# Statistics/Testing Data/Types of Tests

A statistical test is always about one or more parameters of the concerned population (distribution). The appropiate test depends on the type of null and alternative hypothesis about this (these) parameter(s) and the available information from the sample.

## Example

It is conjectured that British children gain more weight lately. Hence the population mean μ of the weight X of children of let's say 12 years of age is the parameter at stake. In the recent past the mean weight of this group of children turned out to be 45 kg. Hence the null hypothesis (of no change) is:

$\,H_{0}:\mu =45$ .

As we suspect a gain in weight, the alternative hypothesis is:

$\,H_{1}:\mu >45$ .

A random sample of 100 children shows an average weight of 47 kg with a standard deviation of 8 kg.

Because it is reasonable to assume that the weights are normally distributed, the appropriate test will be a t-test, with test statistic:

$T={\frac {{\bar {X}}-45}{S}}{\sqrt {100}}$ .

Under the null hypothesis T will be Student distributed with 99 degrees of freedom, which means approximately standard normally distributed.

The null hypothesis will be rejected for large values of T. For this sample the value t of T is:

$t={\frac {47-45}{8}}{\sqrt {100}}=2.5$ .

Is this a large value? That depends partly on our demands. The so called p-value of the observed value t is:

$p=P(T\geq t;H_{0})=P(T\geq 2.5;H_{0})\approx P(Z\geq 2.5)<0.01$ ,

in which Z stands for a standard normally distributed random variable.

If we are not too critical this is small enough, so reason to reject the null hypothesis and to assume our conjecture to be true.

Now suppose we have lost the individual data, but still know that the maximum weight in the sample was 68 kg. It is not possible then to use the t-test, and instead we have to use a test based on the statistic max(X).

It might also be the case that our assumption on the distribution of the weight is questionable. To avoid discussion we may use a distribution free test instead of a t-test.

A statistical test begins with a hypothesis; the form of that hypothesis determines the type(s) of test(s) that can be used. In some cases, only one is appropriate; in others, one may have some choice.

For example: if the hypothesis concerns the value of a single population mean ($\mu$ ), then a one sample test for mean is indicated. Whether the z-test or t-test should be used depends on other factors (each test has its own requirements).

A complete listing of the conditions under which each type of test is indicated is probably beyond the scope of this work; refer to the sections for the various types of tests for more information about the indications and requirements for each test.