In a simple linear regression model, testing whether the slope, $\beta _ { 1 }$ , of the population regression line is zero is the same as testing whether the population coefficient of correlation, $\rho$ , equals zero.

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Testing whether the slope, $\beta_1$, of the population regression line is zero is equivalent to testing whether there is no linear relationship between the two variables, which is the same as testing if the population coefficient of correlation, $\rho$, equals zero. If $\rho = 0$, it indicates no linear relationship, which implies that the slope ($\beta_1$) of the population regression line would also be zero.

In a Simple Linear Regression Model, Testing Whether the Slope β1\beta _ { 1 }β1​

In a Simple Linear Regression Model, Testing Whether the Slope $\beta _ { 1 }$