why t-distribution and normal distribution?

Question

Why do we use the t-distribution when the population standard deviation is not known? And similarly why the normal distribution when standard deviation is known?

Answer 1

The $t$ distribution arises in the following situation. You have a normally distributed population with unknown mean $\mu$ and standard deviation $\sigma$. You draw $n$ independent samples $X_i$ from this population, and then you want to test a hypothesis about the mean of the population using this sample. The null hypothesis for this test is $\mu=\mu_0$. (The alternative hypothesis varies depending on context.)

The test statistic for this is

$$\frac{\overline{X}-\mu_0}{S/\sqrt{n}}$$

where $S$ is the sample standard deviation. Now if $S$ were just a fixed number then this ratio would be normally distributed. Moreover, if it were exactly $\sigma$ then this would be $N(0,1)$ distributed under the null hypothesis (which is why we use the normal distribution for hypothesis tests for the mean when the true standard deviation is known). But in fact $S$ is not fixed, it depends on the sample. The distribution of this ratio, assuming the null hypothesis, is called the Student's $t$ distribution with $n-1$ degrees of freedom.

Intuitively, the difference is that $S$ is occasionally much smaller than $\sigma$, and it is these cases that cause the $t$ distribution to have a longer tail than the normal distribution, especially when the number of degrees of freedom is small.