Hypothesis Tests about a Population Mean

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In this Demonstration, the test statistic is marked in red, the -value is the purple area, the black line marks the boundary of the rejection region when the significance level is , and the blue area is equal to . Samples are drawn from a uniformly distributed population with mean zero and variance controlled by the slider.

Contributed by: Chris Boucher (March 2011)
Open content licensed under CC BY-NC-SA



A statistical hypothesis test about the mean of an unknown population tests one of three alternative or research hypotheses against the null hypothesis that serves as a benchmark of sorts. The statistic , when computed from a random sample drawn from the population, follows approximately a -distribution with degrees of freedom if the null hypothesis is true. The degree to which the value of this statistic obtained from a given sample falls into the tail(s) of the -distribution measures our lack of confidence in the truth of the null hypothesis and support for the research hypothesis. The tail area determined by the statistic is called the -value of the test—the smaller the -value, the greater the support for the research hypothesis. For the sake of a clear decision, sometimes a boundary -value, , is specified. If the -value of the test is smaller than , then the null hypothesis is rejected.

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