Critical Values

Critical Values

Critical values are the cut-off points that separate the critical region from the non-critical region in a hypothesis test or determine the width of a confidence interval.

Choosing Critical Values

Distribution	When to Use	Requirements	Example Uses
z-critical ( $z^{*}$ )	Large samples	$n \geq 30$ , known $σ$	Proportions, large sample means
t-critical ( $t^{*}$ )	Small samples	$n < 30$ , unknown $σ$	Small sample means, differences

Types of Critical Values

z-Critical Values ( $z^{*}$ )

Used with:

Normal Distribution
Proportion tests and intervals

Confidence Level	Area in Tails ( $α$ )	$z^{*}$ Value
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.576

t-Critical Values ( $t^{*}$ )

Used with:

t-distribution
Mean-based tests and intervals

t-critical values depend on:

Confidence level
degrees of freedom (df = $n - 1$ )

Finding Critical Values

Process Comparison

Step	z-Critical Values	t-Critical Values
1. Level	Choose confidence level (C)	Choose confidence level (C)
2. Calculate	$α = 1 - C$	df = $n - 1$
3. Find	$z^{*}$ for area $1 - \frac{α}{2}$	$t^{*}$ for area $1 - \frac{α}{2}$ with df
Example	95%: $z_{0.975} = 1.96$	95%, n=10: $t_{0.975, 9} = 2.262$

Applications in Statistical Analysis

Confidence Intervals

Type	Formula	When to Use
Large Sample Mean	$\bar{x} \pm z^{*} \times \frac{σ}{\sqrt{n}}$	$n \geq 30$ , known $σ$
Small Sample Mean	$\bar{x} \pm t^{*} \times \frac{s}{\sqrt{n}}$	$n < 30$ or unknown $σ$
Proportion	$\hat{p} \pm z^{*} \times \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}}$	$n \hat{p} \geq 10$ , $n (1 - \hat{p}) \geq 10$

Hypothesis Testing Decision Rules

Method	Decision Rule	Equivalent to
Critical Value	Reject $H_{0}$ if $	$t e s t s t a t i s t i c$
p-value	Reject $H_{0}$ if p-value < $α$	Same conclusion

Common Values Reference Table

Confidence Level	$α$	z-critical	t-critical (df=10)	t-critical (df=20)
90%	0.10	1.645	1.812	1.725
95%	0.05	1.96	2.228	2.086
99%	0.01	2.576	3.169	2.845

Key Relationships

Critical values ↔ Confidence level (fixed relationship)
p-value ↔ Test statistic (varies by sample)
Sample size ↔ Type of critical value (z or t)

See also: