to learn:• How to create and interpret a scatterplot• What a regression line is• When and how to compute the Pearson r• How to perform significance testing of the
Pearson r• The logic of predicting scores using linear
regression and 2r
![Page 3: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/3.jpg)
Understanding Correlations
![Page 4: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/4.jpg)
Correlation Coefficient
• A correlation coefficient is a statistic that describes the important characteristics of a relationship
• It simplifies a complex relationship involving many scores into one number that is easily interpreted
![Page 5: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/5.jpg)
Distinguishing Characteristics
• A scatterplot is a graph of the individual data points from a set of X-Y pairs
• When a relationship exists, as the X scores increase, the Y scores change such that different values Y tend to be paired with different values of X
![Page 6: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/6.jpg)
A Scatterplot Showing the Existence of a Relationship Between the Two Variables
![Page 7: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/7.jpg)
Linear Relationships
• A linear relationship forms a pattern following one straight line
• The linear regression line is the straight line that summarizes a relationship by passing through the center of the scatterplot
![Page 8: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/8.jpg)
Positive and Negative Relationships
• In a positive linear relationship, as the X scores increase, the Y scores also tend to increase
• In a negative linear relationship, as the scores on the X variable increase, the Y scores tend to decrease
![Page 9: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/9.jpg)
Scatterplot of a Positive Linear Relationship
![Page 10: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/10.jpg)
Scatterplot of a Negative Linear Relationship
![Page 11: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/11.jpg)
Nonlinear Relationships
In a nonlinear relationship, as the X scores increase, the Y scores do not only increase or only decrease: at some point, the Y scores alter their direction of change.
![Page 12: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/12.jpg)
Scatterplot of a Nonlinear Relationship
![Page 13: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/13.jpg)
Strength of a Relationship
The strength of a relationship is the extent to which one value of Y is consistently paired with one and only one value of X
•The larger the absolute value of the correlation coefficient, the stronger the relationship
•The sign of the correlation coefficient indicates the direction of a linear relationship
![Page 14: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/14.jpg)
Correlation Coefficients
• Correlation coefficients may range between –1 and +1. The closer to ±1 the coefficient is, the stronger the relationship; the closer to 0 the coefficient is, the weaker the relationship.
• As the variability in the Y scores at each X becomes larger, the relationship becomes weaker
![Page 15: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/15.jpg)
Correlation Coefficient
A correlation coefficient tells you• The relative degree of consistency with which
Ys are paired with Xs• The variability in the group of Y scores paired
with each X• How closely the scatterplot fits the regression
line• The relative accuracy of prediction
![Page 16: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/16.jpg)
A Perfect Correlation (±1)
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Intermediate Strength Correlation
![Page 18: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/18.jpg)
No Relationship
![Page 19: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/19.jpg)
The Pearson Correlation Coefficient
![Page 20: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/20.jpg)
Pearson Correlation Coefficient
Describes the linear relationship between two interval variables, two ratio variables, or one interval and one ratio variable.
The computing formula is
])()([])()([
))(()(2222 YYNXXN
YXXYNr
![Page 21: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/21.jpg)
Step-by-Step
Step 1. Compute the necessary components:• • • •
X2)( X
2XXY
•
•
•
•
Y2)( Y
2YN
![Page 22: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/22.jpg)
Step-by-Step
• Step 2. Use these values to compute the numerator
• Step 3. Use these values to compute the denominator and then divide to find r
))(()( YXXYN
])()([])()([ 2222 YYNXXN
![Page 23: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/23.jpg)
Significance Testing of the Pearson r
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Two-Tailed Test of the Pearson r
• Statistical hypotheses for a two-tailed test
• This H0 indicates the r value we obtained from our sample is because of sampling error
• The sampling distribution of r shows all possible values of r that occur when samples are drawn from a population in which = 0
0:;0:0 aHH
![Page 25: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/25.jpg)
Two-Tailed Test of the Pearson r
![Page 26: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/26.jpg)
Two-Tailed Test of the Pearson r
• Find appropriate rcrit from the table based on– Whether you are using a two-tailed or one-tailed
test– Your chosen – The degrees of freedom (df) where df = N – 2,
where N is the number of X-Y pairs in the data
• If robt is beyond rcrit, reject H0 and accept Ha
• Otherwise, fail to reject H0
![Page 27: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/27.jpg)
One-Tailed Test of the Pearson r
• One-tailed, predicting positive correlation
• One-tailed, predicting negative correlation
0:;0:0 aHH
0:;0:0 aHH
![Page 28: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/28.jpg)
An Introduction to Linear Regression
![Page 29: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/29.jpg)
Linear Regression
Linear regression is the procedure for predicting unknown Y scores based on known correlated X scores.•X is the predictor variable•Y is the criterion variable•The symbol for the predicted Y score is (pronounced Y prime)
Y
![Page 30: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/30.jpg)
Linear Regression
The equation that produces the value of at each X and defines the straight line that summarizes the relationship is called the linear regression equation.
Y
![Page 31: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/31.jpg)
Proportion of VarianceAccounted For
• The proportion of variance accounted for describes the proportion of all differences in Y scores that are associated with changes in the X variable
• The proportion of variance accounted for equals 2r
![Page 32: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/32.jpg)
Example 1
For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.
X Y
1 8
2 6
3 6
4 5
5 1
6 3
![Page 33: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/33.jpg)
Example 1 Pearson Correlation Coefficient
• Determine N• Calculate
• Insert each value into the following formula
])()([])()([
))(()(2222 YYNXXN
YXXYNr
,X ,)( 2X ,2XXY
,Y ,)( 2Y,2Y and
![Page 34: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/34.jpg)
Example 1 Pearson Correlation Coefficient
X X 2 Y Y 2 XY
1 1 8 64 8
2 4 6 36 12
3 9 6 36 18
4 16 5 25 20
5 25 1 1 5
6 36 3 9 18
X = 21 X 2 = 91 Y = 29 Y 2 = 171 XY = 81
N = 6
![Page 35: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/35.jpg)
Example 1 Pearson Correlation Coefficient
88.0374.139
123
]185][105[
609486
)29()171(6[])21()91(6[
)29)(21()81(6
)()([])()([
))(()(
22
2222
YYNXXN
YXXYNr
![Page 36: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/36.jpg)
Example 2 Significance Test of the Pearson r
Conduct a two-tailed significance test of the Pearson r just calculated. Use = .05.•df = N – 2 = 6 – 2 = 4•rcrit = 0.811
•Since robt of –0.88 falls beyond the critical value of –0.811, reject H0 and accept Ha.
•The correlation in the population is significantly different from 0
![Page 37: Chapter 10 DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION.](https://reader035.fdocuments.us/reader035/viewer/2022062407/56649cf45503460f949c2c31/html5/thumbnails/37.jpg)
Example 3 Proportion of Variance Accounted For
Calculate the proportion of variance accounted for, using the given data.
Proportion of variance accounted for is
7744.088.0 22 r