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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 Hypothesis Testing
Edexcel AS Level Maths: 7.2 Finding Critical Values (Part 1)Curated by Kepler, not produced by us. Watch it 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
1

Critical region and value

The critical region\textbf{critical region} contains the values of the test statistic for which you would reject H0H_0. Its boundary is the critical value\textbf{critical value}. If your observed value falls in the critical region, you reject H0H_0.

2

Finding the critical value

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

P(X≤c)≤α (lower tail),P(X≥c)≤α (upper tail)P(X \le c) \le \alpha \ \text{(lower tail)}, \quad P(X \ge c) \le \alpha \ \text{(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.

3

Actual significance level

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

Example 1If the critical region is X≥9X \ge 9 and P(X≥9)=0.041P(X \ge 9) = 0.041, the actual significance level is:
1It is the probability of the critical region.
Answer0.0410.041 (4.1%)

Formula recap

critical region⇒reject H0\text{critical region} \Rightarrow \text{reject } H_0
Decision by region.
P(X≤c)≤αP(X \le c) \le \alpha
Finding the critical value (lower tail).
actual sig. level=P(critical region)\text{actual sig. level} = P(\text{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

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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