Week 7 Data Lab: Counting Chinstraps

Part 1 — The Big Picture

This week you learned that a count of successes in a fixed number of independent tries follows the binomial model, and that a smooth normal curve can stand in for that model only when the expected counts are large enough. In this lab you build the model from real data. You count the Chinstrap penguins in the field log to get the chance of picking one, build the binomial table for 10 random picks and chart it, then run 200 simulated rounds of 10 picks and compare what you get with the model. Last, you redo the chart for 100 picks, check the rule of thumb for both sizes, and audit a chatbot's answers about the same counts.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: If you pick penguins at random from the field log and count the Chinstraps, how well does the binomial model describe the counts you get, and is the count of Chinstraps in 10 picks, or in 100 picks, large enough for a normal curve to stand in for it?

Workshop data: palmer-station-penguins

Get the data: palmer-station-penguins landing page or the direct data file — free, no login required.

Source: Measurements of 344 penguins of three species on three islands of Antarctica's Palmer Archipelago, recorded by field researchers and republished in the palmerpenguins package.

• Rows: 344 penguins
• Columns: species, island, bill_length_mm, bill_depth_mm, flipper_length_mm, body_mass_g, sex, year
• Missing values: Numeric measurement columns and sex carry missing values for a small number of penguins

Part 3 — Materials & Procedure

Tools and materials

• Google Sheets (free) or Excel
• The palmerpenguins CSV file
• A document or notebook for your written answers

Ways of working

• interpret-real-data
• compute-and-plot
• structured-observation

Procedure

1. Open a blank sheet and name the first tab Penguins. In cell A1 type =IMPORTDATA("https://raw.githubusercontent.com/allisonhorst/palmerpenguins/main/inst/extdata/penguins.csv") in Google Sheets, or in Excel use Data, then From Web, and paste the same address. Check that the species names sit in column A. Tool: Google Sheets (free) or Excel. Check: Row 1 shows the eight column names, and the penguins fill rows 2 through 345.
2. In an empty cell type =COUNTIF(A2:A345,"Chinstrap") to count the Chinstrap penguins, and in the next cell divide that count by 344 to get p, the chance that one random pick is a Chinstrap. Format p to four decimals and record both numbers in the first scaffold table. Tool: Google Sheets (free) or Excel. Check: The count is a whole number well below half of 344, and p is a decimal between 0 and 1.
3. Open a second tab named Model. Type the counts 0 through 10 in A2 to A12. In B2 type =BINOM.DIST(A2,10,68/344,FALSE) and fill it down to B12. In C2 type =B2*200 and fill it down, which gives the number of rounds out of 200 you would expect at each count. Select A1:B12 and use Insert, then Chart, and choose a column chart. Tool: Google Sheets (free) or Excel. Check: Column B holds eleven chances that add to 1, and the chart has one bar for each count from 0 to 10.
4. Open a third tab named Simulation. In B2 type =INDEX(Penguins!$A$2:$A$345,RANDBETWEEN(1,344)) and copy it across to K2, so the row holds 10 random picks made with replacement. In L2 type =COUNTIF(B2:K2,"Chinstrap"). Copy row 2 down to row 201, so you have 200 rounds of 10 picks. Tool: Google Sheets (free) or Excel. Check: Column L holds 200 whole numbers from 0 to 10, and every edit to the sheet picks new penguins.
5. Before your next edit, count how many rounds gave each number of Chinstraps with =COUNTIF(L2:L201,0), =COUNTIF(L2:L201,1), and so on, and find =AVERAGE(L2:L201) and =STDEV(L2:L201). Copy every value into the second and third scaffold tables right away, because the next edit picks again. Tool: Google Sheets (free) or Excel. Check: The counts for 0 through 10 add to 200, and the average is a decimal between 0 and 10.
6. Back on the Model tab, type the counts 0 through 40 in E2 to E42 and =BINOM.DIST(E2,100,68/344,FALSE) in F2, filled down to F42. Chart E1:F42 as a column chart. Then compute np and n(1 − p) for 10 picks and for 100 picks, and fill in the fourth scaffold table. Tool: Google Sheets (free) or Excel. Check: The new chart has one bar for each count from 0 to 40, and each expected count is p or 1 − p multiplied by the number of picks.
7. Answer the analysis questions in your document, then paste the AI-critique prompt into an approved chatbot, or use the equivalent route, and write your critique report. Tool: A document or notebook for your written answers. Check: Every answer cites a number from your own tables or charts.

Fallback route

When to use it: Use this route if the penguin file will not import or no spreadsheet is available.

Work from the model data below, which reports the species counts from the real file. Use a calculator for np, n(1 − p), and √(np(1 − p)). For the simulation, put 86 slips in a bag, 17 marked Chinstrap and 69 marked other, which is exactly the same share as 68 out of 344; draw a slip, record it, and put it back, ten times per round. Run 40 rounds of 10 draws by hand and scale your expected rounds to 40 instead of 200. For the model charts, sketch the bars from the chances you compute with the binomial probability formula for 0 to 4 Chinstraps. Answer every analysis question and the critique step from these records, and submit the same items for equal credit.

This route earns the same credit as the primary route.

These counts from the real file give you every penguin number the lab needs without importing it.

Fallback model data

Quantity from the file	Value
Penguins in the file	344
Chinstrap penguins	68 (all on Dream)
Adelie penguins	152
Gentoo penguins	124
p = 68 ⁄ 344 for one random pick, to four decimals

Part 4 — The Scaffold

Count the Chinstraps in the real file so you know the chance of one random pick.

Student scaffold table 1

Quantity	Your value
Chinstrap penguins in the file
All penguins in the file
p, the chance one random pick is a Chinstrap (four decimals)

Compare how often each count turned up in your 200 rounds with how often the binomial model says it should.

Student scaffold table 2

Chinstraps in 10 picks, k	Model chance P(X = k) (four decimals)	Expected rounds out of 200	Your simulated rounds out of 200
0
1
2
3
4
5 or more

Put the model's center and spread beside the center and spread of your own simulated counts.

Student scaffold table 3

Quantity	From the binomial model	From your 200 simulated rounds
Mean number of Chinstraps in 10 picks
Standard deviation of the number of Chinstraps in 10 picks

Check the rule of thumb for both sizes before you let a normal curve stand in for the count.

Student scaffold table 4

Number of picks, n	np	n(1 − p)	Both at least 10?	Shape of the model chart, in words
10
100

Part 5 — Analysis Questions

1. Using your second table, how close did your simulated rounds come to the expected rounds for 0, 1, 2, 3, and 4 Chinstraps? Which count was most common in the model and in your simulation, and what is the model chance P(X = 2)?
2. Describe your two model charts in words: where the tallest bar sits, which way the long, thin tail runs, and what changes between 10 picks and 100 picks. Name the shape of each, with the skew named by its tail.
3. Check the four binomial conditions for picking 10 penguins at random with replacement and counting Chinstraps. Then explain what would change if you instead picked 10 different penguins without replacement from a small group of 20, of whom 4 are Chinstraps, and why picking without replacement from all 344 penguins changes the chance only a little.
4. Compare the model mean np and standard deviation √(np(1 − p)) with the mean and standard deviation of your 200 simulated rounds. Then use your fourth table to say whether a normal curve may stand in for the count in 10 picks and in 100 picks, and, where it may, write the normal model you would assume, with its mean and standard deviation.
5. Choose one and answer it in 3–4 sentences. (a) Your 200 rounds did not match the model exactly: what explains the gaps, and what would you expect if you ran 2,000 rounds instead? (b) Every Chinstrap in the file was logged on Dream: what can and cannot the counts tell you about why Chinstraps live there, or about penguins that were never logged?

Part 6 — AI-Critique Moment

Choose one equal-credit critique route. Route A: ask an approved AI tool to critique your completed work, then check its response against your own evidence. Route B is the no-AI equivalent printed below.

Approved tools: Claude, Gemini, ChatGPT, Grok, Microsoft Copilot.

Prompt to paste:

I've pasted my completed Week 7 Data Lab tables above. In the lab I used the Palmer penguins file, where 68 of the 344 penguins are Chinstrap and all 68 were logged on Dream island. I picked 10 penguins at random from the file, with replacement, and counted the Chinstraps. Please do five things. First, give the chance of getting exactly 2 Chinstraps in 10 picks, and show how you got it. Second, describe the shape of the distribution of the number of Chinstraps in 10 picks and name its skew. Third, say whether these counts show that living on Dream makes a penguin a Chinstrap. Fourth, tell me how many Chinstrap penguins are in the file and how you know. Fifth, as a stretch question for later in the course, say what a 95% confidence interval means.

Test the response for these likely failures:

• Chatbots often multiply the chance of one arrangement and stop, reporting 0.1977² × 0.8023⁸, about 0.0067, as the chance of exactly 2 Chinstraps.
• Chatbots often call the 10-pick distribution left-skewed because its tallest bars sit at the low counts, naming the skew by the pile.
• Chatbots sometimes slide from the table to a cause, saying Dream's conditions turn penguins into Chinstraps because every Chinstrap was logged there.
• Chatbots sometimes state a Chinstrap count, a share, or a chance that does not come from the file, or quote a figure for the file that you never gave them.
• On the stretch question, chatbots often say a 95% confidence interval means a 95% chance that the true value is inside this one interval.

Write 3–5 sentences naming one thing the chatbot got right and at least two places it went wrong or overreached, quoting its words and correcting each with evidence from your own tables, your model chart, or the penguin counts.

Equivalent route (no AI required)

No AI account is required for this lab. Instead of the chatbot, critique this flawed analysis, printed here in full: "The file has about 70 Chinstrap penguins, so p is about 0.2. The chance of exactly 2 Chinstraps in 10 picks is 0.1977² × 0.8023⁸, about 0.0067, so getting 2 is very rare. The 10-pick distribution is left-skewed, because the biggest bars are on the left. All the Chinstraps live on Dream, so the island turns penguins into Chinstraps. A 95% confidence interval means there is a 95% chance the true value is inside it." Mark it up: quote each flawed sentence, name what is wrong (an invented or rounded-off statistic, a dropped number of ways, a skew named by the pile, cause read from association, or a misread confidence statement), cite the evidence from your tables, charts, or the penguin counts, and write a corrected sentence beneath it; for the last sentence, say what the course will check later. This route assesses the same objectives (Objective 4 and Objective 5) as the chatbot critique, takes comparable effort (about 30 minutes and 200–300 words), is scored by the same scoring criteria for the same credit, and its own submission artifact is the marked-up flawed analysis, posted in place of the chatbot share link and report.

Part 7 — What to Submit

• All four completed scaffold tables
• Your two model charts, for 10 picks and for 100 picks, or labelled sketches of them
• Written answers to all five analysis questions
• The chatbot critique report with its share link, or the equivalent route's marked-up flawed analysis

Deliverable:

A short Data Lab report on counting Chinstraps: four completed tables (the Chinstrap count and p from the real file; model chances and expected rounds beside your 200 simulated rounds; the model and simulated mean and standard deviation; and the rule-of-thumb check for 10 and 100 picks), two model charts or labelled sketches of them, written answers to five analysis questions, and a critique of a chatbot's answers about the same counts, or, on the equivalent route, a marked-up correction of the printed flawed analysis. Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text. Choose a written, recorded, or visual form for your answers; every form is graded by the same scoring criteria.

Rubric

The rubric below shows how each submitted part is evaluated; bands are full, partial, or none.

Workshop evaluation criteria

Criterion	Full	Partial	None
Counting from the real file	The Chinstrap count and the total come from the file or the fallback table, and p is the count divided by the total, reported to four decimals.	The counts are present but p is missing, divided by the wrong total, or rounded too early.	No count or p is recorded, or the numbers are invented rather than produced.
Building and charting the binomial model	The model chances for 0 to 10 Chinstraps in 10 picks come from the binomial model with the file's p, the expected rounds are the chances times the number of rounds, and both model charts, or labelled sketches, have one bar per count and labelled axes.	The model table or the charts are present, but one or two chances, an expected-round column, or an axis label is missing or wrong.	No model table or chart is submitted.
Running and comparing the simulation	The 200 rounds, or the hand-drawn fallback rounds, are run and recorded; the simulated counts add to the number of rounds; the simulated mean and standard deviation are computed; and analysis question 1 compares the simulated rounds with the expected rounds count by count.	The simulation is run, but some counts, the mean, or the standard deviation are missing, or the comparison with the model is vague.	No simulation results are recorded, or the counts are invented rather than produced.
Reasoning about conditions, shape, and the rule of thumb	Analysis questions 2 to 5 are answered accurately: the shapes are named by their tails, all four binomial conditions are checked and the small-group change is explained, np and n(1 − p) are checked against 10 for both sizes with a normal model stated as an assumption where it may stand in, and the chosen reflection option names a specific limit.	The questions are attempted, but one or two answers name skew by the pile, check the rule on n alone, skip a condition, call real data normal, or give a general reflection.	The reasoning questions are missing or rest on the misconceptions listed above throughout.
Critique of the chatbot or of the printed flawed analysis	On either route, at least two errors are quoted and corrected with evidence from the student's own tables, charts, or the penguin counts, and one correct point is acknowledged; both routes earn this band equally.	On either route, errors are named but corrected without evidence, or only one error is caught.	No critique appears, or pasted output is submitted without any checking.

Quality Gate

• Every count, chance, and summary in the tables comes from your own formulas, your own simulation, or the fallback model data, never from memory or a chatbot.
• p is the Chinstrap count divided by 344, and each model chance comes from the binomial model with that p.
• The simulated rounds in the second table add to the number of rounds you ran, and the expected rounds are the model chances times that number.
• Each chart has one bar per count and labelled axes, and its shape is named by its tail.
• The rule-of-thumb verdict checks both np and n(1 − p) against 10, and any normal curve is stated as a model you assume.
• No statistic, count, or source is invented anywhere in the submission.
• The critique quotes the chatbot or the flawed analysis and corrects at least two errors with evidence from your tables, charts, or the penguin counts.