# Discussion: Power Analysis Case

# Discussion: Power Analysis Case

## Discussion: Power Analysis Case

Discussion: Power Analysis Case

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Module X Discussion Explain how performing a power analysis reduces the possibility of making a Type I and Type II Error. Post your initial response by Wednesday at midnight. Respond to one student by Sunday at midnight. Both responses should be a minimum of 150 words, scholarly written, APA formatted, and referenced. A minimum of 2 references are required (other than your text). Refer to the Grading Rubric for Online Discussion in the Course Resource section.

The **power** of a binary hypothesis test is the probability that the test rejects the null hypothesis (H_{0}) when a specific alternative hypothesis (H_{1}) is true. The statistical power ranges from 0 to 1, and as statistical power increases, the probability of making a type II error (wrongly failing to reject the null hypothesis) decreases. For a type II error probability of β, the corresponding statistical power is 1 − β. For example, if experiment 1 has a statistical power of 0.7, and experiment 2 has a statistical power of 0.95, then there is a stronger probability that experiment 1 had a type II error than experiment 2, and experiment 2 is more reliable than experiment 1 due to the reduction in probability of a type II error. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H_{1}) when it is true—that is, the ability of a test to detect a specific effect, if that specific effect actually exists. That is,

- {displaystyle {text{power}}=Pr {big (}{text{reject }}H_{0}mid H_{1}{text{ is true}}{big )}.}

If {displaystyle H_{1}} is not an equality but rather simply the negation of {displaystyle H_{0}} (so for example with {displaystyle H_{0}:mu =0} for some unobserved population parameter {displaystyle mu ,} we have simply {displaystyle H_{1}:mu neq 0}) then power cannot be calculated unless probabilities are known for all possible values of the parameter that violate the null hypothesis. Thus one generally refers to a test’s power *against a specific alternative hypothesis*.

As the power increases, there is a decreasing probability of a type II error, also referred to as the false negative rate (*β*) since the power is equal to 1 − *β*. A similar concept is the type I error probability, also referred to as the “false positive rate” or the level of a test under the null hypothesis.

Power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size. For example: “how many times do I need to toss a coin to conclude it is rigged by a certain amount?”^{[1]} Power analysis can also be used to calculate the minimum effect size that is likely to be detected in a study using a given sample size. In addition, the concept of power is used to make comparisons between different statistical testing procedures: for example, between a parametric testand a nonparametric test of the same hypothesis.

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