# Effect of sample size in t test? [duplicate]

Possible Duplicate:
Why does t statistic increase with the sample size?

In t test,the test statistic is $$t = \frac{\overline{x} - \mu_0}{s/\sqrt{n}}$$

I wonder what effect of the sample size $n$ has on the t test?

For example, as $n$ increases, I thought $t$ will increases at first, but later I realized $s$ is at the scale of $1/\sqrt{n-1}$. So $s/\sqrt{n}$ seems not to decrease as $n$ increase?

What other effect does $n$ have?

Thanks!

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Possibly related: Why does t statistic increase with the sample size?. –  caracal Oct 11 '12 at 21:20

## marked as duplicate by whuber♦Oct 11 '12 at 21:24

If data is Normal($\mu$,$\sigma^2$), the distribution of the the $t$ statistic is the same for every $n$ (namely, it follows a $t$ distribution). What changes is the number of degrees of freedom. See this plot to see how it changes. If the original data is not $iid$ normal, other assumptions are needed, and the results are asymptotic.
On the contrary, the distribution of $t$ is different for every $n$ because the degrees of freedom matter! –  whuber Oct 11 '12 at 21:25