Meaning of “critical value”
how to find critical value in a hypothesis test: select the null distribution of the test statistic \(T\), decide whether the test is left-, right-, or two-tailed, then compute the cutoff quantile(s) so that the rejection region has total probability \(\alpha\) under \(H_0\).
Step 1: Identify the null distribution of the test statistic
Critical values come from the sampling distribution of the statistic under the null hypothesis \(H_0\). Typical choices:
- z test: \(T=Z \sim N(0,1)\) under \(H_0\).
- t test: \(T \sim t_{\nu}\) under \(H_0\) with degrees of freedom \(\nu\).
- Chi-square tests: \(T \sim \chi^2_{\nu}\) under \(H_0\).
- F tests / ANOVA: \(T \sim F_{\nu_1,\nu_2}\) under \(H_0\).
Step 2: Determine the tail direction from the alternative hypothesis
- Right-tailed: \(H_a\) uses “greater than” (\(>\)); reject for large values of \(T\).
- Left-tailed: \(H_a\) uses “less than” (\(<\)); reject for small values of \(T\).
- Two-tailed: \(H_a\) uses “not equal” (\(\ne\)); reject for extreme values in both tails.
Step 3: Allocate \(\alpha\) to the tail area(s)
The total probability of the rejection region under \(H_0\) is \(\alpha\).
- One-tailed: one tail gets area \(\alpha\).
- Two-tailed: each tail gets area \(\alpha/2\).
Step 4: Compute the critical value(s) as quantiles (inverse-CDF)
Let \(F_T\) be the CDF of \(T\) under \(H_0\). Critical values are quantiles of \(F_T\):
- Left-tailed cutoff \(c_L\): \[ P(T \le c_L)=\alpha \quad \Longrightarrow \quad c_L=F_T^{-1}(\alpha). \]
- Right-tailed cutoff \(c_R\): \[ P(T \ge c_R)=\alpha \quad \Longleftrightarrow \quad P(T \le c_R)=1-\alpha \quad \Longrightarrow \quad c_R=F_T^{-1}(1-\alpha). \]
- Two-tailed cutoffs \(c_L,c_R\): \[ c_L=F_T^{-1}\!\left(\frac{\alpha}{2}\right), \qquad c_R=F_T^{-1}\!\left(1-\frac{\alpha}{2}\right). \]
| Tail type | Rejection region | How to find the critical value(s) |
|---|---|---|
| Left-tailed | \(T \le c_L\) | \(c_L=F_T^{-1}(\alpha)\) |
| Right-tailed | \(T \ge c_R\) | \(c_R=F_T^{-1}(1-\alpha)\) |
| Two-tailed | \(T \le c_L\) or \(T \ge c_R\) | \(c_L=F_T^{-1}(\alpha/2)\), \(c_R=F_T^{-1}(1-\alpha/2)\) |
Visualization: critical values as cutoffs on a null distribution
Worked example: finding a z critical value
Consider a z test for a population mean with \(\alpha=0.01\) and a right-tailed alternative (\(H_a: \mu > \mu_0\)).
- Right-tailed implies a single upper tail of size \(\alpha=0.01\).
- Find the \(1-\alpha=0.99\) quantile of the standard normal distribution.
- The critical value is \[ c_R=z_{0.99}, \] and the rejection region is \(Z \ge z_{0.99}\).
The numerical value of \(z_{0.99}\) is obtained from a standard normal table or an inverse-normal function; the procedure above is the statistical calculation.
How the same steps apply to t, \(\chi^2\), and F
- t critical value: replace \(z_{\cdot}\) by \(t_{\cdot,\nu}\) (degrees of freedom \(\nu\)). For a two-tailed t test, use \(t_{1-\alpha/2,\nu}\) and \(-t_{1-\alpha/2,\nu}\).
- \(\chi^2\) critical value: use \(\chi^2_{1-\alpha,\nu}\) for a right tail and \(\chi^2_{\alpha,\nu}\) for a left tail. The distribution is not symmetric, so the two-tailed cutoffs are \(\chi^2_{\alpha/2,\nu}\) and \(\chi^2_{1-\alpha/2,\nu}\) (not negatives of each other).
- F critical value: for a right-tailed test (common in ANOVA), use \(F_{1-\alpha,\nu_1,\nu_2}\) so that \(P(F_{\nu_1,\nu_2} \ge c_R)=\alpha\).
Checklist
- Confirm the test statistic and its null distribution (including degrees of freedom when needed).
- Read tail direction from \(H_a\).
- Split \(\alpha\) across tails if two-tailed.
- Compute the quantile(s) using \(F_T^{-1}\) at \(\alpha\), \(1-\alpha\), \(\alpha/2\), and \(1-\alpha/2\) as appropriate.