Slope of the Population Regression Line

In general, the sign of [latex]r[/latex] (positive, negative, or [latex]0[/latex]) will be the same as the sign of [latex]b[/latex]. This tells us that in studying and understanding the correlation coefficient, we simultaneously have information about the slope of the line of best fit.

When the line of best fit is estimated, the slope, [latex]b[/latex], is calculated using sample data. The slope, [latex]b[/latex], is an estimate of the slope of the population regression line, [latex]\beta_1[/latex]. This is similar to the relationship between the sample mean, [latex]\bar{x}[/latex], and the population mean, [latex]\mu[/latex], that we previously studied.

Hypothesis Test for Significance of Slope

We want to conduct a hypothesis test to find out whether or not two quantitative variables have a significant linear relationship. When two variables do NOT have a significant linear relationship, the true value of the slope of the population line is [latex]0: \beta_1 = 0[/latex]. That is, the population regression line is a horizontal line, and the value of [latex]y[/latex] in the simple linear regression model does not depend on  [latex]x[/latex].