# Test Statistic

(Redirected from test statistic)

A Test Statistic is a statistic function, $t= g(s)$, that ... can be used to determine the veracity of the statistical hypotheses.

where $\hat{\theta}(X)$ is a (point estimate derived from sample data of the random variable $X$, $\theta_0$ is a population parameter value stated under the null hypothesis (i.e. $H_0:\; \theta=\theta_0$) and $\sigma(X,\theta)$ standard deviation which depends on both sampling distribution and population distribution.
• It is generally defined as sum of observed differences or ranks, $t= \sum f(R_i)$

## References

### 2016

• (Wikipedia, 2016) ⇒ http://en.wikipedia.org/wiki/test_statistic Retrieved 2016-09-11
• QUOTE: A test statistic is a statistic (a quantity derived from the sample) used in statistical hypothesis testing. A hypothesis test is typically specified in terms of a test statistic, considered as a numerical summary of a data-set that reduces the data to one value that can be used to perform the hypothesis test. In general, a test statistic is selected or defined in such a way as to quantify, within observed data, behaviours that would distinguish the null from the alternative hypothesis, where such an alternative is prescribed, or that would characterize the null hypothesis if there is no explicitly stated alternative hypothesis (...) For example, suppose the task is to test whether a coin is fair (i.e. has equal probabilities of producing a head or a tail). If the coin is flipped 100 times and the results are recorded, the raw data can be represented as a sequence of 100 heads and tails. If there is interest in the marginal probability of obtaining a head, only the number T out of the 100 flips that produced a head needs to be recorded. But T can also be used as a test statistic in one of two ways:
Using one of these sampling distributions, it is possible to compute either a one-tailed or two-tailed p-value for the null hypothesis that the coin is fair. Note that the test statistic in this case reduces a set of 100 numbers to a single numerical summary that can be used for testing.

### 2016

Suppose the test statistic in a hypothesis test is equal to S. If the probability of observing a test statistic as extreme as S is less than the significance level, we reject the null hypothesis.

### 1978

• (Rosenthal, 1978) ⇒ Robert Rosenthal. (1978). “Combining results of independent studies." Psychological bulletin 85, no. 1 (1978): 185.
• QUOTE: ... Not simply in connection with combining ps but at any time that test statistics such as t, F, or Z are reported, estimated effect sizes should routinely be reported. The particular effect size d seems to be the most useful one to em- ploy when two groups are being compared. ...