This means that the null hypothesis would be written as

The Brinell hardness scale is one of several definitions used in the field of materials science to quantify the hardness of a piece of metal. The Brinell hardness measurement of a certain type of rebar used for reinforcing concrete and masonry structures was assumed to be normally distributed with a standard deviation of 10 kilograms of force per square millimeter. Using a random sample of n = 25 bars, an engineer is interested in performing the following hypothesis test:

the null hypothesis is rejected when it is true b.

That makes the power of the engineer's hypothesis test 0.6915 as illustrated here:

the null hypothesis is not rejected when it is false c.

Generally speaking, one-tailed tests are often reserved for situations where a clear directional outcome is anticipated or where changes in only one direction are relevant to the goals of the study. Examples of the latter are perhaps more often encountered in industry settings, such as testing a drug for the alleviation of symptoms. In this case, there is no reason to be interested in proving that a drug worsens symptoms, only that it improves them. In such situations, a one-tailed test may be suitable. Another example would be tracing the population of an endangered species over time, where the anticipated direction is clear and where the cost of being too conservative in the interpretation of data could lead to extinction. Notably, for the field of experimental biology, these circumstances rarely, if ever, arise. In part for this reason, two-tailed tests are more common and further serve to dispel any suggestion that one has manipulated the test to obtain a desired outcome.

the research hypothesis is not rejected when it is false722-1

Also, just to reinforce a point raised earlier, greater variance in the sample data will lead to higher -values because of the effect of sample variance on the SEDM. This will make it more difficult to detect differences between sample means using the -test. Even without any technical explanation, this makes intuitive sense given that greater scatter in the data will create a level of background noise that could obscure potential differences. This is particularly true if the differences in means are small relative to the amount of scatter. This can be compensated for to some extent by increasing the sample size. This, however, may not be practical in some cases, and there can be downsides associated with accumulating data solely for the purpose of obtaining low -values (see ).

Let's take a look at another example that involves calculating the power of a hypothesis test.
What is the power of the hypothesis test if the true population mean were μ = 108?

failing to reject the null hypothesis when it is false.

In this scenario, the data are meaningfully paired in that we are measuring GFP levels in two distinct cells, but within a single worm. We then collect fluorescence data from 14 wild-type worms and 14 worms. A visual display of the data suggests that expression of ::GFP is perhaps slightly decreased in the right cell where gene has been inhibited, but the difference between the control and experimental dataset is not very impressive (). Furthermore, whereas the means of GFP expression in the left neurons in wild-type and worms are nearly identical, the mean of GFP expression in the right neurons in wild type is a bit higher than that in the right neurons of worms. For our -test analysis, one option would be to ignore the natural pairing in the data and treat left and right cells of individual animals as independent. In doing so, however, we would hinder our ability to detect real differences. The reason is as follows. We already know that GFP expression in some worms will happen to be weaker or stronger (resulting in a dimmer or brighter signal) than in other worms. This variability, along with a relatively small mean difference in expression, may preclude our ability to support differences statistically. In fact, a two-tailed -test using the (hypothetical) data for right cells from wild-type and strains () turns out to give a > 0.05.

What is the power of the hypothesis test if the true population mean were μ = 112?

rejecting the null hypothesis when it is true.

Most statistical tests culminate in a statement regarding the -value, without which reviewers or readers may feel shortchanged. The -value is commonly defined as the probability of obtaining a result (more formally a ) that is at least as extreme as the one observed, assuming that the is true. Here, the specific null hypothesis will depend on the nature of the experiment. In general, the null hypothesis is the statistical equivalent of the “innocent until proven guilty” convention of the judicial system. For example, we may be testing a mutant that we suspect changes the ratio of male-to-hermaphrodite cross-progeny following mating. In this case, the null hypothesis is that the mutant does not differ from wild type, where the sex ratio is established to be 1:1. More directly, the null hypothesis is that the sex ratio in mutants is 1:1. Furthermore, the complement of the null hypothesis, known as the or , would be that the sex ratio in mutants is different than that in wild type or is something other than 1:1. For this experiment, showing that the ratio in mutants is different than 1:1 would constitute a finding of interest. Here, use of the term “significantly” is short-hand for a particular technical meaning, namely that the result is , which in turn implies only that the observed difference appears to be real and is not due only to random chance in the sample(s). . Moreover, the term significant is not an ideal one, but because of long-standing convention, we are stuck with it. Statistically or statistically may in fact be better terms.

What is the power of the hypothesis test if the true population mean were μ = 116?

rejecting the null hypothesis when the alternative is true.

This is straightforward. The peer review process for RRs differs in two ways from conventional review, both of which can be dovetailed with existing systems. First, unlike standard submissions, RRs have two distinct stages of review – one before data collection and one afterward. This can be integrated into existing software by simply treating each stage as a new submission, linked by a journal editorial assistant, and by treating in-principle acceptance as a technical “rejection”. Second, the review process for RRs at several journals is structured: reviewers are asked to assess the extent to which the manuscript meets a number of fixed criteria. Even if your handling software is unable to implement a structured review mechanism in which reviewers enter text into pre-defined fields, the criteria can be easily incorporated into the reviewer invitation letters. We have found that this works adequately provided the attention of reviewers is drawn specifically to these criteria. Generic templates of reviewer invitation letters and editorial decision letters can be downloaded from our Resources for Editors page (see tabs above). Once adapted to your specific requirements, the technical staff at your publisher should be able to add them to the system in a matter of days. At Cortex, for example, the necessary amendments to the Elsevier Editorial System were implemented in less then a week by a single member of the publishing staff.