# Examples of statistical analysis

As an example, consider determining whether a suitcase contains some radioactive material. Placed under a Geiger counter , it produces 10 counts per minute. The null hypothesis is that no radioactive material is in the suitcase and that all measured counts are due to ambient radioactivity typical of the surrounding air and harmless objects. We can then calculate how likely it is that we would observe 10 counts per minute if the null hypothesis were true. If the null hypothesis predicts (say) on average 9 counts per minute, then according to the Poisson distribution typical for radioactive decay there is about 41% chance of recording 10 or more counts. Thus we can say that the suitcase is compatible with the null hypothesis (this does not guarantee that there is no radioactive material, just that we don't have enough evidence to suggest there is). On the other hand, if the null hypothesis predicts 3 counts per minute (for which the Poisson distribution predicts only % chance of recording 10 or more counts) then the suitcase is not compatible with the null hypothesis, and there are likely other factors responsible to produce the measurements.

A sample concretely represents n experiments in which the same quantity is measured. For example, if X represents the height of an individual and n individuals are measured, X i {\displaystyle X_{i}} will be the height of the i -th individual. Note that a sample of random variables (. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take, formally called random variates ). In other words, X i {\displaystyle X_{i}} is a function representing the measurement at the i-th experiment and x i = X i ( ω ) {\displaystyle x_{i}=X_{i}(\omega )} is the value actually obtained when making the measurement.