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## Chapter 10

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**Chapter 10**Describing Relationships Using Correlation and Regression**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**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**Distinguishing Characteristics**• A scatterplotis 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**A Scatterplot Showing the Existence of a Relationship**Between the Two Variables**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**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**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.**Strength of a Relationship**The strengthof 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**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**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**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**Step-by-Step**Step 1. Compute the necessary components:**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**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 rshows all possible values of r that occur when samples are drawn from a population in which r = 0**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**One-Tailed Test of the Pearson r**• One-tailed, predicting positive correlation • One-tailed, predicting negative correlation**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)**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.**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**Example 1**For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.**Example 1 Pearson Correlation Coefficient**• Determine N • Calculate • Insert each value into the following formula and**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**Example 3 Proportion of Variance Accounted For**Calculate the proportion of variance accounted for, using the given data. Proportion of variance accounted for is