Hypothesis Tests about a Population Mean

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

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A statistical hypothesis test about the mean of an unknown population tests one of three alternative or research hypotheses (Subscript[H, a]) against the null hypothesis (Subscript[H, 0]) which serves as a benchmark of sorts. The statistic (((OverBar[X])-(Subscript[μ, 0]))/(s/(Sqrt[n]))), when computed from a random sample drawn from the population, follows approximately a t-distribution with n-1 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 t-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 p-value of the test—the smaller the p-value, the greater the support for the research hypothesis. For the sake of a clear decision, sometimes a boundary p-value, α, is specified. If the p-value of the test is smaller than α then the null hypothesis is rejected.

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Contributed by: Chris Boucher

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