📚 Chapter 8 — Homework (z, t, χ²)

Three problems using the same student-life context (commute times). Show your work, formulas, and a one-sentence decision in words.

HW 8.Az-interval (σ known)Mean μ

Commute Time — Is the average ≤ 30 minutes? (σ known)

Context. A prior city study established the population SD of one-way student commute times as σ = 8.0 minutes and the distribution is approximately Normal on your campus routes. This semester you sample n = 36 students and observe a sample mean of x̄ = 31.4 minutes. Use a 95% confidence level.

Tasks
  1. Compute the 95% z-interval for the true mean commute time μ.
  2. Report the SE, z-critical, and margin of error (ME).
  3. Decision vs 30: Based on the interval, can you claim μ ≤ 30? Explain in one sentence.
  4. Write a plain-English interpretation of the CI (what “95% confident” means here).
Excel steps (use α = 0.05)
  • Put B2=31.4 (x̄), B3=8 (σ), B4=36 (n), B5=0.05 (α).
  • B6 z-critical → =NORM.S.INV(1-$B$5/2)
  • B7 SE → =$B$3/SQRT($B$4)
  • B8 ME → =$B$6*$B$7
  • B9 Lower → =$B$2-$B$8   B10 Upper → =$B$2+$B$8
Deliverables
  • SE, z*, ME, CI [Lower, Upper]
  • 1-sentence decision vs 30 (≤, >, or “inconclusive”).
  • 1-sentence interpretation of a 95% CI.
Answer check (FYI only) With x̄=31.4, σ=8, n=36, α=0.05 → z*≈1.96, SE=1.3333, ME≈2.613. CI ≈ [28.787, 34.013]. 30 is inside ⇒ inconclusive.
HW 8.Bt-interval (σ unknown)Mean μ

Commute Time — Small sample, σ unknown (use t)

Context. A club survey measures n = 10 students and finds x̄ = 31.4 minutes with s = 8.0 minutes. Assume the population is approximately Normal. Use a 95% confidence level.

Tasks
  1. Compute the 95% t-interval for μ (report df).
  2. Report the SE, t-critical and ME.
  3. Decision vs 30: Based on the interval, can you claim μ ≤ 30? Explain briefly.
  4. Comment: how does this CI compare in width to HW 8.A? Why?
Excel steps (use α = 0.05)
  • B2=31.4 (x̄), B3=8 (s), B4=10 (n), B5=0.05 (α).
  • B6 df → =$B$4-1
  • B7 t-critical → =T.INV(1-$B$5/2,$B$6)
  • B8 SE → =$B$3/SQRT($B$4)
  • B9 ME → =$B$7*$B$8
  • B10 Lower → =$B$2-$B$9   B11 Upper → =$B$2+$B$9
Deliverables
  • df, SE, t*, ME, CI [Lower, Upper]
  • 1-sentence decision vs 30.
  • One sentence comparing width to HW 8.A (reason: heavier tails/small n).
Answer check (FYI only) df=9, t*≈2.262, SE≈2.530, ME≈5.72. CI ≈ [25.677, 37.123]. 30 is inside ⇒ inconclusive. Wider than z-CI because σ unknown and small n ⇒ use t with heavier tails.
HW 8.Cχ²-interval (variance / SD)Spread σ or σ²

Commute Variability — Is the process stable (σ ≤ 6 min)?

Context. For the same n = 10 students, the sample standard deviation is s = 8.0 minutes. Assume approximate Normality. Use a 95% confidence level. Operations defines “stable” as population SD σ ≤ 6 minutes.

Tasks
  1. Compute the 95% χ²-interval for the variance σ² and then for the SD σ (report df and both χ² cutoffs you used).
  2. Decision vs σ ≤ 6: Based on the σ-CI, can you claim the process meets σ ≤ 6?
  3. Write one caution about the Normality assumption for very small n.
Excel steps (left-tail χ² quantiles; α = 0.05)
  • B2=8 (s), B3=10 (n), B4=0.05 (α).
  • B5 df → =$B$3-1
  • B6 χ²α/2,df → =CHISQ.INV($B$4/2,$B$5)
  • B7 χ²1−α/2,df → =CHISQ.INV(1-$B$4/2,$B$5)
  • B8 s² → =$B$2^2
  • B9 Lower var → =$B$5*$B$8/$B$7
  • B10 Upper var → =$B$5*$B$8/$B$6
  • B11 Lower SD → =SQRT($B$9)   B12 Upper SD → =SQRT($B$10)
Deliverables
  • df, χ² cutoffs, CI for σ², CI for σ.
  • Decision vs σ ≤ 6 (supported / not supported / inconclusive).
  • One-sentence caution re: Normality at small n.
Answer check (FYI only) df=9; χ²0.025,9≈2.700; χ²0.975,9≈19.02. Var CI ≈ [30.3, 213.3] ; SD CI ≈ [5.503, 14.605]. Since upper σ bound > 6 and 6 lies inside → cannot claim σ ≤ 6.
Common pitfalls to avoid

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