• A regression line is a straight line that describes how a response variable y changes as an explanatory variable x changes.
• The most common method of fitting a line to a scatterplot is least squares. The least-squares regression line is the straight line that minimizes the sum of the squares of the vertical distances of the observed y-values from the line.
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• You can use a regression line to predict the value of y for any value of x by substituting this x into the equation of the line. Extrapolation beyond the range of x-values spanned by the data is risky.
• The slope b1 of a regression line is the rate at which the predicted response changes along the line as the explanatory variable x changes. Specifically, b1 is the change in when x increases by 1. The numerical value of the slope depends on the units used to measure x and y.
• The intercept b0 of a regression line is the predicted response when the explanatory variable x = 0. This prediction is not particularly useful unless x can actually take values near 0.
• The least-squares regression line of y on x is the line with slope and intercept . This line always passes through the point .
• Correlation and regression are closely connected. The correlation r is the slope of the least-squares regression line when we measure both x and y in standardized units. The square of the correlation r2 is the fraction of the variance of one variable that is explained by least-squares regression on the other variable.