Change in hypothesis testing circumstances and associated p-value variation

For hypothesis Z-testing where a certain p-value is arrived at based on Ho: μ = μo against  H1: μ > μo; if we recalibrate the test to that of  Ho: μ = μo against  H1: μ ≠ μo but utilise the same data set ( x̄ , σ, n, μo) as previously, how would the p-value change?

Student X

You should expect the new p-value to be twice that of the original. 

For the original one-tail test, the p-value is simply the result of computing P( Z > (x̄ -μo) /(σ/√n ) ); adjusting this to a two-tail test using the same set of data would imply the value of the test-statistic Z= (x̄ -μo)/(σ/√n) remains unchanged, and in similar regard  P( Z > (x̄ -μo)/ (σ/√n)) .

Since a two-tail test includes consideration of the left tail, by symmetry it is therefore also known that P( Z < (x̄ -μo)/(σ/√n )) = original p-value. Adding p-values attributable to both sides together, recognize that the overall new p-value is twice that of the original.

Hope this clarifies. Peace.

Best Regards,

Mr Koh