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 increasing the sample size, decreasing the standard error, increasing the difference between the sample statistic and the hypothesized parameter, or increasing the alpha level. Using a directional test (i.e., left- or right-tailed) as opposed to a two-tailed test would also increase power.
When we increase the sample size, decrease the standard error, or increase the difference between the sample statistic and hypothesized parameter, the p value decreases, thus making it more likely that we reject the null hypothesis. When we increase the alpha level, there is a larger range of p values for which we would reject the null hypothesis. Going from a two-tailed to a one-tailed test cuts the p value in half. In all of these cases, we say that statistically power is increased.
There is a 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 (i.e., decreasing the likelihood of both Type I and Type II errors), then we should increase the sample size.
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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 Type I error rate (i.e., alpha level) 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 really 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.