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 \(α\) .