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B_Miner
03-03-2010, 08:50 PM
This is probably staring me in the face, but my face has been staring at it for many hours ... ;)

In basic power analysis of a mean, using the CLT, testing a lower tail hypothesis I have:

X_lower_critical = qnorm(alpha | mu_null, sigma/sqrt(n))

and also
X_lower_critical = qnorm(1-beta | mu_alternative, sigma/sqrt(n))

where qnorm is the quantile with alpha (or 1-beta) area to the left.

The author then says "standardizing X_lower_critical both way":


mu_null - qnorm(alpha | 0,1) * (sigma/sqrt(n)) =
mu_alternative + qnorm(1-beta | 0,1) * (sigma/sqrt(n))


Can anyone help me, how are they getting this last step?

Thanks!

Brian

fed1
03-03-2010, 10:11 PM
I will show you heuristic derivation:
Rearrange to get,


[mu_null - mu_alternative]= delta = qnorm(alpha | 0,1) * (sigma/sqrt(n)) + qnorm(1-beta | 0,1) * (sigma/sqrt(n))

Now consider the numerator of ur test stat, its like T =(ybar1 - ybar2), right.

How is it distributed under the null, Z ~ Norm( 0, var_0 ); say.

var_0 = sigma/sqrt(n) apparently.

Now draw the distribution of t under HO; Where does its 95% point lay?
Hint: Pr[ Z* < z ] = Pr[ T < var_0 * z];Z*~N(0,1)

Now do the same for T under HA. Actually, cheat and assume T has normal distribution with mean del, and va var_1; apparently same as va_0;

Now deduce that the relationship is true. Its tricky but its there, enjoy.:)

This is maybe very important relationship in stats.