P-values & Critical Values

Logic, Geometry, and Decisions

The Big Picture

Core question: Is our test statistic surprising if H₀ were true?

Approach	What we locate	Decision rule
P-value	Area in the tail(s) beyond the test statistic	Reject H₀ if p < α
Critical value	A threshold on the x-axis	Reject H₀ if the test statistic enters the rejection region

Important

These two approaches are equivalent — they are two views of the same geometry.

P-value: Area Beyond the Test Statistic

p-value = P(seeing something this extreme or more | H₀ is true)

The further z* moves into the tail → smaller area → smaller p-value → stronger evidence against H₀

Critical Value: Threshold on the X-axis

z* > CV → z* is inside the rejection region → Reject H₀

The CV cuts off exactly α in the tail; anything past it is “too extreme under H₀”

One-tailed vs Two-tailed

p-value area rejection region — test statistic (z*) – critical value (CV)

Standard Normal Z ~ N(0, 1)

Key features: Symmetric around 0 · Ranges (−∞, +∞) · Area under curve = 1 · No parameters needed

T-Student Distribution

Key features: Symmetric around 0 · df controls tail weight · As df → ∞, t → Z · Always check df first — it changes the CV

Chi-Square Distribution (χ²)

Key features: Values always ≥ 0 · Right-skewed · Only right-tail matters · Uses: goodness-of-fit, independence tests, variance tests

F Distribution

Key features: Values always ≥ 0 · Right-skewed · Two df parameters · Uses: ANOVA, overall regression F-test · F* = ratio of two variance estimates

Summary: All Four Distributions

Distribution	Symmetric?	Range	Typical test direction	# of df
Z	✓	(−∞, +∞)	one or two-tailed	—
t	✓	(−∞, +∞)	one or two-tailed	1
χ²	✗	[0, +∞)	right-tailed	1
F	✗	[0, +∞)	right-tailed	2 (df₁, df₂)

Common Critical Values — Z and t

Z — critical values

Values are the same regardless of sample size.

α	One-tailed	Two-tailed
0.10	1.282	1.645
0.05	1.645	1.960
0.01	2.326	2.576
0.001	3.090	3.291

Common Critical Values — Z and t

t — critical values (two-tailed)

As df grows the t-distribution approaches Z; notice the bottom row matches the Z table.

df	α = 0.10 (2-tail)	α = 0.05 (2-tail)	α = 0.01 (2-tail)
5	2.015	2.571	4.032
10	1.812	2.228	3.169
15	1.753	2.131	2.947
20	1.725	2.086	2.845
30	1.697	2.042	2.750
60	1.671	2.000	2.660
∞ (→ Z)	1.645	1.960	2.576

Common Critical Values — χ² and F

χ² — critical values (right-tailed)

df	α = 0.10	α = 0.05	α = 0.025	α = 0.01
1	2.706	3.841	5.024	6.635
2	4.605	5.991	7.378	9.210
3	6.251	7.815	9.348	11.345
4	7.779	9.488	11.143	13.277
5	9.236	11.070	12.833	15.086
10	15.987	18.307	20.483	23.209
15	22.307	24.996	27.488	30.578
20	28.412	31.410	34.170	37.566

Common Critical Values — χ² and F

F — critical values (right-tailed, α = 0.05)

df₂ df₁	df₁ = 1	df₁ = 2	df₁ = 3	df₁ = 4	df₁ = 5
df₂ = 10	4.96	4.10	3.71	3.48	3.33
df₂ = 15	4.54	3.68	3.29	3.06	2.90
df₂ = 20	4.35	3.49	3.10	2.87	2.71
df₂ = 30	4.17	3.32	2.92	2.69	2.53
df₂ = 60	4.00	3.15	2.76	2.53	2.37

Note

For F, you need both df₁ (numerator) and df₂ (denominator). For χ² and t, only one df. For Z, no df at all.

Decision Rules — Both Approaches

P-value approach

Compute the test statistic
Find the area in the tail(s) beyond it
That area is your p-value
Reject H₀ if p-value < α

Critical value approach

Compute the test statistic
Find the CV that cuts off area α in the tail(s)
Compare test statistic to CV
Reject H₀ if test statistic exceeds CV

The geometry is always the same

A smaller p-value ↔︎ a test statistic further in the tail ↔︎ further past the CV. Setting p-value = α gives you exactly the critical value: CV = F⁻¹(1 − α).