## p-value for t-test

If the test statistic $t$ has $t$ distribution with $n$ degrees of freedom, then the $p$-value of the test for testing

a. left-tailed alternative is $p$-value = $P(t_{n}\leq t)$.

b. right-tailed alternative is $p$-value = $P(t_{n}\geq t)$.

c. two-tailed alternative is $p$-value = $2P(t_{n}\geq |t|)$.

## Formula

The $p$-value of t-test for left tailed alternative is

### $p$-value = $P(t_{n}\leq t)$

The $p$-value of t-test for right tailed alternative is

### $p$-value = $P(t_{n}\geq t)$

The $p$-value of t-test for two tailed alternative is

### $p$-value = $2P(t_{n}\geq |t|)$

where,

• $t$ is the value of test statistic,
• $n$ is the degrees of freedom.