Week 14 Data Lab: Is the Gap Real?

Part 1 — The Big Picture

This week you learned four tests: the one-sample t test, the two-sample t test read from software output, the one-proportion z test, and the two-proportion z test with a pooled proportion. You also learned that a two-sided test and its confidence interval are two views of one question, and that "not significant" is not "no difference". In this lab you run the tests yourself on the real penguin file. You compare two pairs of species on body mass, one pair where the gap is huge and one where it is small, then test the sex split and the Adelie share on two islands. Then you check a chatbot's answers about the same data.

Time budget: Plan for about 90 minutes.

Part 2 — The Task

Guiding question: When the penguin data say a gap is or is not statistically significant, what do the tests and their intervals tell us about how big the real difference could be?

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 Excel, save the file from that address, then use Data, then From Text/CSV, and load it starting in cell A1. Tool: Google Sheets (free) or Excel. Check: Row 1 shows the eight column names, the penguins fill rows 2 through 345, body mass sits in column F, and sex sits in column G.
2. Add a tab named Masses. In A1 type =FILTER(Penguins!F2:F345, Penguins!A2:A345="Adelie", Penguins!F2:F345<>"NA"). In B1 use the same formula with "Chinstrap", and in C1 with "Gentoo". In rows 202 to 204 of each column, below the rows the formulas read, type =COUNT, =AVERAGE, and =STDEV over rows 1 to 200 of that column (for example =COUNT(A1:A200)), and fill the first scaffold table, rounding to two decimals only when you copy a value. Tool: Google Sheets (free) or Excel. Check: The counts are 151, 68, and 123, and the three means sit near 3,700 g, 3,700 g, and 5,000 g.
3. Fill the second scaffold table for each comparison. The difference is one mean minus the other. The standard error is =SQRT(STDEV(C1:C200)^2/COUNT(C1:C200)+STDEV(A1:A200)^2/COUNT(A1:A200)) for Gentoo minus Adelie, and the same with column B for Chinstrap minus Adelie. Divide the difference by the standard error for t. Then =T.TEST(C1:C200, A1:A200, 2, 3) gives the Welch two-sided p-value; use column B for Chinstrap. Build each 95% interval as the difference minus and plus 1.976 times the standard error. Tool: Google Sheets (free) or Excel. Check: The Gentoo-minus-Adelie t matches the course value 23.386 and its p-value appears in scientific notation; the Chinstrap-minus-Adelie t is well under 1.976.
4. Fill the third scaffold table. In any empty cell, =COUNTIF(Penguins!G2:G345,"female") and =COUNTIF(Penguins!G2:G345,"male") count the sexes. Compute p̂ as females divided by their total, the null standard error as =SQRT(0.25/333), z as (p̂ minus 0.5) divided by that standard error, and the two-sided p-value as =2*NORM.S.DIST(-ABS(z), TRUE). Tool: Google Sheets (free) or Excel. Check: The counts are 165 females and 168 males, z matches the course value of about −0.164, and the p-value is about 0.87.
5. Finish the third scaffold table. =COUNTIFS(Penguins!A2:A345,"Adelie",Penguins!B2:B345,"Biscoe") and =COUNTIF(Penguins!B2:B345,"Biscoe") give the Adelie count and the total on Biscoe; repeat with "Dream". Pool the counts, build the standard error from the pooled share, divide the difference in shares by it for z, and use =2*NORM.S.DIST(-ABS(z), TRUE) for the p-value. Tool: Google Sheets (free) or Excel. Check: The counts are 44 of 168 on Biscoe and 56 of 124 on Dream, the pooled share is about 0.342, and z is about −3.38.
6. Draw the two mean-difference intervals on one number line in grams, by hand or with a chart, and mark 0 on it. Label each interval with its pair of species in words. Check: One interval sits far to the right of 0, near 1,400 g; the other straddles 0.

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 summaries and counts from the real file. Copy the counts, means, and standard deviations into the first scaffold table, compute each s² ⁄ n and each standard error with a calculator, and build both intervals by hand with t* = 1.976. For the proportion tests, use the sex and island counts below and compute each z by hand. For the p-values, use the course values given below for the Gentoo, sex, and island tests, and for Chinstrap minus Adelie judge the decision by comparing your t with 1.976. Draw the number line by hand. 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 summaries and counts from the real file give you every input the lab needs without importing it.

Fallback model data

Quantity	Value
Adelie: n, mean, standard deviation	151 birds, 3,700.66 g, 458.57 g
Chinstrap: n, mean, standard deviation	68 birds, 3,733.09 g, 384.34 g
Gentoo: n, mean, standard deviation	123 birds, 5,076.02 g, 504.12 g
Sex counts among 333 sexed penguins	165 female, 168 male
Adelie counts by island	44 of 168 on Biscoe, 56 of 124 on Dream
Course p-values	Gentoo minus Adelie far below 0.001; sex test about 0.87; island test below 0.003
Your Chinstrap-minus-Adelie decision

Part 4 — The Scaffold

Summarize each species so you have every input the two-sample tests need.

Student scaffold table 1

Summary	Adelie	Chinstrap	Gentoo
Count with a recorded mass, n
Sample mean, x̄ (g)
Sample standard deviation, s (g)
s² ⁄ n (g²)

Run each two-sample comparison from the difference to the interval, so you can set the verdict beside the size.

Student scaffold table 2

Step	Gentoo minus Adelie	Chinstrap minus Adelie
Difference in means (g)
Standard error of the difference (g)
t statistic
Welch two-sided p-value from T.TEST
Decision at α = 0.05
95% interval with t* = 1.976 (g)

Run the two proportion tests and record each decision in words.

Student scaffold table 3

Step	Sex split: H₀: p = 0.5	Adelie share: Biscoe versus Dream
Counts used
Sample proportion or proportions
Pooled proportion (two-proportion test only)
Standard error
z statistic
Two-sided p-value and decision at α = 0.05

Part 5 — Analysis Questions

1. Report the Gentoo-minus-Adelie difference, standard error, and t from your second table, and check your t against the course value of 23.386. Explain any small difference as rounding.
2. Report the Chinstrap-minus-Adelie t, its p-value, and your decision at α = 0.05. Look at your number line and describe, in words, where each interval sits relative to 0.
3. The Chinstrap-minus-Adelie test is not significant. Use your interval to explain why that does not show the two species have the same mean mass: name two differences that are still plausible and one that is not.
4. Compare the two proportion tests from your third table. For each, say whether the data are consistent with the null hypothesis or give evidence against it, and explain why the sex test used 0.5 inside its standard error while the island test used the pooled share.
5. Choose one and answer it in 3–4 sentences. (a) The penguins were observed in a field study, and the team chose which colonies to monitor: how far can your island comparison speak for the islands as a whole? (b) Gentoo and Adelie differ with a tiny p-value: does a tiny p-value by itself tell you the difference is large? Use your interval to explain what does.

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 14 Data Lab tables above. In the lab I used the Palmer penguins file to run two-sample t tests on body mass (Gentoo versus Adelie and Chinstrap versus Adelie), a one-proportion z test of whether the share of females among the 333 sexed penguins is 0.5, and a two-proportion z test comparing the share of Adelie penguins on Biscoe and on Dream. Please do five things. First, tell me the difference in mean body mass, Chinstrap minus Adelie, its standard error, and its two-sided p-value, and say what the result means. Second, describe the shape of the body masses of all 342 penguins together and name any skew. Third, say whether living on Dream Island makes a penguin more likely to be an Adelie. Fourth, if each of two penguins picked at random independently has a 0.5 chance of being female, give the chance that exactly one of the two is female. Fifth, tell me whether the Chinstrap-minus-Adelie interval, which contains 0, proves that the two species have the same mean mass.

Test the response for these likely failures:

• Chatbots often state a Chinstrap-minus-Adelie difference, standard error, or p-value from memory, such as a made-up difference of 50 g or a p-value that does not match the file, or they invent the sample sizes.
• Chatbots often say that because the Chinstrap-minus-Adelie test is not significant, or its interval contains 0, the two species have the same mean mass, or that there is a 95% probability the true difference is in the interval.
• Chatbots sometimes call the 342 body masses left-skewed because most penguins are on the lighter side, naming the skew by where the pile is.
• Chatbots often say that living on Dream Island makes a penguin more likely to be an Adelie, reading a cause from an association.
• Chatbots often compute 0.5 × 0.5 = 0.25 for exactly one female and stop, forgetting that the female could be the first pick or the second.

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 or the course's pinned values.

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: "Chinstrap penguins average 50 g more than Adelie penguins, with a p-value of 0.02, so the difference is significant. Because the test was not significant, Chinstrap and Adelie penguins have exactly the same mean mass, and there is a 95% probability that the true difference is in my interval. The body masses of all 342 penguins are left-skewed, because most penguins are on the light side. Living on Dream Island makes a penguin more likely to be an Adelie. If each of two penguins has a 0.5 chance of being female, the chance that exactly one is female is 0.5 × 0.5 = 0.25." Mark it up: quote each flawed sentence, name what is wrong (an invented statistic, a "not significant means the same" claim or a misread confidence statement, a wrong skew claim, cause read from association, or a dropped number of ways), cite the evidence from your tables or the course's pinned values, and write a corrected sentence beneath it. This route assesses the same objectives (Objective 7 and Objective 6) 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 three completed scaffold tables
• Your number line of the two mean-difference intervals with 0 marked, as a chart or a labelled sketch, or a written description of it
• 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 four penguin tests: three completed tables (each species' count, mean, standard deviation, and s² ⁄ n; two two-sample comparisons from the difference to the interval; and two proportion tests from the counts to the decision), a number line of the two mean-difference intervals with 0 marked, as a chart or a labelled sketch or a written description, written answers to five analysis questions, and a critique of a chatbot's answers about the same data, 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
Importing and summarizing the three species	The count, mean, standard deviation, and s² ⁄ n are recorded for all three species with the right group sizes, from the file or the fallback model data.	The summaries are present, but one or two values are missing, use the wrong group, or include the NA entries.	No summaries are recorded, or the values are invented rather than computed.
Running the two mean comparisons	Both comparisons show the difference, the standard error built by adding the two s² ⁄ n terms, t, the p-value, a decision, and the interval with t* = 1.976, and analysis question 1 checks the Gentoo t against 23.386.	The comparisons are mostly complete, but a step is missing or wrong, such as subtracting the standard errors, or the check against 23.386 is left out.	The comparison table is missing or not built from the data.
Running the two proportion tests	Both proportion tests show the counts, the proportions, the correct standard error (the null value for the sex test, the pooled share for the island test), z, the p-value, and a decision in words.	Both tests are attempted, but the sex test uses p̂ in its standard error, the island test averages the percentages, or a decision is missing.	No proportion test is run, or the values are invented.
Interpreting significance and size	Analysis questions 2, 3, and 4 read each decision in context, use the Chinstrap-minus-Adelie interval to name plausible differences without claiming equal means, and explain the two different standard errors in the proportion tests.	The decisions are read, but a "not significant" result is treated as proof of no difference, or the standard-error explanation is missing.	Analysis questions 2 to 4 are missing or do not use the student's own tables.
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 or the course's pinned values, including the "not significant means the same" claim, 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.
Reflection on what the tests can and cannot show	The chosen reflection option is answered in 3–4 sentences that name a specific limit, such as colonies chosen by the team or a tiny p-value that says nothing about size, and tie it to this lab's results.	The reflection names a limit in general terms without tying it to this lab's tests.	No reflection is submitted.

Quality Gate

• Every count, mean, standard deviation, test statistic, and p-value comes from your own formulas or the fallback model data, never from memory or a chatbot.
• Each standard error of a difference adds the two s² ⁄ n terms before taking the square root; nothing is subtracted.
• The one-proportion test puts the null value 0.5 inside its standard error, and the two-proportion test pools the counts, not the percentages.
• No "not significant" result is described as proof of no difference; each one names the values the interval still allows.
• The number line, chart, or description labels each interval and marks 0 in words.
• 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 or the course's pinned values.