Testing for zero slope

To assess whether the explanatory variable affects the response, we test the hypotheses

H0 :   β1  =  0
HA :   β1  ≠  0

The test is based on how far the least squares slope, b1, is from zero. To assess this, we must take into account its standard deviation (standard error),

If we knew the value of σ, we could standardise b1 to get a test statistic,

standardised value,  

If β1 was really zero (H0), the probability of getting a least squares slope as far from zero as that recorded would be the p-value,

Unfortunately σ is usually unknown and the standard deviation of b1 must be estimated from the sample data. We therefore use a test statistic of the form

t ratio,  

and refer to a t distribution with n - 2 degrees of freedom to find the p-value.

The p-value is interpreted in the same way as for other hypothesis tests — a p-value close to zero means that the sample slope is far enough from zero to be inconsistent with H0: β1 = 0.

Examples