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contrast on the mean of a normal population

The following contrasts were performed assuming that the population distribution of our sample is normal. First distinguish two cases, when the variance is known and when the variance is unknown. The contrast to make is as follows a significance level :

• When the population variance, , is known using the following statistic:
  

that is asymptotically distributed as a Normal classified. And therefore reject the null hypothesis that:

• In the event that the population variance is unknown the statistic to use is
  

and reject the null hypothesis when

and if that is rejected when