How does the Pearson correlation coefficient relate to the slope of the least squares regression line?
a) The Pearson correlation coefficient is 0.69. What is the sign of the slope of the least squares regression line?
b) The Pearson correlation coefficient increases from -0.96 to -0.91. Do the points of the scatter plot move toward the regression line or away from it?
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The formula for Pearson's correlation coefficient is
`r = (S_(xy))/sqrt(S_(x'x)S_(y'y))`
(assuming with out loss of generality that the mean of x and y are both zero).
The formula for the slope of the least squares regression line is
`beta = (S_(xy))/(S_(x'x))`
The denominators in both cases are positives as they involve sums of squares which are always positive. The numerators are equal (`S_(xy)`, the sum of `x times y`).
Therefore, if the correlation coefficient `r` is positive, then the regression slope is positive.
If the correlation coefficient of the data changes, then the new points will have in general have moved away from the original regression line towards a line with a different slope. Individual points may have moved towards the original line but that is not the tendency.
a) the sign of the slope of the least squares regression line is positive
b) the points have moved away from the original regression line
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