Introduction to Hypothesis Testing

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Hypothesis testing is used to test whether the value of a mean or standard deviation has changed over time, whether the mean or standard deviation of two samples are equal, whether dogs prefer Dogslop or Kanineswill etc. You are testing always a null hypothesis, labelledagainst an alternative hypothesisThe null hypothesis assumes as little as possible – no change, no difference. The alternative hypothesis involves a a difference in a quantity between two samples – often the mean of the two samples are different – or the manufacturer of Dogslop claims that dogs prefer Dogslop to Kanineswill, or maybe on a stretch of road the number of accidents has increased after the local council put up a sign saying, “Drive Safely!”, and they want to find out whether the sign is less than useless. We write down the null and alternative hypotheses. To test for the example above whether the frequency of accidents has increased, we could write

:The frequency of accidents has not changed.

:The frequency of accidents has increased.

The council is suspicious about the sign. Maybe all the short sighted readers we struggling to read it, so didn’t see large red buses in front of them for example, hence the alternative hypothesis is, “the frequency of accidents has increased”. If they were not suspicious, and they only wanted to find out whether there was any statistical evidence that the frequency of accidents had increased. In that case the hypotheses could be:

:The frequency of accidents has not changed.

:The frequency of accidents has changed.

The difference is important. If we are asking has the frequency of accidents increased then we conduct a “one tailed test hypothesis test”. If we are asking has the frequency of accidents changed, then we conduct a “two tailed test”. The probability of rejecting the null hypothesis is usually higher for a one tailed test. In general if there is a claim or any sentiment involved then conduct a one tailed test. Just because a quantity has increased – or decreased – it does not follow that a one tailed test is necessary. Sometimes it is not clear whether a one tailed or two tailed test is needed. In these circumstances, ask whether there is a claim in the question or whether it is impartial. If the former, conduct a one tailed test. If the latter, conduct a two tailed test.

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