SME
Grade 12 Math · Problem type
Confidence interval for a mean, σ unknown
Use this for “10 plants average 52 cm with s = 8 cm. Find a 95% confidence interval for μ.”
Chart
t curve with df = 9 (solid) over the normal (dashed): lower in the middle, heavier in the tails; t⋆ = 2.262 for 95%, where z⋆ = 1.96.
The curve is centered at x̄ = 52 cm with standard error SE = 2.5298 cm.
Interval: 52 ± 5.7229, from 46.2771 to 57.7229, the middle 0.9500 of the curve.
This is an example. Type your own values to start a new problem.
x̄ (cm)
±
t⋆
×
s (cm)
n
=
(
L (cm)
,
U (cm)
)
C
Confidence level
example
df
Degrees of freedom
worked out (not typed)
SE
Standard error
worked out (not typed)
E
Margin of error
calculated
Formulas
df
=
n
−
1
9 = 10 − 1
t⋆
=
invT(1
−
1 −
C
2,
df)
2.2622
=
invT(1
−
1 − 0.95
2,
9)
SE
=
s
÷
√
n
2.5298
=
8
÷
√
10
E
=
t⋆
×
SE
5.7229 = 2.2622 × 2.5298
L
=
x̄
−
E
46.2771 = 52 − 5.7229
U
=
x̄
+
E
57.7229 = 52 + 5.7229
Assumptions
•
σ is unknown, so the sample’s s stands in for it and t⋆ replaces z⋆, with df = n − 1.
•
t⋆ is larger than z⋆ for small samples, so the interval is wider; they agree as n grows.
•
A random sample from a population close to normal (or n ≥ 30).
Step-by-step
WE KNOW
x̄ = 52 cm, s = 8 cm, n = 10, C = 0.95
FIND
degrees of freedom (df), critical value (t⋆), standard error (SE), margin of error (E), lower end (L), upper end (U)
Step 1 · Find degrees of freedom (df)
Use df = n − 1
One less than the sample size: the mean uses up one degree of freedom.
df = 10 − 1
df = 9
Step 2 · Find critical value (t⋆)
Use t⋆ = invT(1 − (1 − C) ÷ 2, df)
The
middle
C
of
the
t
curve
lies
within
±t⋆,
so
the
area
left
of
t⋆
is
1
−
1 −
C
2
.
t⋆ = invT(0.975, df)
t⋆ = invT(0.975, 9)
t⋆ = 2.2622
Step 3 · Find standard error (SE)
Use SE = s ÷ √n
The
sample’
s
s
stands
in
for
σ:
divide
it
by
the
root
of
n
.
SE
=
8
÷
√
10
SE
=
8
3.162
SE = 2.5298 cm
Step 4 · Find margin of error (E)
Use E = t⋆ × SE
The margin of error is t⋆ standard errors.
E
=
2.2622
×
2.5298
E
=
5.7229
cm
Step 5 · Find lower end (L)
Use L = x̄ − E
Go down from the estimate by the margin.
L
=
52
−
5.7229
L
=
46.2771
cm
Step 6 · Find upper end (U)
Use U = x̄ + E
Go up from the estimate by the margin.
U
=
52
+
5.7229
U
=
57.7229
cm
Check
Put the numbers back in. Both sides should match.
9 = 10 − 1 ✓
2.2622 = invT(0.975, 9) ✓
2.5298
≈
8
÷
√
10
✓
5.7229 ≈ 2.2622 × 2.5298 ✓
46.2771 = 52 − 5.7229 ✓
57.7229 = 52 + 5.7229 ✓
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