STAT 101
M05 · L02
Module 5 · Lesson 2
Confidence Intervals
A point estimate gives a single guess. A confidence interval gives a range — and tells you how much to trust it. Same data, richer story.
01 / 13
STAT 101
M05 · L02
The most common mistake
What a CI is not
“There is a 95% probability that μ is inside this interval.” Wrong. μ is a fixed number — it either is or isn’t in the interval. No probability.
The correct statement
95% of intervals produced by this procedure will contain μ. Not this interval — the procedure.
02 / 13
STAT 101
M05 · L02
What confidence really means
A Procedure, Not a Probability
- Repeat the experiment 100 times
- Each time compute a 95% CI
- About 95 of those intervals contain the true μ
- The other 5 miss — we just don’t know which
- “Confident” = the method succeeds 95% of the time
03 / 13
STAT 101
M05 · L02
When σ is known
The z-Interval
z-CI for μ
\bar{x} \pm z_{\alpha/2}\,\dfrac{\sigma}{\sqrt{n}}
90% CI
z = 1.645
95% CI
z = 1.960
99% CI
z = 2.576
04 / 13
STAT 101
M05 · L02
The real-world pivot
Student’s t-Distribution
- Replace σ with S → pivot is no longer normal
- It follows t with n−1 degrees of freedom
- Heavier tails than normal: more uncertainty
- As n → ∞, tn−1 → N(0,1)
- Derived by W.S. Gosset in 1908 (“Student”)
05 / 13
STAT 101
M05 · L02
When σ is unknown (always)
The t-Interval
t-CI for μ
\bar{x} \pm t_{\alpha/2,\,n-1}\,\dfrac{s}{\sqrt{n}}
Example
n=16, x̄=12.3, s=2.4
t₀.₀₂₅,₁₅ = 2.131
CI: (11.02, 13.58)
t₀.₀₂₅,₁₅ = 2.131
CI: (11.02, 13.58)
06 / 13
STAT 101
M05 · L02
When is the t-interval valid?
Check Your Assumptions
- Normal population: exact for any n
- Large n (≥ 30): CLT makes it approximately valid
- Skewed populations: n ≥ 50 recommended
- Outliers: check with boxplot first — t is sensitive when n is small
07 / 13
STAT 101
M05 · L02
Categorical outcomes
CI for a Proportion
Wald CI for p
\hat{p} \pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}
Poll example
n=400, p̂=0.55
95% CI: 0.55 ± 0.049
= (0.501, 0.599)
95% CI: 0.55 ± 0.049
= (0.501, 0.599)
08 / 13
STAT 101
M05 · L02
Controlling interval width
Three Levers
- More data: margin of error shrinks as 1/√n — quadruple n to halve E
- Lower confidence: smaller z or t — but higher miss rate
- Less spread (σ): better instruments, stricter inclusion criteria
09 / 13
STAT 101
M05 · L02
Plan before you collect
Sample Size for a Mean
Required n
n = \left(\dfrac{z_{\alpha/2}\,\sigma}{E}\right)^{\!2}
Set the desired margin of error E, plug in zα/2 and σ, solve for n. Always round up.
10 / 13
STAT 101
M05 · L02
Planning for proportions
Sample Size for p
Required n
n = \left(\dfrac{z_{\alpha/2}}{E}\right)^{\!2}\hat{p}(1-\hat{p})
Why 1000 for polls?
p̂=0.5, E=0.03, 95% CI
n = (1.96/0.03)² × 0.25 ≈ 1068
n = (1.96/0.03)² × 0.25 ≈ 1068
11 / 13
STAT 101
Summary
Recap
What you learned
- CI = a procedure, not a probability statement about a single interval
- z-interval: x̄ ± z⋅σ/√n (rare; needs known σ)
- t-interval: x̄ ± t⋅s/√n (standard practice; heavier tails)
- Proportion CI: p̂ ± z⋅√(p̂(1−p̂)/n)
- Width = 2E: controlled by n, confidence level, and σ
Up next
13 / 13