Module 6 Project — Statistical Investigation: Study Habits & Performance

Capstone project · 30 points 30 pts Due Oct 2

A Rigorous Statistical Investigation (RSI) integrating every concept from Module 6: scatterplots, correlation, simple linear regression, sets, probability, tree diagrams, sample spaces, data classification, pictographs, and two-way frequency tables. You will analyze a real dataset, build statistical models, compute probabilities, and defend your conclusions.

RSI Framework. This project follows the Rigorous Statistical Investigation cycle: (R) Research question → (S) Statistical analysis → (I) Interpretation & defense. Every claim must be backed by evidence from your analysis, and every model must be checked — not assumed.

Student Learning Outcomes

By completing this project, you will demonstrate the ability to:

SLO 1: Classify variables as categorical or quantitative and identify individuals in a dataset.
SLO 2: Construct and interpret scatterplots to describe the direction, form, and strength of a relationship.
SLO 3: Calculate and interpret the correlation coefficient \(r\) and distinguish association from causation.
SLO 4: Fit a simple linear regression model, interpret slope and intercept, and evaluate residuals.
SLO 5: Use set operations (union, intersection, complement) to describe events in a sample space.
SLO 6: Compute probabilities using the addition rule, multiplication rule, and tree diagrams.
SLO 7: Construct and interpret two-way frequency tables, including marginal, joint, and conditional probabilities.
SLO 8: Create pictographs and assess whether they communicate data proportionally without distortion.
SLO 9: Communicate statistical findings with defensible language and identify limitations of the analysis.

The Dataset: Study Habits & Exam Performance

You will work with the following dataset, collected from 12 students in an introductory statistics course. Each student recorded their weekly study hours, the number of practice problems they completed, whether they attended a review session, and their final exam score (out of 100).

Student	Study Hours / Week	Practice Problems	Attended Review	Exam Score
1	3	15	No	58
2	5	22	No	64
3	4	18	Yes	67
4	7	30	No	72
5	6	25	Yes	75
6	8	35	Yes	81
7	10	40	No	85
8	9	38	Yes	88
9	12	50	Yes	94
10	2	10	No	52
11	11	45	Yes	91
12	6	28	No	70

Why this dataset? Every variable is neutral and observable. No demographic data is collected or implied. The dataset is small enough for hand calculation and large enough to show meaningful patterns.

Project Tasks

TASK 1

Variable Classification & Individuals 3 pts

RSI phase: Research question setup.

Identify the individuals (observational units) in this dataset.
For each variable (Study Hours, Practice Problems, Attended Review, Exam Score), classify it as categorical or quantitative and explain why.
For quantitative variables, identify the units of measurement.
State whether this is an observational study or an experiment. Justify your answer and explain what this means for causal claims.

TASK 2

Scatterplot & Description 4 pts

RSI phase: Exploratory analysis.

Create a scatterplot of Study Hours (x-axis) vs Exam Score (y-axis). You may draw by hand, use a spreadsheet, or use an online tool. Label both axes with units.
Describe the scatterplot using the three key features:
- Direction: positive, negative, or no association?
- Form: linear, curved, or no pattern?
- Strength: strong, moderate, or weak?
Identify any outliers or unusual patterns. If none, state so explicitly.
Create a second scatterplot of Practice Problems (x-axis) vs Exam Score (y-axis). Compare the two scatterplots — which relationship appears stronger, and why?

TASK 3

Correlation Coefficient 4 pts

RSI phase: Statistical measurement.

Calculate the correlation coefficient \(r\) between Study Hours and Exam Score. Show your work using the formula:

\[ r = \frac{1}{n-1} \sum \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right) \]

Interpret \(r\) in context: what does the value tell you about the direction and strength of the linear relationship?
Calculate the correlation coefficient between Practice Problems and Exam Score. Compare the two correlation values.
A classmate claims: "Since \(r\) is high, studying more causes higher exam scores." Explain why this conclusion is not justified by correlation alone. Name at least two possible confounding variables.

TASK 4

Simple Linear Regression 5 pts

RSI phase: Model building.

Fit a simple linear regression line for predicting Exam Score from Study Hours. Calculate the slope \(b_1\) and intercept \(b_0\) using:

\[ b_1 = r \cdot \frac{s_y}{s_x}, \qquad b_0 = \bar{y} - b_1 \bar{x} \]

Write the equation of the regression line in the form \(\hat{y} = b_0 + b_1 x\).
Interpret the slope in context: what does it mean in terms of study hours and exam scores?
Interpret the intercept in context. Is this interpretation meaningful? Why or why not?
Predict the exam score for a student who studies 9 hours per week. Is this prediction extrapolation? Explain.
Compute the residuals for all 12 students. Create a brief residual plot (fitted values vs residuals) and describe what you see. Does the linear model appear appropriate?

TASK 5

Sets, Sample Spaces & Probability 4 pts

RSI phase: Probabilistic reasoning.

Define the following events based on the dataset:

\(A\) = {student scored 75 or above}
\(B\) = {student attended the review session}
\(C\) = {student completed fewer than 25 practice problems}

List the elements of each set: \(A\), \(B\), \(C\), \(A \cup B\), \(A \cap B\), \(A^c\) (complement of \(A\)), and \(B \cap C^c\).
Using the 12 students as your sample space, compute: \(P(A)\), \(P(B)\), \(P(C)\), \(P(A \cap B)\), \(P(A \cup B)\), and \(P(A \mid B)\).
Verify the addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Show the numbers.
Are events \(A\) and \(B\) independent? Check whether \(P(A \mid B) = P(A)\). What does this tell you?

TASK 6

Tree Diagram & Conditional Probability 3 pts

RSI phase: Probabilistic reasoning.

Construct a tree diagram for a randomly selected student, with the first branch being "Attended Review" (Yes/No) and the second branch being "Scored 75+" (Yes/No). Label each branch with its probability.
Use the tree diagram to compute:
- \(P(\text{Scored 75+} \mid \text{Attended Review})\)
- \(P(\text{Scored 75+} \mid \text{Did Not Attend})\)
- \(P(\text{Scored 75+})\) using the law of total probability
Write a one-paragraph interpretation: does attending the review session appear to be associated with higher scores? What caveats apply to this conclusion?

TASK 7

Two-Way Frequency Table 4 pts

RSI phase: Statistical communication.

Construct a two-way (contingency) frequency table with "Attended Review" (Yes/No) as rows and "Scored 75+" (Yes/No) as columns. Include all marginal totals.
From the table, compute:
- The marginal probability \(P(\text{Scored 75+})\)
- The joint probability \(P(\text{Attended} \cap \text{Scored 75+})\)
- The conditional probability \(P(\text{Scored 75+} \mid \text{Attended})\)
- The conditional probability \(P(\text{Scored 75+} \mid \text{Did Not Attend})\)
Convert the frequency table to a relative frequency table (all entries as proportions of the grand total).
Explain what the conditional probabilities tell you. How do they compare to your tree diagram results from Task 6?

TASK 8

Pictograph Design & Critique 3 pts

RSI phase: Communication & visualization ethics.

Create a pictograph that displays the average exam score for students who attended the review session vs those who did not. Use icons (books, stars, or similar) where each icon represents 10 points.
Critique your own pictograph: does the visual area of the icons accurately represent the data proportions? If you used differently-sized icons, could this mislead a viewer?
Find one example (from a news article, textbook, or online source) of a misleading pictograph or bar chart. Describe what makes it misleading and how you would fix it.

Discussion Piece

Discussion: "Correlation Does Not Imply Causation — But What Does It Imply?"

You found a strong positive correlation between study hours and exam scores. You also found that attending a review session is associated with higher scores. Your regression model can predict exam scores from study hours.

But can you conclude that studying more causes higher scores?

In a 200–300 word discussion post, address the following:

The causal boundary: What is the difference between "studying more is associated with higher scores" and "studying more causes higher scores"? Use at least two confounding variables from this dataset or beyond to explain the distinction.
The design question: What study design would you need to make a causal claim — observational or experimental? Describe what such a study would look like for this research question. What ethical or practical concerns might arise?
The communication question: If you were advising a student who wants to improve their exam score, how would you communicate your findings honestly — using correlation and prediction language without overstating causation? Write one sentence you would say to the student.

Response requirements: 200–300 words. Use proper statistical terminology (correlation, confounding, association, prediction, causal). Reference at least one specific number from your analysis. Submit by the due date and respond to at least one classmate's post.

Interactive Self-Checks

Before submitting, verify your understanding with these predict-and-reveal cards. If you get one wrong, review the module reading before continuing.

Scatterplots · predict before reveal

A scatterplot shows points rising from lower-left to upper-right with only minor scatter. What direction, form, and strength should you report?

Direction: positive association. Form: roughly linear. Strength: strong (points are tightly clustered around a straight-line trend with little vertical spread).

Correlation · predict before reveal

If \(r = 0.93\) between study hours and exam scores, does this prove that increasing study hours will cause exam scores to rise?

No. Correlation measures the strength and direction of a linear association, not causation. Confounding variables (motivation, prior knowledge, sleep, course difficulty) may explain part or all of the relationship. A causal claim requires an experimental design with random assignment or a quasi-experimental design with strong controls.

Regression · predict before reveal

You compute a regression line \(\hat{y} = 48 + 4x\). What does the slope of 4 mean in context (study hours predicting exam score)?

For each additional hour of study per week, the model predicts an increase of approximately 4 points on the exam, on average. This is a prediction from the fitted line, not a guarantee for any individual student.

Sets · predict before reveal

In a sample of 12 students, 6 scored 75+ (set \(A\)) and 5 attended review (set \(B\)). If 4 students are in both \(A\) and \(B\), what is \(P(A \cup B)\)?

Using the addition rule: \(P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{6}{12} + \frac{5}{12} - \frac{4}{12} = \frac{7}{12} \approx 0.583\). So about 58.3% of students scored 75+ or attended review (or both).

Tree diagrams · predict before reveal

A tree diagram shows \(P(\text{Review}) = 0.5\) and \(P(\text{75+} \mid \text{Review}) = 0.83\). What is \(P(\text{Review} \cap \text{75+})\)?

Using the multiplication rule: \(P(\text{Review} \cap \text{75+}) = P(\text{Review}) \times P(\text{75+} \mid \text{Review}) = 0.5 \times 0.83 = 0.417\). About 41.7% of students (5 out of 12) both attended review and scored 75+.

Two-way tables · predict before reveal

In a two-way table, the conditional probability \(P(\text{75+} \mid \text{Attended}) = 0.83\) and \(P(\text{75+} \mid \text{Did Not Attend}) = 0.17\). Are the events "Attended" and "Scored 75+" independent?

No. If the events were independent, the conditional probability would equal the marginal probability regardless of attendance. Since \(P(\text{75+} \mid \text{Attended}) \neq P(\text{75+} \mid \text{Did Not Attend})\), attendance and scoring 75+ are associated (dependent). This association does not prove attendance causes higher scores — confounding is still possible.

Submission Checklist

Task 1: Variable classification with justifications (3 pts)
Task 2: Two labeled scatterplots with written descriptions (4 pts)
Task 3: Two correlation calculations with interpretation and causal-boundary discussion (4 pts)
Task 4: Regression line equation, interpretations, prediction, and residual analysis (5 pts)
Task 5: Set listings and probability calculations with addition-rule verification (4 pts)
Task 6: Tree diagram with labeled probabilities and conditional-probability interpretations (3 pts)
Task 7: Two-way frequency table with marginal, joint, and conditional probabilities (4 pts)
Task 8: Pictograph, self-critique, and misleading-graph example (3 pts)
Discussion post (200–300 words) + response to one classmate
All work uses neutral examples — no demographic-based claims
All statistical language is defensible (association vs causation)

Grading Rubric (30 points)

Criterion	Excellent (Full)	Proficient (Partial)	Developing (Half)	Beginning (Zero)
Variable Classification (3)	All variables correctly classified with units; study type correctly identified with justification	Most correct, minor errors in justification	Classification errors but some understanding shown	Missing or fundamentally incorrect
Scatterplots (4)	Both scatterplots correctly drawn with labels; direction, form, strength accurately described; outliers noted	Scatterplots drawn, descriptions mostly accurate	One scatterplot or vague descriptions	Missing or unreadable
Correlation (4)	Both \(r\) values correctly calculated; interpretation precise; causal-boundary discussion names 2+ confounders	Calculations correct, interpretation vague	Calculation errors but method shown	Missing or no causal discussion
Regression (5)	Correct slope/intercept, equation, context interpretations, valid prediction, residual analysis with plot	Correct equation, partial interpretations	Equation errors but method attempted, minimal residuals	Missing or no residual analysis
Sets & Probability (4)	All sets listed, all probabilities correct, addition rule verified, independence checked with reasoning	Most probabilities correct, minor set errors	Several errors but approach shown	Missing or fundamentally incorrect
Tree Diagram (3)	Correctly constructed tree with labeled probabilities; all conditional probabilities computed; thoughtful interpretation	Tree correct, minor probability errors	Tree incomplete or errors throughout	Missing
Two-Way Table (4)	Complete table with marginals, all probabilities correct, relative frequency table included, clear comparison to tree	Table complete, most probabilities correct	Table incomplete, several errors	Missing
Pictograph (3)	Accurate pictograph, thoughtful self-critique, valid misleading-graph example with explanation	Pictograph correct, critique superficial	Pictograph errors, minimal critique	Missing
Discussion (as above)	200–300 words, references data, uses correct terminology, distinguishes association/causation, responds to classmate	Mostly correct, minor terminology issues	Short, vague, or missing causal discussion	Missing or no response to classmate

How to Submit

Format: Submit a single PDF or document containing all 8 tasks and your discussion post. Include all calculations, tables, and graphs. You may hand-draw graphs and scan them, or use software.

Discussion: Post your 200–300 word response in the discussion forum by the due date. Respond to at least one classmate's post with a substantive comment (50+ words).

Due: October 2 · Total: 30 points (project) + discussion participation

Academic honesty note. You may use calculators, spreadsheets, and AI tools to assist with calculations and check your work. However, all written interpretations, explanations, and discussion responses must be in your own words. Copying AI-generated text without understanding or attribution violates course policy. If you use AI to check your work, include a brief note describing how you used it.

QuantegyAI · Learn Smarter. Pass Faster. · This project aligns with Module 6 competencies: scatterplots, correlation, regression, sets, probability, tree diagrams, sample spaces, data classification, pictographs, and two-way frequency tables.

Student	Study Hours / Week	Practice Problems	Attended Review	Exam Score
1	3	15	No	58
2	5	22	No	64
3	4	18	Yes	67
4	7	30	No	72
5	6	25	Yes	75
6	8	35	Yes	81
7	10	40	No	85
8	9	38	Yes	88
9	12	50	Yes	94
10	2	10	No	52
11	11	45	Yes	91
12	6	28	No	70

Student	Study Hours / Week	Practice Problems	Attended Review	Exam Score
1	3	15	No	58
2	5	22	No	64
3	4	18	Yes	67
4	7	30	No	72
5	6	25	Yes	75
6	8	35	Yes	81
7	10	40	No	85
8	9	38	Yes	88
9	12	50	Yes	94
10	2	10	No	52
11	11	45	Yes	91
12	6	28	No	70

Student	Study Hours / Week	Practice Problems	Attended Review	Exam Score
1	3	15	No	58
2	5	22	No	64
3	4	18	Yes	67
4	7	30	No	72
5	6	25	Yes	75
6	8	35	Yes	81
7	10	40	No	85
8	9	38	Yes	88
9	12	50	Yes	94
10	2	10	No	52
11	11	45	Yes	91
12	6	28	No	70