I know that when sample size goes large, they basically give the same result.

My question is that

what happens if I use z-test instead of t-test when sample size is not large.

I guess the type I error increases and power increases. Am I correct? Thank you.

#### Best Answer

There is little to add to @Glen_b expert and eloquent discussion; rather, there is only room for making concepts less precise with the excuse or intention of appealing to intuition. That being said, here're a couple of plots that helped me come to terms with these concepts:

### 1. t-Student Distributions have "fatter" tails:

These tend towards the normal distribution as the sample size (or degrees of freedom) increase. Consequently, there are more points lying in the asymptotes, and to determine a certain risk alpha, the cut-off point of the test statistic would have to be slid towards the right (let's leave aside two-tail tests). The comparison is thus:

### 2. The farther away the cut-off, the lower the power:

So we slide the cutoff value to the right, and in doing so, we stay within the *NULL* a longer stretch before we reject it. Looking at it from the *alternative* there is a larger area of its corresponding curve sub-tending to the left of the cut-off value (*beta*), and a smaller slice to the right (the *power*). Just like this:

Notice the funny looking *t-distribution* under the *alternative*, which is a tentative approximation to a *non-central t* with a *delta* parameter of $2$. Under the *NULL* the distribution is central with $2,df$. The cut-off value for a one-sided risk alpha of $0.5,%$ is shown as vertical straight lines on both the Z-test (above in blue) and the t-test (below in red).

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