S7.2StatisticsStretch

Finding Critical Values

The critical region is the set of outcomes extreme enough to reject the null hypothesis. Finding its boundary — the critical value — lets you test simply by checking whether your result lands inside it.

30 min Video by Zeeshan Zamurred Hypothesis Testing

Edexcel AS Level Maths: 7.2 Finding Critical Values (Part 1)Watch the full walkthrough before the notes below.

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What you'll be able to do

Define the critical region and critical value
Find a critical value from cumulative probabilities
Understand the actual significance level
Use the critical region to reach a conclusion

Critical region and value

The $critical region$ contains the values of the test statistic for which you would reject $H_{0}$ . Its boundary is the $critical value$ . If your observed value falls in the critical region, you reject $H_{0}$ .

Finding the critical value

For a binomial test, find the value where the cumulative probability first passes the significance level. You want the largest $c$ with $P (X \leq c) \leq α$ (lower tail) — calculators/tables make this quick.

P (X \leq c) \leq α (lower tail), P (X \geq c) \leq α (upper tail)

The boundary of the critical region.

Tip — Test each candidate against the cumulative probability; the critical value is the first one inside the significance level.

Actual significance level

Because the binomial is discrete, the critical region’s probability is usually $less than$ the nominal level (e.g. not exactly 5%). The $actual significance level$ is the true probability of being in the critical region.

1It is the probability of the critical region.

Answer

0.041

(4.1%)

Formula recap

critical region \Rightarrow reject H_{0}

Decision by region.

P (X \leq c) \leq α

Finding the critical value (lower tail).

actual sig. level = P (critical region)

True probability.

Common mistakes to avoid

Assuming the actual significance level equals the nominal 5%.

The discrete binomial usually gives an actual level below the nominal one.

Choosing the critical value on the wrong side of the boundary.

Pick the value so the tail probability stays within the significance level.

Key takeaways

The critical region is where you reject H₀; its edge is the critical value.
Find it from cumulative probabilities vs the significance level.
The actual significance level is the true probability of the critical region.

Test yourself

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