I have a little problem with power calculations. I have calculated the power for a one sample Z test, which was fine. I found the critical value under H0:

[tex] X_{a}=M_{0} plus+ Z_{\alpha }\cdot \frac{\sigma }{\sqrt{n}} [/tex]

then I calculated the Z value under a specific value for H1, and the following probability:

[tex] Z_{a}=\frac{X_{a}-M_{1}}{\frac{\sigma }{\sqrt{n}}} [/tex]

[tex] Power = 1-\Phi (Z_{_{\alpha }}) [/tex]

Now I want to do the same with the one sample t-test, when the population standard deviation is unknown. I thought it would be the same procedure, however, this is the required procedure:

[tex] X_{a}=M_{0} plus+ t_{\alpha }\cdot \frac{S }{\sqrt{n}} [/tex]

[tex] t_{a}=\frac{X_{a}-M_{1}}{\frac{S }{\sqrt{n}}} plus+ \lambda [/tex]

[tex] \lambda = \frac{M_{1}-M_{0}}{\frac{S }{\sqrt{n}}} [/tex]

"lambda" is a non centrality parameter, and my question is why do I need it ? Why isn't the procedure the same except for replacing Z with t and sigma with S ?

Comment: for some reason latex didn't accept my + sign, so I wrote "plus" every time I wanted to use it.