Complete Analysis of a Linear Model

You will be able to:

• Run a full six-step analysis of a linear model
• Classify predictions and spot model breakdown
• Explain why association is not causation

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Write the equation and interpret slope and intercept in context
2. Make interpolation and extrapolation predictions
3. Identify when a model breaks down
4. Explain why association does not prove causation

Do Absences Cause Lower Scores?

More absences go with lower scores: . Causation? Predict 25 absences?

Six Moves Make One Workflow

A complete analysis runs these in order:

1. Write the equation
2. Interpret slope
3. Interpret intercept
4. Predict (interpolation)
5. Predict (extrapolation)
6. Association or causation?

Step 1: Write the Equation

Line through and :

Slope:

Intercept: , so

Step 2: Interpret the Slope

• Slope
• For each additional absence, the model predicts the exam score is associated with a 4-point decrease

Step 3: Interpret the Y-Intercept

• Intercept
• At absences, predicted score is 95
• Meaningful: perfect attendance, and is in our data

Step 4: An Interpolation Prediction

Predict at 6 absences:

6 is inside the 0-to-12 range — trustworthy.

Step 5: Extrapolation and Breakdown

• At 20 absences: — beyond the data
• At 25 absences:
• A score of is impossible — the model breaks down

Quick Check: Which Kind of Prediction?

Is predicting the score at 6 absences interpolation or extrapolation?

Decide and justify before advancing.

Step 6: Association Versus Causation

• The data show absences and scores move together
• That is an association
• It does not prove absences cause lower scores

Lurking Variables Can Explain Both

• A hidden third factor can drive both variables
• It creates an association without direct cause

A Classic Example of a Lurking Variable

• Ice cream sales and drowning rates rise together
• Does ice cream cause drowning? No.
• Summer heat drives both

Back to the Absences Question

• Maybe absences directly hurt scores — possibly
• Or illness, or home challenges, affect both
• The data alone cannot tell which

Your Turn: Find the Error

A classmate writes: "This proves that missing class causes lower scores."

1. What is the logical error?
2. Propose a lurking variable

Fix the claim before advancing.

Guided Practice: Walk the Workflow

A new data set: hours of sleep vs. reaction time, line through and .

Step 1: Find the slope — what is the rise over run?

Work step 1, then advance for the rest.

Your Turn: Run a Complete Analysis

Temperature vs. pool visitors, line through and .

Run all six steps: equation, slope, intercept, two predictions, causation.

Work the whole analysis on your own first.

The Complete Story of an Analysis

✓ The equation gives the rule; slope and intercept give meaning
✓ Predictions are safe inside the data, reckless far outside
✓ A strong association still does not prove causation

Coming Up Next: High School Statistics

In high school, you'll fit lines formally with least-squares regression, measure fit with correlation, and learn how experiments can actually establish causation.