Introduction to Power

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Prerequisites

Define power

  • Suppose you work for a foundation whose mission is to support researchers in mathematics education and your role is to evaluate grant proposals and decide which ones to fund.
  • You receive a proposal to evaluate a new method of teaching high-school algebra.
  • The research plan is to compare the achievement of students taught by the new method with those taught by the traditional method.
  • The proposal contains good theoretical arguments why the new method should be superior and the proposed methodology is sound.
  • In addition to these positive elements, there is one important question still to be answered: Does the experiment have a high probability of providing strong evidence that the new method is better than the standard method even if, in fact, the new method is actually better?
  • It is possible, for example, that the proposed sample size is so small that even a fairly large population difference would be difficult to detect.
  • That is, if the sample size is small, then even a fairly large difference in sample means might not be significant.
  • If the difference is not significant, then no strong conclusions can be drawn about the population means.
  • It is not justified to conclude that the null hypothesis that the population means are equal is true just because the difference is not significant.
  • Of course, it is not justified to conclude that this null hypothesis is false. Therefore, when an effect is not significant, the result is inconclusive.
  • You may prefer that your foundation's money be used to fund a project that has a higher probability of being able to make a strong conclusion.


  • Power is defined as the probability of correctly rejecting a false null hypothesis.
  • In terms of our example, it is the probability that given there is a difference between the population means of the new method and the standard method, the sample means will be significantly different.
  • The probability of failing to reject a false null hypothesis is often referred to as β.
  • Therefore power can be defined as:
:
               power = 1 - β.


Identify situations in which it is important to estimate power

  • It is very important to consider power while designing an experiment.
  • You should avoid spending a lot of time and/or money on an experiment that has little chance of finding a significant effect.


Questions

1 Power is:

The probability that the null hypothesis is true.
The probability that the null hypothesis is false.
The The probability a false null hypothesis will be rejected.
The The probability a true null hypothesis will be rejected

Answer >>

It is the probability of correctly rejecting a false null hypothesis.


2 If the power of an experiment is low then

The experiment will likely be inconclusive.
Any significant findings obtained are suspect.
The results are skewed

Answer >>

With low power, the null hypothesis is unlikely to be rejected. When the null hypothesis is not rejected, the experiment is inconclusive.



Power | Example Calculations >