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Week 7 · Data Lab

Week 7 — Data Lab · "Two Hundred Rounds of Four"

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objective: Objective 4 — the binomial model: setting, probabilities, mean & SD · SLO A (reason quantitatively from data)
Worth 50 points · Data labs group = 15% of the grade · Data Lab 7 · Due: end of Week 7
Format: a hands-on data lab — this week you generate the data yourself: a spreadsheet simulation of a 50-50 free-throw shooter, run 200 times, set against the binomial model's exact predictions; then you catch the AI's mistakes when it interprets your results.

This is the course's signature weekly component. Every instructional week has a Data Lab: real, free, linked public data — or, on probability weeks like this one, data you generate live with the spreadsheet's own random engine — 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's chapter handed you a bold claim: if a situation passes the B·I·N·S checklist, then a formula — not an experiment — tells you exactly how often each count of successes occurs. That should feel slightly suspicious. A formula, predicting the future?

Today you put the claim on trial. Your shooter: a player who makes exactly 50% of her free throws, each shot independent — as pure a binomial machine as exists (n = 4 shots per round, p = 0.5). The theory says a 4-shot round ends in 2 makes with probability 0.375, in 0 makes with probability 0.0625, and so on. Your job: simulate 200 rounds of 4 shots in a spreadsheet, count what actually happens, and compare reality to the formula's forecast — then explain the gaps like a statistician instead of panicking about them. (Spoiler: the gaps are Week 1's old friend, sampling variability, and after the midterm they become the engine of the whole course.)

Part 2 — The Task

The guiding question: When 200 real (simulated) rounds of four 50-50 shots are tallied, how closely do the observed counts, the observed mean, and the observed SD track the binomial model's predictions — and what explains the daylight between them?

Before you build anything, write a one-line hypothesis: "If the binomial model is right, then in 200 rounds I expect the counts of 0/1/2/3/4 makes to land near ___ / ___ / ___ / ___ / ___." (You'll compute the exact expected counts in Step 2 — write your gut guesses first. If… then… because thinking starts now.)

No dataset link this week — you are the data source. Everything happens in a blank spreadsheet: Google Sheets (free) or Excel. The random engine is the spreadsheet's own =RANDBETWEEN().

Part 3 — Materials & Procedure

You need: a spreadsheet — Google Sheets (free) or Excel. Nothing else.

Warm-up (5 min, no spreadsheet needed). Before simulating, compute the model's centerpiece by hand and write it down: P(exactly 2 makes in 4 shots at p = 0.5) = C(4, 2) × 0.5² × 0.5² = 6 × 0.0625 = 0.375. Also note why the 6: with 4 shots there are 2⁴ = 16 equally likely make/miss sequences, and exactly 6 of them contain two makes — so 6/16 = 0.375. You'll check the machine against this number in Step 2.

Now build, step by step:

  1. Set up headers. Open a blank sheet. In row 1, type: Shot 1 (A1), Shot 2 (B1), Shot 3 (C1), Shot 4 (D1), Makes (E1). Then, off to the right: k (H1), P(X = k) (I1), Expected of 200 (J1), My count (K1).
  2. Build the theory table first. In H2:H6 enter 0, 1, 2, 3, 4. In I2 type =BINOM.DIST(H2, 4, 0.5, FALSE) and fill down to I6 — you should see 0.0625, 0.25, 0.375, 0.25, 0.0625 (your warm-up 0.375 sitting in the middle). In J2 type =I2*200 and fill down: 12.5, 50, 75, 50, 12.5. Sanity checks: =SUM(I2:I6) must equal 1, and =SUM(J2:J6) must equal 200. If not, re-check the formula arguments before continuing.
  3. Simulate the shots. In A2 type =RANDBETWEEN(0,1) — a fresh 50-50 shot, where 1 = make, 0 = miss. Fill A2 across to D2, then fill A2:D2 down to row 201. You now have 200 rounds × 4 shots.
  4. Count makes per round. In E2 type =SUM(A2:D2) and fill down to E201 — each round's make-count, 0 through 4.
  5. Tally your 200 rounds. In K2 type =COUNTIF($E$2:$E$201, H2) and fill down to K6. Check =SUM(K2:K6) equals 200 (every round counted exactly once).
  6. Compute your simulation's mean and SD. In two labeled cells: =AVERAGE(E2:E201) and =STDEV(E2:E201). The model's targets: mean np = 4 × 0.5 = 2 and SD √(np(1−p)) = √1 = 1 exactly.
  7. Watch it wobble (the point of the lab). Type anything into an empty far-away cell and delete it — every =RANDBETWEEN recalculates and your 200 rounds are replayed. Do this two or three times and watch column K and your mean/SD shimmer around the theory values without ever matching them exactly.
  8. Optional but recommended: select H1:H6 and K1:K6 and insert a column chart — your simulated histogram, which you can eyeball against the theory column.

Fallback (no spreadsheet available): flip any coin 4 times per round (heads = make) for 20 rounds and tally your counts of 0–4 makes by hand against expected counts of 1.25 / 5 / 7.5 / 5 / 1.25 (that's 20 × each probability) — OR analyze this pre-run model simulation of 200 rounds: counts 14 / 47 / 78 / 49 / 12 for k = 0/1/2/3/4 (mean 1.99, SD ≈ 1.00). State in your submission that you used the fallback.

Part 4 — The Scaffold (complete both tables)

Table 1 compares the binomial model's predictions to your simulation, one row per possible make-count.

k (makes in a round) Ways (out of 16 sequences) P(X = k) from =BINOM.DIST Expected count (of 200) Your simulated count Difference (yours − expected)
0 1 ______ ______ ______ ______
1 4 ______ ______ ______ ______
2 6 ______ ______ ______ ______
3 4 ______ ______ ______ ______
4 1 ______ ______ ______ ______

Table 2 captures the summary comparison.

Quantity Model (theory) Your simulation
Mean makes per round np = 2 ______
SD of makes per round √(np(1−p)) = 1 ______
Total rounds 200 ______

Part 5 — Analysis Questions

Answer in 1–3 sentences each:

  1. Which make-count k occurred most often in your 200 rounds, and does the ranking of your counts (most common to least) match the theory column's ranking? Note any place it doesn't.
  2. Your simulated mean isn't exactly 2 (almost certainly). Is that an error in your spreadsheet? Name the Week-1 phenomenon at work, and say what would happen to the gap if you ran 2,000 rounds instead of 200.
  3. Exactly 6 of the 16 equally likely make/miss sequences produce 2 makes. Use that fact to explain — in plain words — why 2 makes beats 0 makes so decisively in your tally, even though any single sequence is as likely as any other.
  4. Model vs. reality: your simulated shooter is a perfect binomial machine, but a real shooter might not be. Name one B·I·N·S assumption a real player could violate (and how), and one limitation of using =RANDBETWEEN as your source of randomness.
  5. Look at the shape of the theory column (and your chart if you made one): symmetric, peaked in the middle. Why is it symmetric for this shooter — and based on Segment 6 of this week's lecture, what happens to the shape of binomial histograms as the number of trials grows large? (One sentence of preview: what's coming next week?)

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 completed Table 1 and Table 2 into your chatbot and ask: "Check my theoretical probabilities and expected counts. Then tell me: do the differences between my simulated counts and the expected counts mean I made a mistake?"
  2. Check everything it says against your own work:
    - Did it recompute the probabilities correctly? (Chatbots often drop the ways factor — reporting P(2 makes) as 0.5⁴ = 0.0625 instead of 6 × 0.0625 = 0.375 — or quietly switch to the cumulative P(X ≤ 2) = 0.6875 and present it as P(X = 2).)
    - Did it get the expected counts right? (Each is just probability × 200 — redo one yourself.)
    - Did it explain your simulation-vs-theory gaps as sampling variability — or did it call your honest wobble an "error" that needs fixing?
  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. You now hold three weapons that must agree — the formula, =BINOM.DIST, and your own 200 rounds — and a chatbot that contradicts all three is wrong, however confident it sounds.

Part 7 — What to Submit

One document or text entry containing: your hypothesis line (Part 2), your warm-up computation, 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 7 · 50 points (rubric below the key).

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

Students generate their own random rounds, so simulated counts, means, and SDs are graded on method and interpretation, not on matching one number. Every theoretical value asserted below is pre-computed and independently re-verified by script.

Warm-up (fixed — every student should match): C(4, 2) = 6; 0.5⁴ = 0.0625; P(X = 2) = 6 × 0.0625 = 0.375 ✓. The "why 6": 2⁴ = 16 equally likely sequences, 6 contain exactly two makes.

Theory table (fixed — every student should match):

k Ways of 16 P(X = k) Expected of 200
0 1 0.0625 12.5
1 4 0.25 50
2 6 0.375 75
3 4 0.25 50
4 1 0.0625 12.5

Probabilities sum to 1; expected counts sum to 200. (Equivalently: 1/16, 4/16, 6/16, 4/16, 1/16.) Cumulative check the AI-critique may surface: P(X ≤ 2) = 0.0625 + 0.25 + 0.375 = 0.6875 — cumulative, not P(X = 2). Model targets: mean = np = 2; SD = √(np(1−p)) = √1 = 1.

Simulated results (variable — grade the method): with 200 rounds, essentially all honest runs land inside these ±2-SD ranges — counts for k = 0: roughly 6–19 (SD √(200 × 0.0625 × 0.9375) ≈ 3.42); k = 1: roughly 38–62 (SD √37.5 ≈ 6.12); k = 2: roughly 61–89 (SD √46.875 ≈ 6.85); k = 3: roughly 38–62; k = 4: roughly 6–19. Simulated mean typically 1.86–2.14 (2 ± 2 × 1/√200); simulated SD typically about 0.90–1.10. A value modestly outside a range on one run is unremarkable (about 1 row in 20 will be); wildly outside every range means a formula error — check their =COUNTIF anchoring and that =SUM of counts is 200. Grade the analysis, not a specific count. The embedded fallback model run — 14 / 47 / 78 / 49 / 12 (sum 200, mean 1.99, SD ≈ 1.00) — is itself inside every range; students who used it should be graded on the same analysis questions.

Part 5 expected answers:
1. Most often: almost always k = 2 (expected 75 of 200); full ranking usually 2, then 1 and 3, then 0 and 4 — matching theory. Small inversions (e.g., 47 vs. 49 between k = 1 and k = 3, as in the model run) are normal and worth naming, not fixing.
2. Not an error — sampling variability: 200 honest rounds wobble around the model. With 2,000 rounds the proportions would hug the theory tighter (long-run relative frequency — the Week 5 idea); the mean would land even closer to 2.
3. Any answer connecting count-of-ways to probability: each single sequence has probability (1/2)⁴ = 1/16, but 6 sequences deliver "2 makes" while only 1 delivers "0 makes," so 2 makes happens ~6× as often. The ways factor is the reason the middle towers over the ends.
4. B·I·N·S violations for a real shooter: independence (confidence/rattle after makes or misses — the "hot hand" debate, this week's discussion) or same p (fatigue, pressure, defensive changes across a game). =RANDBETWEEN limitation: it's a pseudo-random algorithm (and every recalc replays it) — excellent for teaching, but not literally a coin. Any one violation + any reasonable limitation earns full credit.
5. Symmetric because p = 0.5 — makes and misses are interchangeable, so k and 4−k have equal probability. As n grows, binomial histograms smooth into a bell shape — next week that bell gets its name (the normal distribution) and takes over the course.

AI-critique full credit: names a specific claim checked or corrected — most commonly the chatbot dropping the ways factor (0.0625 for P(2)), presenting cumulative 0.6875 as "the probability of 2," botching an expected count, or calling the simulation gaps "errors" — OR documents how each claim was verified against the formula/=BINOM.DIST/their own tally. 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
Hypothesis + warm-up — if/then line written before simulating; warm-up lands on 0.375 with the 6-of-16 reasoning (5) 5 3 0–1
Theory table — =BINOM.DIST values 0.0625/0.25/0.375/0.25/0.0625, expected counts 12.5/50/75/50/12.5, ways column correct, sums verified (10) 10 5–8 0–4
Simulation build — 200 × 4 =RANDBETWEEN grid, Makes column, =COUNTIF tally summing to 200 (10) 10 5–8 0–4
Comparison + summary — differences computed correctly from their own counts; simulated mean and SD reported against 2 and 1 (10) 10 5–8 0–4
Analysis questions — Q1–Q5 hit the expected ideas, especially sampling variability (Q2) and the ways explanation (Q3) (10) 10 5–8 0–4
AI-critique — specific catch or verification reported in 2–3 sentences (5) 5 3 0–2

Quality gate (self-checked): all five theoretical probabilities re-verified from the formula (they sum to 1; expected counts sum to 200); mean np = 2 and SD √1 = 1 verified; the fallback model run's tally sums to 200 with mean 1.99 and SD ≈ 1.00, and sits inside every stated ±2-SD range; simulated results stated as ranges, never exact targets; no external link needed (self-generated data); no graded-item leakage (quiz, practice, assignment, and chapter self-checks all use different scenario surfaces and numbers); rubric sums 5 + 10 + 10 + 10 + 10 + 5 = 50. ✓