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Week 6 · Data Lab

Week 6 — Data Lab · "Run the Raffle: What a Ticket Is Really Worth"

Introduction to Statistics Generic evergreen edition

Course: Introduction to Statistics (18-week generic edition)
Objective: Objective 4 — random variables; expected value; SD of a random variable · SLO A (reason quantitatively from data)
Worth 50 points · Data labs group = 15% of the grade · Data Lab 6 · Due: end of Week 6
Format: a hands-on data lab — this week you generate the data yourself: you'll build a raffle's probability distribution in your spreadsheet, compute what a ticket is worth by formula, then simulate 1,000 ticket draws and watch the long-run average find your answer; 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 — or, on simulation weeks like this one, data your own spreadsheet generates live — nothing to buy, download from a store, or sign up for. Plan on 60–90 minutes.

Part 1 — The Big Picture

This week you learned that an uncertain number — a payout, a count — has a worth (the expected value) and a wobble (the standard deviation), both computed from its probability distribution. Today you'll test that machinery against brute force: first compute E(X) and SD(X) exactly, by formula, then make your spreadsheet actually draw 1,000 raffle tickets and check whether the simulated average agrees with the theory. It will — almost, and the "almost" is the week's deepest lesson: chance is wild in the short run and dependable in the long run.

The setup: a community club runs a fundraiser raffle. Exactly 100 tickets are sold at $5 each. The printed prize list: 1 ticket wins $100, 4 tickets win $25, 15 tickets win $5, and the other 80 win nothing. X = the payout of one randomly drawn ticket. Everything in this lab flows from that one sentence.

Part 2 — The Task

The guiding question: What is one $5 raffle ticket actually worth — and when your spreadsheet draws 1,000 tickets, how close does the simulated average land to the formula's answer?

The data: generated live in your own spreadsheet — the club's prize list above is the complete "dataset," and =RANDBETWEEN() will play the role of the raffle drum. There is nothing to download and no login; you need only a browser and Google Sheets (free) or Excel.

Part 3 — Materials & Procedure

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

Warm-up (3 min, no spreadsheet needed). Before building anything, answer on paper: out of 100 tickets, how many win something? (1 + 4 + 15 = 20 — so P(win anything) = 0.20.) Keep that number; you'll check your simulation against it later.

Stage 1 — build the distribution and compute the theory (steps 1–5):

  1. Open a blank sheet. In A1:D1 type headers: x (payout), P(x), x*P(x), (x-2.75)^2*P(x). In A2:A5 enter the payouts: 0, 5, 25, 100. In B2:B5 enter their probabilities: 0.80, 0.15, 0.04, 0.01 (they come straight from the prize counts: 80/100, 15/100, 4/100, 1/100).
  2. Legitimacy check first, always. In an empty cell type =SUM(B2:B5) — it must display exactly 1. If it doesn't, fix a typo before going on; nothing downstream can be trusted until this shows 1.
  3. Expected value. In C2 type =A2*B2 and fill down to C5. In an empty cell (label it "E(X)") type =SUM(C2:C5) — you should see exactly 2.75. One-cell shortcut that must agree: =SUMPRODUCT(A2:A5,B2:B5).
  4. Variance and SD. In D2 type =(A2-2.75)^2*B2 and fill down to D5. In an empty cell (label it "Var(X)") type =SUM(D2:D5) — you should see 121.1875. Beside it (label "SD(X)") type =SQRT() of that cell — about 11.01.
  5. Pause and read your own numbers: a $5 ticket returns $2.75 on average, with a standard deviation of about $11 — record both in Table 2 before simulating.

Stage 2 — simulate 1,000 ticket draws (steps 6–9):

  1. In F1 type the header drum (1-100); in G1 type payout. In F2 type =RANDBETWEEN(1,100) and fill down to F1001 (that's 1,000 draws — the fastest fill: select F2, copy, then select F2:F1001 and paste).
  2. In G2 type the prize map: =IF(F2=1,100,IF(F2<=5,25,IF(F2<=20,5,0))) and fill down to G1001. Read the logic out loud once — drum number 1 is the $100 ticket; 2–5 are the four $25 tickets; 6–20 are the fifteen $5 tickets; 21–100 win nothing. The mapping must mirror the prize counts exactly.
  3. Summaries. In labeled cells: simulated mean =AVERAGE(G2:G1001) · share of $0 draws =COUNTIF(G2:G1001,0)/1000 · biggest prize drawn =MAX(G2:G1001).
  4. Every edit re-rolls all 1,000 draws (that's RANDBETWEEN working). Watch the simulated mean wobble as you recalculate a few times — then freeze your final numbers for the write-up: select F2:G1001, copy, and paste-as-values (Sheets/Excel: right-click ▸ Paste special ▸ Values only). Record your frozen results in Table 2.

Fallback (no spreadsheet available): use the 40 pre-drawn payouts below as your "simulation." Compute their mean by hand (sum ÷ 40), count the $0 draws, and complete Table 2 with n = 40 instead of 1,000. State in your submission that you used the fallback.

The table below lists 40 pre-drawn ticket payouts, in draw order (read across).
0 0 5 0 0 25 0 0 0 5
0 0 0 0 5 0 0 0 0 0
5 0 0 0 0 0 25 0 0 0
0 5 0 0 0 0 0 0 5 0

Part 4 — The Scaffold (complete both tables)

Table 1 is your distribution build — copy your four computed columns into it (or paste a screenshot-free transcription).

x (payout) P(x) x·P(x) (x − 2.75)²·P(x)
$0 0.80 ______ ______
$5 0.15 ______ ______
$25 0.04 ______ ______
$100 0.01 ______ ______
Totals ______ ______ ______

Table 2 captures the theory-vs-simulation comparison.

Quantity Value
Legitimacy check: =SUM of P(x) column ______
E(X) — by formula (must be exact) ______
Var(X) — by formula ______
SD(X) — by formula ______
YOUR simulated mean payout (1,000 draws; or 40 if fallback) ______
Difference (simulated mean − E(X)) ______
YOUR share of $0 draws ______

Part 5 — Analysis Questions

Answer in 1–3 sentences each:

  1. A ticket costs $5.00 and E(X) = $2.75. Interpret both numbers for a ticket-buyer in long-run language, then explain why the club's side isn't really a gamble at all once all 100 tickets are sold. (Hint: with every ticket sold, the club takes in a fixed $500 and pays out a fixed total — what is it?)
  2. Your simulated mean didn't land exactly on 2.75 (almost certainly). Is that an error? Name the phenomenon, and connect it to Week 5's simulation lab, where the running proportion wobbled before settling at its long-run value.
  3. Error-source question. Suppose a classmate mistypes the prize map as IF(F2<=21,5,...) instead of <=20 — one extra drum number becomes a $5 winner. Which direction would their simulated mean drift, and why is this kind of error different from the ordinary run-to-run wobble? (One never shrinks as draws pile up; say which, and connect to Week 1's "method beats size.")
  4. The SD is about $11, yet the most common experience is winning $0 — and no prize is anywhere near "typical ± $11." What is inflating the SD, and what should a buyer actually take away from a big SD on a cheap raffle ticket? (Look at your Table 1: which single row dominates the (x − 2.75)²·P(x) column?)
  5. Connect to this week's transformation rules. The club debates doubling every prize (2X) while keeping the $5 ticket price. What do E(X) and SD(X) become, and should the treasurer approve? Show the one-line computations.

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 your Table 2 numbers into your chatbot and ask: "Check my expected-value and SD computations for this raffle distribution. Then tell me: is the difference between my simulated mean and 2.75 a mistake I should fix?"
  2. Check everything it says against your own sheet:
    - Did it compute E(X) by weighting each payout by its probability — or did it average the four payout values ((0 + 5 + 25 + 100) ÷ 4 = 32.50, wildly wrong)? Chatbots make exactly this slip.
    - Did it keep variance and SD straight (121.1875 is the variance; ≈ $11.01 is the SD), and did it distinguish the distribution's SD from the SD of your 1,000 simulated payouts (close, but not the same thing)?
    - Did it explain your simulated-mean gap as ordinary sampling variability that shrinks with more draws — or did it call your work an "error," or the raffle "fair because someone always wins"?
  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. A chatbot will confidently average away the probabilities or hand you a variance dressed up as an SD — catching it is the point.

Part 7 — What to Submit

One document or text entry containing: your warm-up count (tickets that win something), 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 6 · 50 points (rubric below the key).

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

The distribution-side numbers are fixed — every correct sheet shows the identical E(X), Var(X), and SD(X). The simulation-side numbers are random by design, so simulated means and counts are graded on method and interpretation against the stated ranges, never on matching one number. Every asserted value below is re-verified in this week's arithmetic script.

Warm-up: winning tickets = 1 + 4 + 15 = 20 of 100 → P(win anything) = 0.20.

Table 1 model values (fixed):

x P(x) x·P(x) (x − 2.75)²·P(x)
0 0.80 0.00 6.05
5 0.15 0.75 0.759375
25 0.04 1.00 19.8025
100 0.01 1.00 94.575625
Totals 1.00 2.75 121.1875

Table 2 fixed rows: legitimacy sum = 1 (exact) · E(X) = $2.75 (exact — SUM(C) and SUMPRODUCT must agree) · Var(X) = 121.1875 dollars² · SD(X) = √121.1875 = $11.01 (11.0085…; accept 11.0–11.01).

Table 2 random rows (grade on method + range): with 1,000 draws, the simulated mean has SE = 11.0085/√1000 ≈ 0.35, so about 95% of correct simulations land between ≈ $2.05 and $3.45 — and the share of $0 draws between ≈ 0.77 and 0.83 (SE ≈ 0.0127). A value modestly outside these bands can still be honest chance (or a re-roll before freezing); a value far outside (a mean near 32, a $0-share near 0.25) almost always means the prize map's thresholds are wrong — regrade the formula, not the luck. The difference row must be computed correctly from the student's own two numbers (re-do their subtraction — that arithmetic is what's graded).

Fallback set (fixed): the 40 listed payouts contain 32 zeros, six $5s, and two $25s: sum = 6(5) + 2(25) = 30 + 50 = 80 → mean = 80 ÷ 40 = $2.00 exactly; share of $0 draws = 32/40 = 0.80. Note the teaching gift: the $100 ticket never appears in 40 draws, so the fallback mean sits below 2.75 — exactly the "rare big prize hasn't shown up yet" behavior Q2/Q4 discuss.

Part 5 expected answers:
1. Buyer, long-run language: per $5 ticket, the average return is $2.75 — an average loss of $2.25 per ticket over many raffles; no single ticket ever pays $2.75. Club side: all 100 tickets sold means revenue is exactly 100 × $5 = $500 and payouts are exactly 100 + 4(25) + 15(5) = $275 — the club keeps $225 with certainty (equivalently, $2.25 × 100). The buyer faces chance; the seller of the whole distribution doesn't.
2. Not an error — sampling variability: 1,000 random draws produce an average near E(X), not equal to it. Same phenomenon as Week 5's lab, where the running proportion wobbled before settling at its long-run value — the settling is a long-run promise, never an exact short-run one.
3. The <=21 slip makes drum numbers 6–21 (sixteen numbers) pay $5, so the simulated mean drifts up (theoretical value becomes 0.01(100) + 0.04(25) + 0.16(5) = $2.80). It's a method error (bias), not chance wobble: more draws make the mean settle ever more confidently on the wrong 2.80. Chance error shrinks with n; method error doesn't — Week 1's lesson, method beats size, in spreadsheet form.
4. The $100 prize dominates: its row contributes 94.58 of the 121.19 total variance — one rare value sitting ≈ $97 from the mean, and squaring rewards distance. Takeaway for a buyer: a big SD here signals a skewed, jackpot-driven ticket — the typical experience (lose $5) is nothing like "mean ± SD," so read the distribution, not just its two summary numbers.
5. Doubling is the transformation Y = 2X: E(Y) = 2(2.75) = $5.50; SD(Y) = 2(11.0085) ≈ $22.02. At a $5 price the club would now lose an average $0.50 per ticket (−$50 on a sold-out run) — the treasurer should refuse (or re-price tickets above $5.50).

AI-critique full credit: names a specific claim checked or corrected — most commonly the chatbot averaging the payouts to 32.50, blurring variance vs. SD (calling 121.19 the "SD"), calling the simulated-mean gap a fixable mistake, or declaring the raffle "fair" — OR documents verification of each claim. 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
Setup & distribution table — warm-up count 20; probabilities from prize counts; legitimacy =SUM shows exactly 1; Table 1 columns computed (8) 8 4–6 0–3
Theory by formula — E(X) = 2.75 exact (both routes agree); Var = 121.1875; SD ≈ 11.01, with SD (not variance) reported as the spread (10) 10 5–8 0–4
Simulation build — 1,000 draws via =RANDBETWEEN(1,100); prize map thresholds mirror the prize counts; mean, $0-share, max summarized; numbers frozen (12) 12 6–10 0–5
Analysis questions — Q1–Q5 hit the expected ideas, especially long-run interpretation (Q1–Q2), bias vs. chance error (Q3), and the doubling computation (Q5) (15) 15 8–12 0–7
AI-critique — specific catch or verification reported in 2–3 sentences (5) 5 3 0–2

Quality gate (self-checked): every fixed number in the key re-verified by the week's arithmetic script (prize counts sum to 100; probabilities to 1; E(X) = 2.75; Var = 121.1875; SD = 11.0085…; fallback mean = 2.00 exactly from the 40 listed values; doubling values 5.50/22.02; club take 500 − 275 = 225); simulated quantities stated as ranges (mean ≈ 2.05–3.45; $0-share ≈ 0.77–0.83), never exact targets; the lab runs on in-sheet functions only — nothing to buy, download, or log into; no graded-item leakage (the raffle surface appears in no quiz, practice, assignment, chapter self-check, or exam item); rubric sums 8 + 10 + 12 + 15 + 5 = 50. ✓