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Power of a Hypothesis Test

In statistics, whenever a hypothesis test is conducted, a statistician would always like to determine whether the test is of sufficiently good quality or not. Evaluation of the power of hypothesis test is a way of measuring its quality. The power determines if the hypothesis test is powerful enough. In statistics, it is very important to learn what is the meaning of the power of a hypothesis test and how to find the size of the sample which is required to ensure that our hypothesis test has a high power. Let us go ahead and learn this in our lesson today.

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Definition

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The power of a hypothesis test is also known as its sensitivity which is defined as the probability of correctly rejecting the hypothesis H$_{0}$ provided that the alternative hypothesis H$_{1}$ is true. In other words, we can say that the power is said to be the probability of accepting alternative hypothesis when it is true. This is referred to the ability of our test that it detects an effect when the effect exists in actual. The power can be expressed by the following relation:
$Power$ = $P\ (reject\ H$$_{0}$ $:\ H$$_{1}$ $is\ true)$
With the increase in the value of power, the chances of occurring Type II error (or false negative) increase. So, we can write power as:
$Power$ =  $1\ -\ P(\ type\ II\ error\ )$ = $1\ -$ $\beta$

How to Find the Power of a Hypothesis Test

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In order to compute the power of a hypothesis test, the following procedure is to be used.

Step 1: First determine the area of acceptance for the given hypothesis test.

Step 2: Now determine the value of the critical parameter. This value acts as an alternative to the value in the null hypothesis. Find the difference between critical parameter value and value obtained from the null hypothesis. This difference is known as the effect size.

Step 3: For computing power, suppose that the actual population parameter is equal to the value of the critical parameter, rather than the value given in null hypothesis. Using this assumption, find the probability of falling sample estimate of the population parameter outside the area of acceptance. This probability is said to be the required power of the hypothesis test.

Effect Size

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The effect size plays an important role in the process of finding the power of a hypothesis test. It is defined as the difference between true value and the value in the null hypothesis.

Effect size = True value - Value specified in null hypothesis

Let's take a simple example - Null hypothesis is assumed to be mean equal to 100, i.e. $H_{0}$ : $\mu$ = $100$. If the question arises what would be the probability of rejecting null hypothesis, provided that true population mean is 90, then in this case,
Effect size90 - 100 -10

Factors That Affect Power

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There are many factors that affect statistical power. Few of them are dependent upon the particular testing situation. In general, the power is influenced by the three factors discussed below:

(i) Sample Size

In smaller samples, it is more difficult to detect the effects. In order to boost the power of a hypothesis test, the sample size has been increased while other variants are same.

(ii) Magnitude of the Effect

In the population, the magnitude of the required effect can be measured in the form of an effect size. The higher statistical power tends to detect greater effects.

(
iii) Statistical Significance Criterion

The significance criterion is said to be the statement of the unlikelihood of a positive result. The criteria often used are $5%$ or $0.05$ , $1%$ or $0.01$ and $0.1%$ or $0.001$. In order to increase the power of a test, one should use a larger significance criterion.

Sample Size Calculation

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The sampling error in test result is determined by the sample size. When other factors are equal, the effects are little hard to find in smaller sample sizes. The statistical power can be boosted by increasing the size of the sample. The higher power can be obtained by an increased sample size.

Thus, the sample size plays a vital role in the power analysis of a hypothesis test. However, a researcher can first select a required power of the test and then calculate the appropriate sample size that should be used for the experiment. It is better than doing power analysis afterwards.

Examples

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An example based on the power of a hypothesis test is discussed below.

Example 1: In a hypothesis test:

Null hypothesis $H_{0}$ : $\mu$ = $4$

Alternative hypothesis $H_{1}$ : $\mu$ = $6$

Standard deviation being $2$. Determine the power of a sample size $n$ = $4$, given that the sample mean value be $5.645$. Discuss the result.

Solution:
 Calculation of $Z$ statistic assuming alternative hypothesis to be true.

Z = $\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Z = $\frac{5.645 - 6}{\frac{2}{\sqrt{4}}}$

Z = $\frac{-0.355}{\frac{2}{2}}$ = $-0.355$

For  $P(Z\ >\ -0.355)$ = $0.6387$ $\approx$ 64%

The power of this test is not considered higher, since the tests that have 80% or more power are said to be statistically powerful enough.
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