More generally,under the null hypothesis the truepopulation slopeβ1takeson some specific value,β1,0. Under thetwo-sidedalternative,β1does notequalβ1,0.That is,thenullhypothesisandthe two-sided alternative hypothesisareH0:β1=β1,0H1:β1≠ β1,0,(5.2)The first step is to compute the standard error of^β1,SE(^β1). The standard error of^β1is an estimator of^σ^β1, the standard deviation of the sampling distribution of^β1. Specifically,SE(^β1)=√σ^β12,(5.3)whereσ^β12=1n×1n−2∑i=1n(Xi−´X)2^ui2[1n∑i=1n(Xi−´X)2]2,(5.4)Although the formula forσ^β12is complicated, in applications the standard error iscomputed by regression software so that it is easy to use in practice.The second step is to compute thet-statistic,t=^β1−β1,0SE(^β1),(5.5)The third step is to compute thep-value, the probability of observing a value of^β1atleast as different fromβ1,0as the estimate actually computed^β1act, assuming that thenull hypothesis is correct. Stated mathematically,p−value=PrH0[|^β1−β1,0|]>PrH0[|^β1act−β1,0|]¿PrH0[|^β1−β1,0SE(^β1)|]>PrH0[|^β1act−β1,0SE(^β1)|]=PrH0(|t|>|tact|),(5.6)wherePrH0denotes the probability computed under the null hypothesis, the secondequality follows by dividing bySE(^β1), andtactis the value of thet-statistic actuallycomputed.