# Power of a one sample t-test

#### NN_STAT

##### New Member
Hi all,

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:
$$X_{a}=M_{0} plus+ Z_{\alpha }\cdot \frac{\sigma }{\sqrt{n}}$$

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

$$Z_{a}=\frac{X_{a}-M_{1}}{\frac{\sigma }{\sqrt{n}}}$$

$$Power = 1-\Phi (Z_{_{\alpha }})$$

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:

$$X_{a}=M_{0} plus+ t_{\alpha }\cdot \frac{S }{\sqrt{n}}$$

$$t_{a}=\frac{X_{a}-M_{1}}{\frac{S }{\sqrt{n}}} plus+ \lambda$$

$$\lambda = \frac{M_{1}-M_{0}}{\frac{S }{\sqrt{n}}}$$

"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.