Critical values

Summary

Critical values from the standard normal

• If $Z ~ N(0,1)$ then for $0<α<1$ we define $z_α$ by

$$P\left( Z>z_α \right)=α$$

• $z_α$ is called the critical value for the standard normal with level of significance $α$ .
• The critical value $z_{α/2}$ for $α=0.05$ should be memorized, $z_{0.025}=1.96$ .
• There is an inverse relationship between $z_α$ and $α$ .

Critical values from the t -distribution

• If $T ~ t_k$ then for $0<α<1$ we define $t_{α,k}$ by

$$P\left( T>t_{α,k} \right)=α$$

• $t_{α,k}$ is called the critical value for the t-distribution with $k$ degrees of freedom with level of significance $α$ .
• There is an inverse relationship between $t_{α,k}$ and $α$ .
• $t_{α,k}>z_α$ for all $α$ (the $t$ -distribution has “ fatter tails ” than the standard normal)
• $t_{α,k}$ converges to $z_α$ as $k→∞$ (the $t$ -distribution looks more like the standard normal as the degrees of freedom increases)

Critical values from the $χ^2$ -distribution

• If $Y ~ χ_k^2$ then for $0<α<1$ we define $χ_{α,k}^2$ by

$$P\left( Y>χ_{α,k}^2 \right)=α$$

• $χ_{α,k}^2$ is called the critical value for the chi-square distribution with $k$ degrees of freedom with level of significance $α$ .

Critical values from the F -distribution

• If $F ~ F_{k,l}$ then for $0<α<1$ we define $F_{α,k,l}$ by

$$P\left( Y>F_{α,k,l} \right)=α$$

• $F_{α,k,l}$ is called the critical value for the F -distribution with $k,l$ degrees of freedom with level of significance $α$ .