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E. Culminating Data Management Investigation · Culminating Data Management Investigation

Culminating Data Management Investigation: Unit Review

E1.1 E1.2 E1.3 E1.4 E1.5 E2.1 E2.2 E2.3 E2.4About 40 minutes
Learning goals
  • I can describe every stage of a culminating investigation, from question to critique.
  • I can check my own project against the expectations of each stage.
  • I can find the lesson to revisit when a stage of my project needs work.

Words to know

investigation cycle
Pose a question, plan, gather and organize data, analyse, conclude, then report, present, answer questions and critique.
strength of evidence
How convincing the results are, judged by effect size, consistency, precision, bias and whether chance could explain them.

Let's review!

In this unit you followed one complete investigation, Amira's study of part-time work and sleep (with example data made up for teaching) plus her carnival dice game, from the first idea to the final critique. You learned to pose a significant question, do background research and write hypotheses, plan the sample and analysis, design an ethical survey, gather, clean and organize data, analyse it with one-variable and two-variable statistics and probability models, draw conclusions with honest limitations, and then report, present, defend and critique investigations.

Each practice set below matches one lesson. Answers are complete, so you can check yourself.

PoseLessons 1-2E1.1PlanLessons 3-4E1.2GatherLesson 5E1.3AnalyseLessons 6-8E1.4ConcludeLesson 9E1.5ReportLesson 10E2.1PresentLesson 11E2.2Defend + critiqueLessons 12-13E2.3, E2.4E1: design and carry out (Pose to Conclude).E2: present and critique (Report onward).
The investigation cycle and the lessons of this unit. Read the top row left to right, then the bottom row right to left.

How to use this review

Work through one or two sections at a time. Use the example data where a question asks for a calculation (the data table is in Gathering, Cleaning and Organizing the Data). For each question you miss, note which stage it belongs to and reread that lesson. Then use the same questions as a checklist for your own project: could you answer each one about your investigation? Finish with the 10-question quiz.

The model investigation at a glance (example data, made up for teaching)QuestionDo Grade 12 students with part-time jobs sleep less on school nights?Samplesystematic random, 48 of 240; 43 returned (89.6%); 40 validJob group (n = 22)mean 7.43 h, median 7.5, s = 0.68; 27.3% meet 8 hNo-job group (n = 18)mean 8.57 h, median 8.5, s = 0.59; 88.9% meet 8 hRelationship (jobs)r = −0.62, r² = 0.39, ŷ = 8.330 − 0.0679xChance?0 of 10 000 shuffles reached the 1.14 h differenceGame of chanceE(prize) = 1.25 tickets; profit 0.75/play; 230 in 300 playsConclusionstrong evidence of association; not cause; one school only
Key results of the model investigation on one card. Your own project should be summarizable in a card like this.

Posing a Significant Question

1

Turn the topic "weather" into an investigable question that uses secondary data.

2

List the five tests for a good culminating question.

3

Classify as statistics, probability or both: "Is a spinner game fair, and how do 200 plays compare with theory?"

Background Research and Hypotheses

1

Write an operational definition for "time on homework".

2

Write a hypothesis and null statement: "Is outdoor temperature related to the number of people at a skating rink?"

3

Why must the hypothesis be written before looking at the data?

Planning the Investigation: Population, Sample Size and Bias

1

Find the sample size for a ±8% margin of error at 95% confidence, p unknown.

2

A list of 360 students is sampled systematically to get 30. Find k and the possible random starts.

3

Name one bias Amira planned for and her fix.

Designing an Ethical Survey

1

Fix: "How much do you exercise and eat healthy food?"

2

Name three things the survey introduction must say.

3

Why are pilot-test answers not included in the data?

Gathering, Cleaning and Organizing the Data

1

Of 75 surveys handed out, 63 came back. Find the response rate.

2

Write a spreadsheet formula that labels sleep in C2 as "Short" if under 7 hours, otherwise "OK".

3

Why should an unusual but possible value stay in the data set?

4

How many example-data students recorded between 7.5 and 8.0 hours (7.5 ≤ sleep < 8.0)?

Analysing the Data I: Summaries, Graphs and the Normal Model

1

Find Q1, Q3 and the IQR of: 5, 6, 6, 7, 8, 9, 10.

2

For the no-job group, N(8.569, 0.593), find the z-score of 7.5 hours.

3

Why must group sizes be labelled beside box plots?

Analysing the Data II: Relationships Between Variables

1

Using ŷ = 8.330 − 0.0679x, predict sleep for 12 hours of work and find the residual for a student who works 12 hours and sleeps 8.0 hours.

2

r = −0.62. Describe the relationship and find r².

3

From the contingency table, find P(job | does not meet the guideline).

Analysing the Data III: Probability Models and Expected Value

1

A game: roll one die; win 6 tickets on a 6, otherwise nothing; cost 1 ticket. Find the player's expected net gain.

2

Find the mean and standard deviation of the number of 6s in 120 rolls of a die.

3

In 120 rolls, 29 sixes appeared. Is this unusual?

Drawing Conclusions: Strength of Evidence, Limitations and Next Steps

1

In a shuffle test, 2400 of 10 000 shuffles gave a difference at least as large as observed. Conclusion?

2

Rewrite to match survey evidence: "Working makes students tired."

3

Give a follow-up investigation that addresses the "one school" limitation.

Writing the Investigation Report

1

Put in order: results, abstract, method, references, introduction, discussion.

2

Write a method sentence about data cleaning for Amira's report.

3

What makes a report reproducible?

Presenting Your Investigation

1

Write a headline for a slide showing the dice game's expected and observed counts.

2

Your 10-minute talk ran 13 minutes in rehearsal. Give two cuts.

3

What are the four steps for talking through a graph?

Answering Questions and Defending Your Reasoning

1

Answer: "Why didn't you use the mean of all 40 students?"

2

A critic says "your sample is biased". What should you ask or do first?

3

What structure helps with difficult questions?

Critiquing Investigations Constructively

1

Write constructive feedback on: "Graph starts at 50%; no n shown."

2

A project concludes "Gamers have faster reaction times because r = 0.5" from 12 volunteers. Name two issues.

3

Name the five areas to check when critiquing.

Just for fun

You have now designed, analysed, defended and critiqued an investigation. The probability that you'll never again read "a survey shows..." without asking "who was surveyed?" is approximately 1.

Check your understanding

Quiz

1.

Which is the best culminating question?

2.

The 95% sample size for a margin of error of ±5% (p unknown) is:

3.

Which survey question is neutral and measurable?

4.

A respondent answered "no job" but listed 12 hours of work. With an anonymous survey, the best choice is to:

5.

In the example data, the job group's mean and median sleep are about:

6.

r² = 0.39 for work hours and sleep means:

7.

A dice game pays 20 tickets with probability 1/36 and 5 tickets with probability 5/36, and costs 2 tickets. The organizer's expected profit per play is:

8.

Which statement best describes the strength of Amira's evidence for Hypothesis 1?

9.

In a report, the sentence "The correlation coefficient was −0.62 (n = 22)" belongs in the:

10.

Which feedback on a classmate's project is most constructive?

For parents and teachers

Use the review to find what to revisit:

  • Question too vague, opinion-based or not feasible → Lesson 1 (Posing a Significant Question).
  • Vague definitions, hypotheses with no direction or written after the data → Lesson 2 (Background Research and Hypotheses).
  • Non-random samples, sample-size or margin-of-error errors, no analysis plan → Lesson 3 (Planning the Investigation).
  • Leading, double-barrelled or sensitive survey questions; missing consent → Lesson 4 (Designing an Ethical Survey).
  • Missing values entered as 0, unrecorded exclusions, overlapping intervals → Lesson 5 (Gathering, Cleaning and Organizing the Data).
  • Wrong quartiles or standard deviation, box plots without n, normal model on skewed data → Lesson 6 (Analysing the Data I).
  • Mixing up r and r², extrapolating, conditional percentages in the wrong direction → Lesson 7 (Analysing the Data II).
  • Probabilities not adding to 1, forgetting the cost in expected net gain, binomial conditions → Lesson 8 (Analysing the Data III).
  • Claiming cause, generalizing too far, vague limitations → Lesson 9 (Drawing Conclusions).
  • Missing report sections, unlabelled graphs, missing citations → Lesson 10 (Writing the Investigation Report).
  • Talks that run over time or slides full of text → Lesson 11 (Presenting Your Investigation).
  • Defensive or vague answers to questions → Lesson 12 (Answering Questions and Defending Your Reasoning).
  • Vague or unkind feedback → Lesson 13 (Critiquing Investigations Constructively).

The big picture: this unit draws on every other MDM4U unit. If a statistics tool itself is shaky (for example, standard deviation or regression), revisit the sibling unit that teaches it: One-Variable Data Analysis, Two-Variable Data Analysis and Validity, Discrete Probability Distributions or Continuous Distributions and the Normal Distribution.

About the model data: Amira, Daniel and Pinecrest Secondary School are fictional, and their data are made up for teaching.