Chapter 1 · §1.3

Guess the parameter. Build the net.

Point vs. interval estimates, what "95% confident" really means, and how confidence trades off against precision.

Point vs. interval

One guess, or a net?

A point estimate is a single value (x̄ for μ). An interval estimate gives a lower and upper value, a confidence interval, and is usually more useful.

x̄
405060708090100

x̄ = 68.4 · 95% CI [63.5, 73.3]

Point landed on μ (to 0.1)
0 / 0
Interval caught μ
0 / 0

Hidden truth: μ = 65, σ/√n = 2.5. Draw several samples: the single guess almost never hits exactly; the net usually catches μ.

Coverage simulator

How often does the net catch μ?

Repeated samples from a population with μ = 60, σ = 10, n = 16. In repeated sampling, about 100(1 − α)% of intervals built this way contain the parameter.

μ = 60
45505560657075
Current batch
—
Cumulative · target 95%
—
SE = σ/√n
2.50
Margin E = z·SE
4.90
Width 2E
9.80

Switching the level re-scores the same samples with wider or narrower nets. The more you draw, the closer the cumulative rate gets to 95%.

Eq. 1.1 vs. Eq. 1.2

Probability vs. Confidence

The most common CI misconception: mixing up the interval before the sample is drawn with the interval after.

Eq. 1.1 · probabilistic

P(Lθ < θ < Uθ) = 1 − α

Lθ and Uθ are random variables that change from sample to sample, so this is a genuine probability statement.

✔ Legitimate probability statement

Eq. 1.2 · confidence

C(60 < μ < 80) = 95%

Once numbers are plugged in, nothing is random. The interval either contains θ or it doesn't, so we say confident, not “probable”.

⚠ A confidence statement, not a probability

Quick check

You computed a 95% interval of [52.4, 67.6]. Is there a 95% probability that μ lies between 52.4 and 67.6?

Sensitivity analysis

More confidence = a wider net

LU
x̄

anchored at x̄ = 60, SE = 2.5

z*
1.960
Margin E
±4.90
Width 2E
9.80

↑ More confidence: fewer missed parameters, but a less precise estimate.

↓ Narrower net: sharper estimate, but the risk of missing μ goes up.

Up next

§1.4 Measurement Scales →