Ch 4.1 & 4.2 Two dimensions concept I. Scatter plot

exponential regression linear regression quadratic regression power regression exponential regression Linear Regression perfect correlation positive correlation negative correlation no correlation As x increases, y increases As x increases, y decreases

II. Correlation coefficient (r) 1. Covariance between X and Y The covariance is a measure of how two variables vary together. 2. When = 0, there is no correlation. If is large, it is difficult to interpret. Pearson used a formula to standardize the covariance, which is called the Pearson Correlation Coefficient . where is the sample standard deviation of the explanatory variable x. where is the sample standard deviation of the response variable y.

3. r measures the strength of the linear association between X and Y. 4. 1 r = -1 r = 0 r = 1 perfect negative correlation no correlation perfect positive correlation If > the critical value for Correlation Coefficient obtained from Table II, there is a positive/negative linear correlation between X and Y.

III. Least-Squares Regression Line If there is positive/negative correlation between X and Y, find the best fitted line for the data. The least-square regression line, , is the line that minimizes the sum of the squared errors (residuals). Residual = observed y – predicted y

The least squares regression line is where is the slope of the least-squares regression line and is the y-intercept of the least-squares regression line

Summary: 1. Use StatCrunch to plot a scatter plot 2. Use StatCrunch to calculate r 3. Determine whether there is a positive/negative linear correlation between X and Y. If > the critical value for Correlation Coefficient obtained from Table II, there is a positive linear correlation between X and Y for r is positive and there is a negative linear correlation between X and Y for r is negative. 4. If there is a linear correlation between X and Y, use StatCrunch to find the least squares regression line. Otherwise, do not find the least squares regression line.

When a value is assigned to X  if there is a correlation between X and Y, use the least squares regression line to find the best predicted Y. When a value is assigned to X  if there is no correlation between X and Y, use StatCrunch to find and the best predicted Y is for any X.