Introduction to Critical Values and Critical Regions

When we conduct a hypothesis test, we need a principled way to decide when our evidence is strong enough to reject the null hypothesis $H_{0}$ . This is where critical values and critical regions come in.

Critical Value

A critical value is a threshold value that marks the boundary of the critical region. It is determined by the chosen significance level $α$ and whether the test is one-tailed or two-tailed.

Critical Region

The critical region (also called the rejection region) is the set of values of the test statistic for which we reject $H_{0}$ . The total area of the critical region equals the significance level $α$ .

For a normal distribution, the standard critical z-values are:

Test Type	Significance Level $α$	Critical Value(s)
Two-tailed	0.05	$z = \pm 1.960$
Two-tailed	0.01	$z = \pm 2.576$
One-tailed (right)	0.05	$z = 1.645$
One-tailed (left)	0.05	$z = - 1.645$

Note

If the calculated test statistic falls inside the critical region (beyond the critical value), we reject $H_{0}$ . If it falls outside the critical region, we fail to reject $H_{0}$ . Notice the language — we never "accept" $H_{0}$ .

Choosing Between the z-Test and the t-Test

The most fundamental decision in a one-sample mean test is whether to use a z-test or a t-test. This depends entirely on whether the population standard deviation $σ$ is known.

When $σ$ is known — use the z-test:

The test statistic is:
$z = \frac{x ˉ - μ _{0}}{σ / n}$

Where $\overset{x}{ˉ}$ is the sample mean, $μ_{0}$ is the hypothesised population mean, $σ$ is the known population standard deviation, and $n$ is the sample size. This statistic follows a standard normal distribution $Z \sim N (0, 1)$ .

When $σ$ is unknown — use the t-test:

The test statistic is:
$t = \frac{x ˉ - μ _{0}}{s / n}$

Where $s$ is the sample standard deviation (used to estimate $σ$ ). This statistic follows a t-distribution with $n - 1$ degrees of freedom.

Exam Tip

The decision rule is simple: known $σ$ → z-test; unknown $σ$ → t-test. This applies regardless of sample size. For large samples ( $n > 30$ ), the t-distribution approximates the normal distribution closely, but you should still use the t-test if $σ$ is unknown.

Warning

A very common mistake is choosing the test based on sample size alone. Sample size does not determine whether you use a z-test or t-test — knowledge of $σ$ does.

Assumptions for the z-test:

Population follows a normal distribution (or $n$ is large enough for the Central Limit Theorem to apply)
$σ$ is known
Data points are independent

Assumptions for the t-test:

Population follows a normal distribution (especially critical for small samples)
$σ$ is unknown
Data points are independent

Free preview

AHL 4.18—T and Z test, type I and II errors

Introduction to Critical Values and Critical Regions

Choosing Between the z-Test and the t-Test

11 more sections in this topic