Homework — Do Dorm Students Average 10 Minutes to to Classroom #288?

Context & Research Question (Realistic)

The university states that dorm students average μ₀ = 10 minutes one-way to class. You surveyed n = 50 dorm students. Use a normal-based test (historical σ = 3.0 minutes).

Data (50 one-way commute times, minutes)

Copy into Excel (one per line → B2:B51)

Suggested Excel cells
• Put μ0 = 10 in E2 • Put σ = 3.0 in E3 • Put α = 0.05 in E4 • Optionally put μ1 in E5 when computing β/Power
Why choose μ₁?

Pick a practically meaningful shift δ = |μ₁ − μ₀|. For operations, +1 minute (μ₁=11) is small but meaningful; +2 minutes (μ₁=12) is clearly material.

What to Deliver (Student Tasks)

  1. Compute sample mean and st.dev. (x̄, s) from the 50 values.
  2. Test H₀: μ=10 vs H₁: μ≠10 at α=0.05 using a z-test with σ=3.0. Report z, p-value, decision, and a short interpretation in context.
  3. Compute a 95% CI for μ using z with σ = 3.0 and comment on whether μ₀ lies inside.
  4. Error rates. State Type I (α). For a meaningful shift: (a) take μ₁=11 → compute β and Power, (b) repeat for μ₁=12. Interpret.
  5. Sample size planning. Find n for 80% power (two-sided, α=0.05) to detect δ=1 minute. Round up.
  6. One-paragraph report (2–4 sentences) summarizing decision, CI, and power for μ₁=11.

Excel Cheat-Sheet (normal z-methods)

/* Put data in B2:B51; set μ0 in E2, σ in E3, α in E4; optionally μ1 in E5 */

Mean (xbar)      = AVERAGE(B2:B51)
Stdev (s)        = STDEV.S(B2:B51)

SE (z-method)    = E3/SQRT(COUNT(B2:B51))
z* (two-sided)   = NORM.S.INV(1 - E4/2)

Test statistic z = ( xbar - E2 ) / SE

Two-sided p-value:
= 2*(1 - NORM.S.DIST( ABS(z), TRUE ))

95% CI on μ (z, known σ):
Lower = xbar - z* * SE
Upper = xbar + z* * SE

Critical bounds on x̄ (for β/Power):
L = E2 - z* * SE
U = E2 + z* * SE

β at μ1 (two-sided, z-approx):
= NORM.DIST(U, E5, SE, TRUE) - NORM.DIST(L, E5, SE, TRUE)

Power:
= 1 - β

Sample size for power (two-sided):
/* Detect δ = |μ1 - μ0| with power pow at α */
= ROUNDUP( ((NORM.S.INV(1 - E4/2) + NORM.S.INV(pow)) * E3 / δ)^2 , 0 )
Show instructor solution (No Excel Functions)
Using the provided dataset (n=50), historical σ=3.0, α=0.05 (two-sided):
  • x̄ = 10.934, s = 2.826
  • SE = σ/√n = 3/√50 = 0.4243
  • z = (10.934 − 10) / 0.4243 = 2.200
  • Two-sided p-value = 0.0277
  • Decision: p < 0.05 → Reject H₀. There is evidence the true mean commute differs from 10 minutes (sample mean is slightly higher).
  • 95% CI for μ: x̄ ± z*SE = 10.934 ± 1.96·0.4243 → [10.102, 11.766] (10 is outside).
  • Critical bounds on x̄ (for α=0.05): L,U = 10 ± 1.96·0.4243 → L=9.168, U=10.832.
  • β/Power (planning, two-sided):
    • μ₁=11 ⇒ β ≈ 0.346, Power ≈ 0.654
    • μ₁=12 ⇒ β ≈ 0.003, Power ≈ 0.997
  • Sample size for 80% power to detect δ=1 at α=0.05 (two-sided): n = ((z1−α/2 + zpower)·σ/δ)² = ((1.96 + 0.842)·3/1)² ≈ 70.64 → 71.
One-paragraph template:
We tested H₀: μ = 10 vs H₁: μ ≠ 10 at α = 0.05 using σ = 3.0 and n = 50. The sample mean was x̄ = 10.934 (SE = 0.4243), giving z = 2.200 and p = 0.0277, so we reject H₀. A 95% CI for μ is [10.102, 11.766], which excludes 10. For a +1 minute shift (μ₁ = 11), Power ≈ 65%; for +2 minutes (μ₁ = 12), Power ≈ 99.7%. To get 80% power to detect +1 minute, we would need n ≈ 71.
20 Quick Q&A (with answers)
  1. What does H₀: μ=10 mean? The average dorm commute time is exactly 10 minutes.
  2. Why two-sided? We care about any difference from 10 (shorter or longer), not just increases.
  3. What does α=0.05 mean? A 5% chance of falsely rejecting a true H₀ (Type I error).
  4. What are x̄ and s? From the 50 values, x̄ = 10.934, s = 2.826.
  5. Why z-test? We’re using a historical σ=3.0 and n=50 is reasonably large; z-approx is standard here.
  6. What is SE with known σ? SE = σ/√n = 3/√50 = 0.4243.
  7. Compute z and interpret sign. z = (10.934−10)/0.4243 = 2.200; positive ⇒ sample mean > 10.
  8. Compute p-value and decide. p = 0.0277 < 0.05 ⇒ Reject H₀.
  9. Interpret in context. Evidence that the true mean commute differs from 10 minutes (here, slightly higher).
  10. 95% CI and interpretation. [10.102, 11.766]; since 10 is not in the interval, it supports rejecting H₀.
  11. Define Type I error. Rejecting H₀ when the true mean really is 10 minutes.
  12. Define Type II error. Failing to reject H₀ when the true mean is not 10 (e.g., 11).
  13. Power for μ₁=11. β ≈ 0.346 ⇒ Power ≈ 0.654 (about 65% chance to detect a +1 minute shift).
  14. Power for μ₁=12. β ≈ 0.003 ⇒ Power ≈ 0.997 (very likely to detect +2 minutes).
  15. Trade-off: α vs Power. Lowering α raises the critical bar (larger z*), which usually reduces Power for fixed n, δ, σ.
  16. How does n affect Power? Larger n ↓ SE, ↓ β, ↑ Power. Small n does the opposite.
  17. n for 80% power (δ=1, α=0.05)? ≈ 71 students.
  18. What if α=0.01? The same z gives p=0.0277 > 0.01 ⇒ we would not reject H₀ at 1% (stricter test).
  19. If σ larger than 3? SE increases, making it harder to detect shifts ⇒ Power decreases (β increases).
  20. How to choose μ₁? Pick a practically meaningful shift (e.g., +1 minute), not one chosen to “game” Power.

Turn-in Checklist