## Statistics a Gentle Introduction Ch_3 | Standard Deviation | Median

The test could be wrong. Given the p-value, we could make an error in our interpretation. In this context, we can think of the significance level as the probability of rejecting the null hypothesis if it were true. That is the probability of making a Type I Error or a false positive. Statistical power, or the power of a hypothesis test is the probability that the test correctly rejects the null hypothesis.

That is, the probability of a true positive result. It is only useful when the null hypothesis is rejected. Statistical power has relevance only when the null is false. The higher the statistical power for a given experiment, the lower the probability of making a Type II false negative error. That is the higher the probability of detecting an effect when there is an effect. In fact, the power is precisely the inverse of the probability of a Type II error. More intuitively, the statistical power can be thought of as the probability of accepting an alternative hypothesis, when the alternative hypothesis is true.

When interpreting statistical power, we seek experiential setups that have high statistical power. Experimental results with too low statistical power will lead to invalid conclusions about the meaning of the results. Therefore a minimum level of statistical power must be sought.

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All four variables are related. For example, a larger sample size can make an effect easier to detect, and the statistical power can be increased in a test by decreasing the significance level. A power analysis involves estimating one of these four parameters given values for three other parameters. This is a powerful tool in both the design and in the analysis of experiments that we wish to interpret using statistical hypothesis tests.

For example, the statistical power can be estimated given an effect size, sample size and significance level. Alternately, the sample size can be estimated given different desired levels of significance. Perhaps the most common use of a power analysis is in the estimation of the minimum sample size required for an experiment. Power analyses are normally run before a study is conducted. A prospective or a priori power analysis can be used to estimate any one of the four power parameters but is most often used to estimate required sample sizes. As a practitioner, we can start with sensible defaults for some parameters, such as a significance level of 0.

We can then estimate a desirable minimum effect size, specific to the experiment being performed. A power analysis can then be used to estimate the minimum sample size required. In addition, multiple power analyses can be performed to provide a curve of one parameter against another, such as the change in the size of an effect in an experiment given changes to the sample size. More elaborate plots can be created varying three of the parameters.

This is a useful tool for experimental design. The assumption, or null hypothesis, of the test is that the sample populations have the same mean, e. The test will calculate a p-value that can be interpreted as to whether the samples are the same fail to reject the null hypothesis , or there is a statistically significant difference between the samples reject the null hypothesis. The size of the effect of comparing two groups can be quantified with an effect size measure.

## Statistics a Gentle Introduction Ch_3

It calculates a standard score that describes the difference in terms of the number of standard deviations that the means are different. For a given experiment with these defaults, we may be interested in estimating a suitable sample size. That is, how many observations are required from each sample in order to at least detect an effect of 0. In our case, we are interested in calculating the sample size.

This tells the function what to calculate.

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A note on sample size: the function has an argument called ratio that is the ratio of the number of samples in one sample to the other. If both samples are expected to have the same number of observations, then the ratio is 1. If, for example, the second sample is expected to have half as many observations, then the ratio would be 0. Running the example calculates and prints the estimated number of samples for the experiment as This would be a suggested minimum number of samples required to see an effect of the desired size. Power curves are line plots that show how the change in variables, such as effect size and sample size, impact the power of the statistical test.

He completed a two-year postdoctoral fellowship in clinical neuropsychology at Shands Teaching Hospital in Gainesville, Florida. He has been awarded three Fulbright Fellowships to India , , and He has also won three teaching awards at the University of Colorado , , and , including the lifetime title of University of Colorado Presidential Teaching Scholar.

Coolidge conducts research in behavioral genetics and has established the strong heritability of gender identity and gender identity disorder. He also conducts research in lifespan personality assessment and has established the reliability of posthumous personality evaluations, and also applies cognitive models of thinking and language to explain evolutionary changes in the archaeological record. Tellige see raamat tutvumiseks meie kauplusesse!

## Statistics

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