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Week 14 · Data Lab

Week 14 — Data Lab · "Put the Field Guide on Trial"

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objective: Objective 7 — one-sample t-test: conditions, mechanics, interpretation · SLO A (reason quantitatively from data)
Worth 50 points · Data labs group = 15% of the grade · Data Lab 14 · Due: end of Week 14
Format: a hands-on data lab — real research data, a spreadsheet, and your own randomly drawn sample; then you catch the AI's mistakes when it interprets your work.

This is the course's signature weekly component. Every instructional week has a Data Lab: real, free, linked public data — nothing to buy, download from a store, or sign up for — worked in your spreadsheet, finished with an AI-critique step. Plan on 60–90 minutes.

Part 1 — The Big Picture

This week you learned to put a claimed average on trial: hypotheses, standard errors, a t cutoff, a verdict. Today you run that trial on real animals. Suppose a field guide lists the typical adult Adélie penguin body mass as 3,700 grams. Your own random sample of 10 Adélie penguins gets to interrogate that number with a one-sample t-test — and because every classmate draws a different sample, the class becomes a live demonstration of the deepest fact about testing: even when a claim is essentially true, about 1 test in 20 will convict it anyway. You met that idea as Type I error in Week 13. Today you may personally commit one, honestly.

Then you'll aim the same reference number at the wrong species — Gentoo penguins — and watch the test do exactly what it was built to do.

The data: the same 344 penguins measured at Palmer Station, Antarctica (2007–2009) that this course has used since Week 1 — three species across three islands, released as the freely available palmerpenguins teaching dataset.

Part 2 — The Task

The guiding question: Does your random sample of penguins provide convincing evidence against the reference value of 3,700 g — for the species the number describes, and for a species it doesn't?

The dataset (both links are free, no login):
- About the data — palmerpenguins documentation
🔗 https://allisonhorst.github.io/palmerpenguins/
- The data itself (CSV file, opens as plain text)
🔗 https://raw.githubusercontent.com/allisonhorst/palmerpenguins/main/inst/extdata/penguins.csv

Part 3 — Materials & Procedure

You need: a web browser and a spreadsheet — Google Sheets (free) or Excel. Nothing else. Keep this week's t-table beside you: at df = 9, the two-sided 5% cutoff is t* = 2.262.

Warm-up (10 min, no dataset needed). A past student's random sample of 10 Adélie body masses (grams):

The table below lists one body mass per sampled penguin.

Penguin 1 2 3 4 5 6 7 8 9 10
Body mass (g) 3990 3570 3910 3650 3870 3690 3830 3730 3800 3760

By hand or calculator: (a) the sample mean (add the 10 values, divide by 10); (b) given that a spreadsheet reports s ≈ 126.5 g for these values, compute SE = s ⁄ √10 (√10 ≈ 3.162); (c) the t-statistic against the reference μ₀ = 3,700; (d) your verdict at α = 0.05, two-sided, df = 9. Write all four down — the key values are checked in your submission.

Now the real data:

  1. Import the dataset. Open the CSV link in your browser, select all (Ctrl/Cmd-A), copy, paste into cell A1 of a blank sheet, then Data ▸ Split text to columns if needed. (Excel: Data ▸ From Web, paste the CSV URL.) You should end with 8 columns plus a header row.
  2. Count your rows. In an empty cell, =COUNTA(A2:A400) — it should say 344. If not, re-import before continuing.
  3. Set up the trial. In a scratch cell, note the hypotheses you are testing for Adélie penguins: H₀: μ = 3700 vs. Hₐ: μ ≠ 3700 (two-sided — you're auditing the number, not gunning for a direction).
  4. Isolate the Adélie penguins. Sort the sheet by the species column (Data ▸ Sort range), or filter species = Adelie (the CSV spells it without the accent). The Adélie rows form one block.
  5. Draw YOUR random sample of 10. In the first empty column beside the Adélie block only, type =RAND() in the top cell and fill down the block. Select the Adélie block (all its columns including the RAND column) → Data ▸ Sort range by the RAND column. Your sample is the top 10 Adélie rows with a numeric body_mass_g — if one of the top rows shows NA for body mass, skip it and take the next row down, so you have 10 numbers.
  6. Copy your 10 masses to a scratch area (say H2:H11), then compute, each in its own labeled cell:
    - =AVERAGE(H2:H11) → your x̄
    - =STDEV(H2:H11) → your s
    - =STDEV(H2:H11)/SQRT(10) → your SE
    - =(AVERAGE(H2:H11)-3700)/(STDEV(H2:H11)/SQRT(10)) → your t
  7. Reach your verdict. Compare |t| to 2.262 (df = 9, two-sided, α = 0.05) and record reject / fail to reject.
  8. Perspective check. Compute the full-data Adélie mean: =AVERAGEIF(A2:A345, "Adelie", F2:F345) (adjust letters if your columns differ). How close is the reference 3,700 to it?
  9. Now the wrong species. Repeat steps 5–7 for the Gentoo block: a fresh =RAND() sample of 10 numeric Gentoo body masses, same hypotheses (μ₀ = 3,700), same cutoff 2.262. Record x̄, s, SE, t, verdict.

Fallback (no import available): if you can't get the CSV into a spreadsheet, use the 10-value warm-up table as your Adélie sample (its full test is the warm-up), and for the Gentoo part use this fallback sample a past student drew: 5000, 5550, 4750, 5200, 4900, 5400, 4650, 5700, 5100, 5250 (mean 5,150 g). Compute its mean by hand, take s ≈ 339 g as given, and run the test of μ₀ = 3,700. State in your submission that you used the fallback.

Part 4 — The Scaffold (complete both tables)

Table 1 captures your Adélie trial (and the warm-up you checked by hand).

Quantity Warm-up sample YOUR Adélie sample
Sample mean x̄ (g) ______ ______
Sample SD s (g) ≈126.5 (given) ______
SE = s ⁄ √10 ______ ______
t = (x̄ − 3700) ⁄ SE ______ ______
Verdict vs. t* = 2.262 ______ ______

Table 2 captures the perspective check and the Gentoo trial.

Quantity Value
Rows of data (from =COUNTA) ______
Full-data Adélie mean (from =AVERAGEIF) ______
YOUR Gentoo sample mean x̄ (g) ______
Gentoo SE and t ______
Gentoo verdict vs. t* = 2.262 ______

Part 5 — Analysis Questions

Answer in 1–3 sentences each:

  1. State the hypotheses you tested for the Adélie penguins, and explain in plain words why the test is two-sided here.
  2. Write your Adélie verdict as one careful sentence in context — and then state one thing your verdict does not mean. (If you failed to reject: did you prove the mean is 3,700? If you rejected: is the reference necessarily wrong?)
  3. The full Adélie data sit very close to the reference, so most correctly-run class samples should fail to reject. If your whole class runs this test honestly at α = 0.05, about what fraction of classmates do you expect to reject anyway — and what is that phenomenon called? (Week 13 named it.)
  4. Your Gentoo test almost certainly rejected, loudly. Looking at t = (x̄ − 3700) ⁄ SE, explain which part of the formula exploded and why. (Next week we stop comparing a group to a fixed number and compare two groups to each other — say in one line why that's the more natural question for Adélie vs. Gentoo.)
  5. Error sources: name two real limitations of your Adélie trial — consider who these 344 penguins are a sample of (three islands, 2007–2009), the NA cells you skipped, and your n of 10 — and one concrete improvement.

Part 6 — AI-Critique Moment (required — the BYOAI step)

Now bring in your chatbot and be the analyst who checks its work.

  1. Paste your 10 Adélie masses and this request: "Run a one-sample t-test of whether the mean body mass is 3700 g, at the 5% level, two-sided. Show the standard error, the t-statistic, the degrees of freedom, the critical value, and the verdict."
  2. Check everything it says against your own spreadsheet run:
    - Did it divide by s instead of s ⁄ √10? (The classic — its t will be √10 ≈ 3.16 times too small.)
    - Did it use 1.96 (the z cutoff) instead of 2.262 (t, df = 9)? With a borderline sample, that flips verdicts.
    - Did it recompute s with the population formula (dividing by n instead of n − 1), quietly shifting your numbers?
    - If it failed to reject, did it then claim the mean "is confirmed to be 3700"? (That's the week's #1 trap — you know better.)
  3. Write 2–3 sentences reporting what the AI got right and at least one thing you had to correct, question, or verify. If it happened to get everything right, say how you verified each claim — that's the skill.

The habit all term: the tool drafts, you judge. This week the stakes are a verdict, not just a number — catching a wrong cutoff is catching a wrong conclusion.

Part 7 — What to Submit

One document or text entry containing: your warm-up answers (a)–(d), your completed Table 1 and Table 2, your Part 5 answers (1–5), and your Part 6 AI-critique paragraph. Due at the end of Week 14 · 50 points (rubric below the key).

Instructor answer key & model values — REMOVE BEFORE PUBLISHING TO STUDENTS

Students draw their own random samples, so sample means, t-statistics, and verdicts are graded on method and interpretation, not on matching one number — recompute each student's SE and t from their own reported x̄ and s (that arithmetic is what's graded). Everything asserted below is pre-computed and verified by the Week 14 math script against the linked source.

Warm-up (fixed — every student should match): sum = 37,800 → x̄ = 3,780 g exactly. With s = √16,000 ≈ 126.49 (the sheet's =STDEV value; the "≈126.5" given in the lab), SE = 126.49 ⁄ √10 = 40.0 g exactly (√(16,000 ⁄ 10) = √1,600 = 40). t = (3,780 − 3,700) ⁄ 40 = 2.0. |2.0| < 2.262 → fail to reject at α = 0.05, two-sided (technology gives p ≈ 0.077). Full credit requires the verdict sentence, not just "fail."

Dataset facts (verified against the linked documentation and CSV this session): 8 columns with headers exactly species, island, bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g, sex, year; 344 data rows (the documentation page states "data for 344 penguins"); first data row is an Adélie from Torgersen (mass 3750 g, male, 2007); species row counts Adelie 152 · Gentoo 124 · Chinstrap 68; body_mass_g has 2 NA cells (one Adélie, one Gentoo), leaving numeric masses Adélie 151 · Gentoo 123 · Chinstrap 68 (342 total).

Model values (script-verified from the full data):
- Full-data Adélie mean body mass = 3,700.66 g (SD 458.57, n = 151) — the reference 3,700 is essentially true for Adélie, which is the pedagogical point of Part 5 Q3.
- A typical Adélie n = 10 sample: SE ≈ 458.57 ⁄ √10 ≈ 145 g, so typical |t| lands well under 1; expect roughly 95% of correctly-run Adélie tests to fail to reject, and about 1 student in 20 to reject honestly (Type I error, live).
- Full-data Gentoo mean = 5,076.02 g (SD 504.12, n = 123). A typical Gentoo n = 10 sample: SE ≈ 504.12 ⁄ √10 ≈ 159 g, typical t ≈ (5,076 − 3,700) ⁄ 159 ≈ +8.6 — student values commonly land roughly between +4 and +15, and every correctly drawn Gentoo sample rejects decisively. A Gentoo "fail to reject" means an arithmetic or sampling error — regrade the method.
- Fallback Gentoo sample (fixed): sum = 51,500 → mean 5,150 g; its actual =STDEV is ≈ 339.1 (script-verified: Σ(x − x̄)² = 1,035,000, s = √115,000), so SE ≈ 107.2 and t ≈ +13.5 → reject.

Part 5 expected answers:
1. H₀: μ = 3700 vs. Hₐ: μ ≠ 3700, where μ is the mean body mass of the population of adult Adélie penguins these data represent; two-sided because the reference is being audited in either direction — too heavy is as wrong as too light.
2. A fail-to-reject verdict worded as "no convincing evidence the mean differs from 3,700 g" — plus the non-meaning: it does not prove μ = 3700 (nearby values like 3,600 or 3,800 would also survive). A reject verdict worded with "convincing evidence" — plus: it does not guarantee the reference is wrong (could be the ~5% false alarm, or a sampling quirk).
3. About 5% — roughly 1 in 20 — reject; that's Type I error (α is exactly the false-alarm rate the class agreed to). Full credit needs the α ↔ class-fraction link, not just the phrase.
4. The numerator exploded: x̄ − 3700 is ≈ +1,376 g for Gentoos while SE stays ≈ 159, so t lands near +9. The natural Week 15 framing: compare Adélie and Gentoo means to each other (two-sample), rather than each to a fixed number.
5. Any two of: the 344 penguins are three islands' study animals from 2007–2009 (scope — not "all Adélie penguins ever"); NA masses were skipped (tiny nonresponse echo); n = 10 is small, so one heavy penguin moves everything (conditions!). Improvements: larger n, fresh/multi-site data, checking the sample's dot plot for outliers before testing — any concrete, sensible fix earns credit.

AI-critique full credit: names a specific claim checked or corrected — most commonly the chatbot dividing by s instead of s ⁄ √10, citing 1.96 instead of 2.262, silently using the population-SD formula, or "confirming" the null after a fail-to-reject — OR documents verification of each claim (recomputed SE, t, cutoff). Zero credit for "the AI agreed with everything" with no evidence of checking.

Grading rubric — 50 points

The rubric below maps each part to its points; bands are full / partial / none.

Criterion Full Partial None
Warm-up trial — x̄ = 3,780; SE = 40; t = 2.0; correct fail-to-reject verdict at 2.262 (6) 6 3–4 0–2
Import + Adélie SRS — 344 rows confirmed; species isolated; =RAND() sort done as specified; 10 numeric masses (8) 8 4–6 0–3
Adélie test executed — hypotheses stated; SE, t, df, cutoff all correct from their own sample values; verdict follows from their t (12) 12 6–10 0–5
Gentoo contrast test — same machinery re-run correctly; decisive rejection reached and stated in context (8) 8 4–6 0–3
Analysis questions — Q1–Q5 hit the expected ideas, especially the ~5% Type I answer in Q3 and the exploded-numerator reasoning in Q4 (12) 12 6–10 0–5
AI-critique — specific catch or verification reported in 2–3 sentences (4) 4 2–3 0–1

Quality gate (self-checked): warm-up arithmetic re-verified (sum 37,800; mean 3,780; Σ(x − x̄)² = 144,000; SE exactly 40; t exactly 2.0); dataset links live and free (documentation page + raw CSV both load without login, fetched this session); column names, 344-row count, first data row, and species counts verified; full-data species means/SDs and the fallback Gentoo values recomputed in w14_math.py; no numeric claim in the key depends on an unverified dataset-wide value; no graded-item leakage (no quiz/assignment/chapter item uses the penguins data); rubric sums 6 + 8 + 12 + 8 + 12 + 4 = 50. ✓