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Power of a t-test

  1. Sep 15, 2014 #1
    I'm looking over my notes, and I'm puzzled by this non-central t-distribution, and why it is the alternative hypothesis.

    Non-central t-distribution: If [itex] X \sim (\delta, 1) [/itex] and [itex] Y \sim \chi^2_{r}[/itex], in addition if X and Y are independent random variables, then [itex] \frac {X}{\sqrt{\frac {Y}{r}}}[/itex] has a t-distribution with non-centrality parameter, [itex] \delta [/itex]

    [itex] \frac {\frac {\bar{X} - \mu}{\frac {\sigma}{\sqrt{n}}}}{\sqrt{\frac {(n-1)s^2}{\sigma^2 (n-1)}}} = \frac {\bar{X} - \mu_0}{\frac {s}{\sqrt{n}}} \sim t_{\delta = \frac {\bar{X} - \mu}{\frac {\sigma}{\sqrt{n}}}}[/itex]

    [itex]H_0 : \mu = \mu_0 [/itex], [itex]H_1 : \mu \neq \mu_0 [/itex]

    [itex] 1 - \beta = P(\frac {|\bar{X} - \mu_0|}{\frac {s}{\sqrt{n}}} > t_{1 - \frac {\alpha}2} | \delta = \frac {\mu - \mu_0}{\frac {\sigma}{\sqrt{n}}}, n-1) [/itex]

    I do see that the general form for the power of a test is P{null is rejected | alternative is true}, but why is it that the alternative hypothesis is this crazy looking distribution?
  2. jcsd
  3. Sep 20, 2014 #2


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    Science Advisor

    The noncentral t distribution arises as you are testing a special alternative, namely that X is distributed normally around delta, and not around 0 as in the null hypothesis.
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