Suppose that the company researchers care about finding any effect on blood pressure

that is 3 millimeters of mercury or larger cersus the standard medication.

What is the power of the test that can detect this effect?

In other words, 3 millimeters of mercury is the minimum effect size of interest and

we want to know how likely we are to detect this size of an effect

in this study.

If the treatment is indeed effective enough to result in a 3 millimeters of

mercury drop in blood pressure on average, then it means the observed distribution

of differences in average blood pressures between the two groups will be shifted

from the null by 3 millimeters of mercury, as shown in this plot here.

We also know that we can only reject the null hypothesis if the observed difference

is less than negative 3.332 millimeters of mercury.

Putting all of these together, the probability of

being able to reject the null hypothesis if the true effect size is negative 3,

is equal to the green shaded area under this curve.

We've been able to simplify this task of calculating the power,

to just calculating an area under the normal curve.

We calculate a Z score as the difference in sample means,

negative 3.332 minus the mean of that distribution,

negative 3 divided by the standard error we calculated earlier.

This yields a Z score of negative 0.20 and

the shaded green area is approximately 0.4207.

Therefore, the power of the test is about 42%

when the effect size is negative 3 and each group has a sample size of 100.

Obviously, this is much lower than the 80% power

we set out to attain at the beginning of this video.

It highlights how important it is to not just arbitrarily select a sample size and

risk being left with an under powered study.

How can we fix things?

We can work backwards from the desired power to determine the minimum required

sample size instead.

Note that the effect size is still negative 3 since that's what the drug

company is interested in, however, the standard error will now be different

since it changes when the sample size changes.

Let's sketch our distributions again and mark the power on the green shaded area.