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**Describing Relationships Using Correlation and Regression**

Chapter 10 Describing Relationships Using Correlation and Regression

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**Going Forward Your 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

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**Understanding Correlations**

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**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

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**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

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**A Scatterplot Showing the Existence of a Relationship Between the Two Variables**

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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

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**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

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**Scatterplot of a Positive Linear Relationship**

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**Scatterplot of a Negative Linear Relationship**

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**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.

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**Scatterplot of a Nonlinear Relationship**

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**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

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**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

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**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

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**A Perfect Correlation (±1)**

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**Intermediate Strength Correlation**

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No Relationship

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**The Pearson Correlation Coefficient**

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**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

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Step-by-Step Step 1. Compute the necessary components:

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**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

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**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 r = 0

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**Two-Tailed Test of the Pearson r**

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**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 a 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

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**One-Tailed Test of the Pearson r**

One-tailed, predicting positive correlation One-tailed, predicting negative correlation

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**An Introduction to Linear Regression**

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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)

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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.

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**Proportion of Variance Accounted 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

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Example 1 For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient. X Y 1 8 2 6 3 4 5

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**Example 1 Pearson Correlation Coefficient**

Determine N Calculate Insert each value into the following formula and

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**Example 1 Pearson Correlation Coefficient**

Y Y 2 XY 1 8 64 2 4 6 36 12 3 9 18 16 5 25 20 SX = 21 SX 2 = 91 SY = 29 SY 2 = 171 SXY = 81

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**Example 1 Pearson Correlation Coefficient**

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**Example 2 Significance Test of the Pearson r**

Conduct a two-tailed significance test of the Pearson r just calculated. Use a = .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

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**Example 3 Proportion of Variance Accounted For**

Calculate the proportion of variance accounted for, using the given data. Proportion of variance accounted for is

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