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Linear Coefficient R

- A numerical measure of the strength of the relationship between two variables representing quantitative data. (x and y). IN A SAMPLE.

- Using paired sample data (sometimes called bivariate data), we find the value of r (usually using technology), then we use that value to conclude that there is (or is not) a linear correlation between the two variables.

- We can often see the relationship between the two variables by constructing a scatter plot.

Correlation

- Exists between two variables when the values of one are somehow associated with the values of the other in some way.

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Requirements

1) The sample of paired (x,y) data is a simple random sample of quantitative data.

2.) Visual examination of the scatter plot must confirm that the points approximate a straight-line pattern.

3.) Outliers must be removed if they are known to be errors. The effects of any other outliers should be onsidered by calculating r with and without the outliers included.

Interpreting R

Using Table A-6: If the absolute value of the computed value of r exceeds the value in Table A-6, conclude that there is a linear correlation. Otherwise, there is not sufficient evidenceto support the conclusion of a linear correlation.

Using Software: If the computed P-value is less than or equal to the significance level, conclude that there is a linear correlation. Otherwise, there is not sufficient evidence to support the conclusion of a linear correlation.

Rounding the Linear Correlation Coefficient R

Round to three decimal places so that it can be compared to critical values in A-6.

Use calculator or computer if possible.

Properties of the Linear Correlation Coefficient R

1.) r is between -1 and 1.

2.) If all values of either variable are converted to a different scale, the value of r does not change.

3.) The value of r is not affected by the choice of x any y. Interchange all x and y values and the value of r will not change.

4.) r measures the strength of a linear relationship.

5.) r is very sensitive to outliers, they can dramatically affect its value.

Interpreting R: Explained Variation

The value of r squared is the proportion of the variation in y that is explained by the linear relationship between x and y.

Common Errors Involving Correlation

1.) Causation: It is wrong to conclude that correlation implies causalty.

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 linear correlation.

Regression Line

- The best-fitting straight line.

- Line of best fit/Least Squares Line.

Regression Equation

- The equation for the best-fitting line.

- Algebraically describes the relationship between two variables.

X

Explanatory/Predictor/Independent Variable.

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Y

Response/Dependent Variable.

Requirements

1.) The sample of paired (x,y) data is a random sample of quantitative data.

2.) Visual examination of the scatterplot shows that the points approximate a straight-line pattern.

3.) Any outliers must be removed if they are known to be errors. Consider the effects of any outliers that are not known to be errors.

Rounding the Y-intercept and the Slope

Round to three significant digits.

Using the Regression Equation for Predictions

1.) Use the regression equation for predictions only if the graph of the regression line on the scatterplot confirms that the regression line fits the points reasonably well.

2.) Use the regression equation for predictions only if the linear correlation coefficient r indicates that there is a linear correlation between the two variables.

Continued

3.) Use the regression line for prediction only if the data do not go much beyond the scope of the available sample data. (Predicting too far beyond the scope of the available sample data is called extrapolation, and it could result in bad predictions).

4.) If the regression equation does not appear to be useful for making predictions, the best predicted value for a variable is its point estimate, which is its sample mean.

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