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
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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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