SME
Grade 12 Math · Problem type
What “95% confident” means
Use this for “Draw 50 samples: how many of the 95% intervals should capture μ?”
Chart
X is normal with μ = 50 and σ = 10.
Each interval is x̄ ± 1.96 × SE.
SE
=
10/
√
25
=
2.
94 of 100 intervals capture μ = 50 (94%).
The red ones miss it.
At 95% confidence, about 95 of 100 would.
This is an example. Type your own values to start a new problem.
C
Confidence level
example
n
Sample size
example
N
Samples drawn
example
K
Intervals expected to capture μ, on average
worked out (not typed)
Formulas
K
=
N
×
C
95 = 100 × 0.95
Assumptions
•
The level is how often the method captures μ over many samples.
•
Any one interval either captures μ or doesn’t; 95% is not the chance for that one.
•
The picture draws N random samples of n and the interval from each: 20, 50 or 100.
Step-by-step
WE KNOW
C = 0.95, n = 25, N = 100
FIND
intervals expected to capture μ, on average (K)
Step 1 · Find intervals expected to capture μ, on average (K)
Use K = N × C
Over
many
samples
the
share
C
of
intervals
capture
μ,
so
expect
C
of
the
N
.
K
=
100
×
0.95
K
=
95
Check
Put the numbers back in. Both sides should match.
95 = 100 × 0.95 ✓
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