9.1.2 — Type I & Type II Errors - ICE (with the Student-Commute Story)

Story first — what are we trying to decide?

Your department claims the average one-way student commute time is μ₀ minutes. You collect a class sample and test that claim at significance level α. Two things can go wrong:

Video — How to Remember Type I vs Type II

If the embed is blocked, open on YouTube: https://www.youtube.com/watch?v=985KQG-8QV8

Step A — Use our class sample to estimate σ (or type your own)

From sample (click “Use this sample…”):
n = —, x̄ = —, s = — (used as σ)
Or type your own σ directly:

Step B — Choose the claim, the risk (α), the tail, and a “true” μ₁ to check β

Excel mirror — critical region (L,U), β, and Power (z-approx)

Step C — Numbers, rules, and plain-English explanation

SE = σ/√n
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Critical region on ȳ
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Risk & power
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Step D — Picture (α tails under H₀, β inside acceptance under H₁)

α under H₀ β under H₁ Dashed = bounds L, U
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Step E — Decision vs. Truth (connect to commute)

H₀ true (μ = μ₀)H₀ false (μ = μ₁)
Fail to reject H₀✔️ No error (≈ 1−α)Type II β
Reject H₀Type I α✔️ Correct (Power = 1−β)

Copy-paste script for reports

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Worked Example (Beautiful Math): β and Power from a Two-Sided z-Test

Step-by-Step (with your numbers)

Click Compute to render the math.
Excel mirrors (stepwise, no LET): copy these lines 
B1 = μ0 (e.g., 30)   B2 = σ (e.g., 6.10)   B3 = n (e.g., 23)   B4 = α (e.g., 0.05)
B5 = B2/SQRT(B3)                               ← SE
B6 = NORM.S.INV(1-B4/2)                        ← z*
B7 = B1 - B6*B5                                ← L
B8 = B1 + B6*B5                                ← U
B9 = μ1 (e.g., 33.5)
B10 = NORM.DIST(B8,B9,B5,TRUE) - NORM.DIST(B7,B9,B5,TRUE)  ← β
B11 = 1 - B10                                               ← Power

Textbook Pack — 9.1.2 (ALL worked examples, α, β, Power)

Setup used below: σ = 2.5 (cm/s). For n = 10, SE = 2.5/√10 ≈ 0.79. “Original” critical region: L = 48.5, U = 51.5 for testing \(H_0:\mu=50\) (two-sided).

Uncheck to use exact SE (0.7906)
Rendering…
Excel mirrors (stepwise, no LET): copy these lines 
B1 = μ0 (50)   B2 = σ (2.5)   B3 = n (10 or 16)
B5 = B2/SQRT(B3)                               ← SE
B7 = 48.5;  B8 = 51.5                          ← L, U (given)
B9 = μ1 (e.g., 52 or 50.5)
B10 = NORM.DIST(B8,B9,B5,TRUE) - NORM.DIST(B7,B9,B5,TRUE)  ← β
B11 = 1 - B10                                               ← Power

Effect Size Explorer — β and Power vs μ₁ (Why pick 52? Try any μ₁.)

β and Power are defined at a particular true mean μ₁. Choose what difference (δ = |μ₁−μ₀|) is meaningful for your problem. Bigger |δ| ⇒ smaller β ⇒ larger Power. Use the controls to explore; defaults match the textbook.

SE
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Critical region on ȳ
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β and Power at μ₁
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Sample β at common μ₁ choices (two-sided, textbook n=10):
μ₁βPower
Tip: If your “meaningful” shift is δ*, plan n for target power: =ROUNDUP(((NORM.S.INV(1-α/2)+NORM.S.INV(power))*σ/δ*)^2,0)

Practice Q&A — 20 quick checks (click to reveal)

1) What does α mean?
Chosen false-alarm rate when H₀ is true. Example: α=0.05 ⇒ 5% chance to reject a true H₀.
2) What does β mean?
Miss rate when H₀ is false at a specific μ₁. Power = 1−β.
3) Why does β depend on μ₁?
Because the acceptance region [L,U] is fixed by μ₀, α, n, σ; how much of the μ₁-distribution lies inside it changes with μ₁.
4) If we increase n, what happens to SE and β?
SE ↓ ⇒ acceptance band narrower in z-units ⇒ β ↓ ⇒ Power ↑.
5) If σ is larger, what happens to Power?
SE ↑, so harder to detect a shift ⇒ Power ↓ (β ↑) for the same μ₁.
6) Two-sided vs one-sided: which has higher Power for the same shift direction?
One-sided (when the direction is correct) because all α is in one tail.
7) Is α the probability H₀ is true?
No. α is the long-run false-alarm rate under H₀, not a posterior probability.
8) Power at μ₁ very close to μ₀?
Low. Small effects are hard to detect unless n is large.
9) Why did the book use μ₁=52?
Just an illustrative +2 shift from 50. Any μ₁ can be used; β/Power will change.
10) Is β symmetric for μ₁=50+δ and μ₁=50−δ in a two-sided test?
Yes, because the bounds are symmetric around μ₀.
11) What target Power is common in planning?
80% or 90% are typical choices.
12) How do I get L and U for two-sided α?
L=μ₀−z*SE, U=μ₀+z*SE with z*=z₁₋α/₂ and SE=σ/√n.
13) What does “fail to reject” mean?
Evidence is insufficient at level α; it does NOT prove H₀ is true.
14) If α is made smaller (e.g., 0.01), what happens to Power (holding n, σ, μ₁)?
Critical region shrinks ⇒ β ↑ ⇒ Power ↓.
15) Do CIs relate to α tests?
Yes. For two-sided: reject H₀ iff μ₀ is outside the (1−α) CI.
16) Does rounding SE change answers?
Slightly. Using SE=0.79 vs 0.7906 makes small differences; the app lets you match the book.
17) If you care about δ*=1, which is better: n=10 or n≈50?
n≈50; it gives ~80% power for σ≈2.5 at α=0.05 to detect a 1-unit shift.
18) Can Power be > 95% for small δ with n=10?
No. With small n and small δ, power stays low. Increase n.
19) When should I prefer a one-sided test?
When only one direction is scientifically meaningful and you would not act on evidence in the opposite direction.
20) Does a significant result always mean a large effect?
No. With very large n, even tiny effects can be significant. Always report effect size and CI.