SME
Grade 12 Math · Statistics
Hypothesis tests for one and two samples
Use this for “60 of 100 flips land heads. Is the coin unfair at the 0.05 level?” with a one-proportion z-test.
Chart
The curve is centered at p₀ = 0.5 with standard error SE = 0.05.
Hₐ uses ≠: both tails count, past −|z| and |z|.
Red outline: the rejection region, area α = 0.05, past z = −1.96 and z = 1.96.
Test statistic z = 2.
Amber: the p-value, 0.0455 < α = 0.05: reject H₀.
This is an example. Type your own values to start a new problem.
Hₐ:
p
p₀
p̂
=
k
n
z
=
−
SE
P
p-value
worked out (not typed)
α
Significance level
example
Formulas
p̂
=
k
n
0.6
=
60
100
SE
=
√(
p₀
×
1 −
p₀
n
)
0.05
=
√(0.5
×
1 − 0.5
100
)
z
=
p̂
−
p₀
SE
2
=
0.6 − 0.5
0.05
P
=
the
tail
past
z
on
Hₐ’s
side
0.0455 = the tail past 2 on ≠’s side
Assumptions
•
H₀: p = p₀; Hₐ says p < p₀, p > p₀ or p ≠ p₀ (tap the sign in Hₐ).
•
The sample is random, with np₀ ≥ 10 and n(1 − p₀) ≥ 10, so p̂ is close to normal.
•
Reject H₀ when the p-value is below α; “fail to reject” never proves H₀ true.
Step-by-step
WE KNOW
p₀ = 0.5, Hₐ: p ≠ p₀, n = 100, k = 60, α = 0.05
FIND
sample proportion (p̂), standard error if H₀ is true (SE), test statistic (z), p-value (P)
Step 1 · Find sample proportion (p̂)
Use p̂ = k ÷ n
The share of the sample that are successes.
p̂
=
60
100
p̂
=
0.6
Step 2 · Find standard error if H₀ is true (SE)
Use SE = √(p₀ × (1 − p₀) ÷ n)
If
H₀
is
true,
p̂
spreads
by
√(
p₀
(1 −
p₀
)
n
):
use
p₀
,
not
p̂
.
SE
=
√(0.5
×
1 − 0.5
100
)
SE
=
√(0.5
×
0.5
100
)
SE
=
√(
0.25
100
)
SE
=
√
0.0025
SE = 0.05
Step 3 · Find test statistic (z)
Use z = (p̂ − p₀) ÷ SE
How many standard errors the estimate is from the value H₀ claims.
z
=
0.6 − 0.5
0.05
z
=
0.1
0.05
z
=
2
Step 4 · Find p-value (P)
Use P = the tail past z on Hₐ’s side
Hₐ
<
takes
the
area
left
of
z
;
Hₐ
>
the
area
right
of
z
;
Hₐ
≠
both
tails
past
|
z
|.
P
=
2
×
(1
−
Φ(|
z
|))
P
=
2
×
(1
−
Φ(|2|))
P
=
0.0455
→
0.0455
<
α
=
0.05:
reject
H₀
Check
Put the numbers back in. Both sides should match.
0.6
=
60
100
✓
0.05
=
√(0.5
×
1 − 0.5
100
)
✓
2
=
0.6 − 0.5
0.05
✓
0.0455 = 2 × (1 − Φ(|2|)) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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