6.5 - Power
The probability of rejecting the null hypothesis, given that the null hypothesis is false, is known as power. In other words, power is the probability of correctly rejecting \(H_0\).
- \(Power = 1-\beta\)
- \(\beta\) = probability of committing a Type II Error.
The power of a test can be increased in a number of ways, for example (1) increasing the sample size; (2) when testing a mean or difference in means, decreasing the sample standard deviation(s); (3) increasing the effect size; or (4) increasing the alpha level.
The relationship between \(\alpha\) and \(\beta\): If the sample size is fixed, then decreasing \(\alpha\) will increase \(\beta\). If we want both \(\alpha\) and \(\beta\) to decrease, then we should increase the sample size.
On Your Own Section
The probability of committing a Type II error is known as \(\beta\).
If power increases then \(\beta\) must decrease. So, if the power of a statistical test is increased, for example by increasing the sample size, the probability of committing a Type II error decreases.
No. When we perform a hypothesis test, we only set the size of Type I error (i.e., \(\alpha\)) and guard against it. Thus, we can only present the strength of evidence against the null hypothesis. We can sidestep the concern about Type II error if the conclusion never mentions that the null hypothesis is accepted. When the null hypothesis cannot be rejected, there are two possible cases:
1) the null hypothesis is true
2) the sample size is not large enough to reject the null hypothesis (i.e., statistical power is too low)
The result of the study was to fail to reject the null hypothesis. In reality, the null hypothesis was false. This is a Type II error.