Think of it in this way: Whenever we make a decision concerning Ho, there are 4 possible scenarios:
1)Ho is true and we reject
2)Ho is true and we fail to reject
3)Ho is false and we reject
4)Ho is false and we fail to reject
From these scenarios, 1) is referred to as a Type I error. The probability of a Type I error translates to the probability of being in the rejection region, which is the area under the curve for our given region (one tail: < or >, two tail: not equal). This is affectionately known as "alpha."
4) from above describes "beta." If the true mean is really different from Ho, and we fail to reject, then we have committed a Type II error. In terms of the normal curve, this is represented as the area under a new curve (new mean) that falls into the fail to reject region of the old curve (claimed mean from Ho).
The rest of the area under this new curve (the probability that we reject when Ho is really false) is known as the power (Scenario 3 from above) of the test, calcualted as "1-beta."
In terms of problem solving, you must first separate your reject region from your fail to reject region based on the claimed mean. Once you have identified these regions, then you must draw a new curve according to your new mean. Shade the area under the new curve that falls within the fail to reject region of the original curve. Use your calculator or z-scores to calculate this probability. Again, this is beta. If you subtract this value from 1, then you will have calculated power.
