Converting Hypotheses into a Decision Rule (DR)

The form for our decision rule is always dependent on the form of our alternative hypothesis, H1. Our alternative hypothesis will always* be one of the following forms: p > 0.50, p < 0.50, or p ≠ 0.50. (Our parameter might actually be something other than p, and the statistic in the DR itself might be something other than z, but the forms will still be the same.)

If H1 is p > 0.50, then our decision rule will be of this form:
DR: If zp-hat > (some number) then reject H0. (The number will be positive in this case.)

If H1 is p < 0.50, then our decision rule will be of this form:
DR: If zp-hat < (some number) then reject H0. (The number will be negative in this case.)

If H1 is p ≠ 0.50, then our decision rule will be of this form:
DR: If zp-hat > (some number) or zp-hat < (some number) then reject H0. (The first number will be positive while the second will be negative.)

*Later we will see some exceptions to this rule, but it is true for now.