🎯⭐ INTERACTIVE LESSON

Coefficient of Determination

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Coefficient of Determination - Complete Interactive Lesson

Part 1: Scatterplots and Correlation

📈 Scatterplots and Correlation

Part 1 of 7 — Exploring Bivariate Relationships

Describing Scatterplots

When examining a scatterplot, describe:

Feature	Options
Direction	Positive, negative, or none
Form	Linear, curved, or no pattern
Strength	Strong, moderate, or weak
Outliers	Any points that don't fit the pattern

Correlation Coefficient $r$

The correlation $r$ measures the strength and direction of a linear relationship:

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

Value of $r$	Interpretation
$r = 1$	Perfect positive linear
$r = -1$	Perfect negative linear
$r = 0$

Important Properties of $r$

$-1 \leq r \leq 1$ always
$r$ has no units
$r$ is not affected by changes in units (e.g., inches to cm)
$r$ measures only linear association — a strong curved relationship can have

🔑 Correlation does NOT imply causation. A strong correlation between two variables does not mean one causes the other.

Correlation Check 🎯

Correlation Practice 🧮

1) If $r = 0.72$ , what is $r^2$ ? (Round to 2 decimal places)

2) What percentage of variation in $y$ is explained by the linear relationship with $x$ ? (Use from #1, express as a whole number)

Part 2: Least-Squares Regression Line

📊 Least-Squares Regression Line

Part 2 of 7 — The LSRL

Topics in This Part

Section
📐 What the LSRL Minimizes
🧮 The Equation & Slope/Intercept
📝 Interpreting Slope and Intercept
📊 Predictions & Extrapolation

🔑 Key Concept: The least-squares regression line (LSRL) is the line that minimizes the sum of the squared residuals — the best-fit line through a scatterplot.

The LSRL Equation

$\boxed{\hat{y} = a + bx}$

Part 3: Residuals and Residual Plots

📊 Residuals and Residual Plots

Part 3 of 7 — Assessing the Fit of a Linear Model

Topics in This Part

Section
📐 What Is a Residual?
📊 Residual Plots
✅ Good vs. Bad Patterns
📝 Worked Example

🔑 Key Concept: A residual is the vertical distance from a data point to the regression line. Residual plots help us assess whether a linear model is appropriate.

Residual Formula

$\boxed{e_i = y_i - \hat{y}_i = \text{observed} - \text{predicted}}$

Part 4: Coefficient of Determination

📊 Coefficient of Determination

Part 4 of 7 — Understanding $r^2$

Topics in This Part

Section
📐 What $r^2$ Measures
🧮 Calculating $r^2$ from

Part 5: Influential Points and Outliers

📊 Influential Points and Outliers

Part 5 of 7 — Leverage, Influence, and Unusual Observations

Topics in This Part

Section
⚠️ Outliers in Regression
📐 High-Leverage Points
🔄 Influential Points
🧪 Diagnosing Unusual Points

🔑 Key Concept: Not all unusual points are equally problematic. Some change the regression line dramatically (influential), while others are just far from the pattern (outliers).

Three Types of Unusual Points

1. Outlier (in $y$ -direction)

A point whose $y$ -value is far from the predicted $\hat{y}$ (large residual)

Part 6: Problem-Solving Workshop

📊 Problem-Solving Workshop

Part 6 of 7 — Full Regression Analysis Problems

Workshop Goals

Skill
📐 Compute and interpret the LSRL
📝 Interpret slope, intercept, $r$ , and $r^2$ in context
📉 Analyze residuals and residual plots
⚠️ Identify unusual/influential points
🎯 Recognize the limits of the model

🔑 AP Tip: Free-response regression questions typically ask you to interpret slope/ $^{}$ in context, describe the residual plot, and discuss whether the model is appropriate.

Part 7: Review & Applications

📊 Review & Applications

Part 7 of 7 — Comprehensive Linear Regression Review

Complete Formula Reference

Concept	Formula
LSRL	$\hat{y} = a + bx$
Slope

r

is sensitive to outliers

3) If every data point falls exactly on the line $y = 3x + 2$ , then $r =$ ?

$b = r \cdot \frac{s_y}{s_x}$ (slope)
$a = \bar{y} - b\bar{x}$ (intercept)
The line always passes through the point $(\bar{x}, \bar{y})$

$\text{LSRL minimizes } \sum (y_i - \hat{y}_i)^2 = \sum e_i^2$

This is the sum of squared residuals — hence "least squares."

Template: "For each additional [1 unit of $x$ ], the predicted [y variable] changes by [ $b$ units], on average."

Example: $\hat{y} = 12 + 3.5x$ where $x$ = hours studied, $y$ = exam score.

✅ "For each additional hour studied, the predicted exam score increases by 3.5 points, on average."

⚠️ AP Tip: Include "predicted" and "on average" for full credit.

Interpreting Intercept

Template: "When $x = 0$ , the predicted [y variable] is [ $a$ ]."

Example: $a = 12$ in $\hat{y} = 12 + 3.5x$ .

✅ "When a student studies 0 hours, the predicted exam score is 12 points."

⚠️ Caution: The intercept often has no practical meaning (e.g., studying 0 hours). State the interpretation but note if $x = 0$ is outside the data range.

Predictions & Extrapolation

Term	Definition
Interpolation	Predicting within the range of observed $x$ values ✓
Extrapolation	Predicting outside the range of observed $x$ values ⚠️

⚠️ Extrapolation is unreliable. The linear relationship may not hold outside the data range.

LSRL Concepts 🎯

LSRL Calculations 🧮

Given: $\bar{x} = 10$ , $\bar{y} = 25$ , $s_x = 4$ , $s_y = 8$ , $r = 0.85$ .

1) What is the slope $b$ ?

2) What is the intercept $a$ ?

3) What is $\hat{y}$ when $x = 12$ ?

Interpretation Practice 🔍

$\hat{y} = 50 - 0.8x$ where $x$ = temperature (°F), $y$ = hot chocolate sales.

Exit Quiz — LSRL ✅

Sign	Meaning
$e > 0$	Point is above the line — model underestimates
$e < 0$	Point is below the line — model overestimates
$e = 0$	Point is exactly on the line

Properties of Residuals

$\sum e_i = 0$ (residuals always sum to zero)
The mean of residuals = 0
$\sum e_i^2$ is minimized by the LSRL

A residual plot plots residuals ( $e$ ) on the $y$ -axis vs. the explanatory variable ( $x$ ) or fitted values ( $\hat{y}$ ) on the $x$ -axis.

Reading Residual Plots

Pattern	Interpretation
Random scatter around $e = 0$	✅ Linear model is appropriate
Curved pattern (U or ∩)	❌ Relationship is nonlinear — use a transformation
Fan shape (spread changes)	❌ Non-constant variance — predictions are less reliable at some $x$ values
Outliers	⚠️ Individual points far from $e = 0$ — investigate

🔑 AP Tip: The residual plot is your most important diagnostic tool. ALWAYS examine it before trusting a regression.

$\hat{y} = 10 + 2x$ . Data point: $(5, 23)$ .

$\hat{y} = 10 + 2(5) = 20$ $e = 23 - 20 = 3$

The residual is $+3$ : the observed value is 3 units above the predicted value.

Residual Concepts 🎯

Residual Calculations 🧮

LSRL: $\hat{y} = 15 + 4x$

1) Point $(3, 30)$ . Residual $e =$

2) Point $(5, 33)$ . Residual $e =$

3) Point $(2, 23)$ . The model ___ (enter "overestimates" or "underestimates").

Residual Plot Patterns 🔍

Exit Quiz — Residuals ✅

🔑 Key Concept: $r^2$ tells you the fraction of variability in $y$ that is explained by the linear relationship with $x$ .

$\boxed{r^2 = \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\text{SSE}}{\text{SST}}}$

SST = total sum of squares = $\sum(y_i - \bar{y})^2$ (total variability in $y$ )
SSE = sum of squared errors = $\sum(y_i - \hat{y}_i)^2$ (unexplained variability)
SSR = regression sum of squares = SST $-$ SSE (explained variability)

Or simply: $r^2 = r \times r$ (square the correlation coefficient).

Interpretation Template

$\text{"[}r^2 \times 100\text{]\% of the variability in [y context] is explained by the linear relationship with [x context]."}$

Example: $r^2 = 0.72$ , $x$ = hours studied, $y$ = exam score.

✅ "72% of the variability in exam scores is explained by the linear relationship with hours studied."

⚠️ AP Tip: Always say "variability in [y]" and "linear relationship with [x]." Do not say "caused by" or "due to."

Statistic	Measures	Range
$r$	Direction and strength of linear relationship	$-1 \leq r \leq 1$
$r^2$	Proportion of variability explained	$0 \leq r^2 \leq 1$

$r$	$r^2$	Strength
$\pm 0.9$	$0.81$	Strong
$\pm 0.7$	$0.49$	Moderate
$\pm 0.5$	$0.25$	Weak
$\pm 0.3$	$0.09$	Very weak

🔑 Key Insight: Even a "moderate" $r = 0.7$ only explains 49% of the variability. Much variation remains unexplained.

$r^2$ Concepts 🎯

$r^2$ Calculations 🧮

1) $r = 0.9$ . What is $r^2$ ?

2) $r^2 = 0.49$ . What percentage of variability is explained?

3) SST = 500, SSE = 125. What is $r^2$ ?

Interpretation Practice 🔍

Exit Quiz — $r^2$ ✅

Has an unusually large

|\text{residual}|

Does NOT necessarily change the regression line much

2. High-Leverage Point (in $x$ -direction)

A point whose $x$ -value is far from $\bar{x}$
Has the potential to influence the regression line
May or may not actually change the line — depends on where it falls

3. Influential Point

A point that, when removed, substantially changes the slope, intercept, or $r^2$
High-leverage points that are also outliers are the most influential
Test: Fit the LSRL with and without the point. If slope/intercept/ $r^2$ changes a lot, the point is influential.

Visualizing the Distinction

Scenario	Large Residual?	Far from $\bar{x}$ ?	Influential?
Regular point near center	No	No	No
Outlier near center of $x$	Yes	No	Usually no
Point at extreme $x$ , on the line	No	Yes	Usually no
Point at extreme $x$ , off the line	Yes	Yes	Yes

A researcher collects data on advertising spending ( $x$ , in thousands) and sales ( $y$ , in thousands) for 10 stores:

Most stores spend $2K–$8K. One store spent $25K (high leverage).

Scenario A: That store had $50K in sales, fitting the overall pattern → high leverage but NOT influential.
Scenario B: That store had $5K in sales, far below the trend → high leverage AND influential. Removing it would substantially change the slope.

⚠️ AP Tip: On the AP exam, "influential" specifically means removing the point changes the regression equation meaningfully. Always describe the effect on slope, intercept, or $r^2$ .

What to Do with Unusual Points

Investigate — is there a data-entry error or special circumstance?
Report both analyses — with and without the point
Never silently delete data — explain your reasoning
Check the residual plot — unusual points often show up clearly

Identifying Unusual Points 🎯

Diagnosing Points 🧮

1) The LSRL is $\hat{y} = 10 + 3x$ . A point has $x = 5, y = 35$ . What is the residual?

2) $\bar{x} = 12$ . A point has $x = 45$ . Is this point high-leverage? (yes/no)

3) With all points: slope $= 1.8$ . Without point A: slope $= 1.7$ . Without point B: slope $= 4.5$ . Which point is more influential? (A/B)

Leverage and Influence Concepts 🔍

Exit Quiz — Influential Points & Outliers ✅

Worked Example 1 — Temperature and Ice Cream Sales

A manager records daily high temperature ( $x$ , °F) and ice cream sales ( $y$ , $100s) for 15 summer days.

Computer output:

Predictor	Coef	SE Coef	T	P
Constant	$-3.50$	$1.12$	$-3.13$	$0.008$
Temperature	$0.15$	$0.013$	$11.54$	$< 0.001$

$S = 0.96 \quad R\text{-}sq = 91.1\%$

Step 1 — Write the LSRL: $\hat{y} = -3.50 + 0.15x$

Step 2 — Interpret the slope: "For each additional degree Fahrenheit increase in daily high temperature, the predicted ice cream sales increase by $15 (0.15 hundreds)."

Step 3 — Interpret $r^2$ : "91.1% of the variability in ice cream sales is explained by the linear relationship with daily high temperature."

Step 4 — Predict: At $x = 85°$ F: $\hat{y} = -3.50 + 0.15(85) = -3.50 + 12.75 = 9.25 \text{ (\$925 in sales)}$

Step 5 — Check appropriateness:

Residual plot shows no obvious pattern → linear model is appropriate
$r^2 = 0.911$ → strong linear fit
No influential points observed in the residual plot

Worked Example 2 — Study Hours and GPA

A sample of 30 college students. $x$ = weekly study hours, $y$ = GPA.

LSRL: $\hat{y} = 1.85 + 0.052x$ , $r = 0.68$ , $r^2 = 0.462$

One student studies 42 hours/week (most study 5–25 hours) and has a GPA of 3.9.

Slope interpretation: "For each additional hour of weekly studying, GPA is predicted to increase by 0.052 points."
$r^2$ interpretation: "46.2% of the variability in GPA is explained by the linear relationship with weekly study hours."
The 42-hour student:
- $\hat{y} = 1.85 + 0.052(42) = 4.034$ — predicted GPA is 4.034
- Residual $= 3.9 - 4.034 = -0.134$ — small residual
- $x = 42$ is far from $\bar{x}$ → high leverage
- But residual is small → likely not influential (on the trend line)
Prediction for 50 hours: $\hat{y} = 1.85 + 0.052(50) = 4.45$
- This is extrapolation (beyond data range) and the prediction exceeds 4.0 (max GPA) — unreliable!

Common Mistakes on the AP Exam

Mistake	Correction
"Temperature causes sales to increase"	Use "is associated with" or "predicts"
"91.1% of the data falls on the line"	" $r^2$ measures variability explained, not % of points on the line"
Interpreting the intercept literally when $x = 0$ is outside the data	"The intercept has no practical interpretation because $x = 0$ is outside the range of data"
Forgetting units in slope interpretation	"For each additional [unit of x], [y] is predicted to [increase/decrease] by [slope] [units of y]"

Regression Analysis Practice 🎯

Computations 🧮

LSRL: $\hat{y} = 5.2 + 1.3x$ , $r = 0.85$

1) Predict $y$ when $x = 10$ .

2) What is $r^2$ ? (two decimal places)

3) Observed $y = 22$ when $x = 10$ . What is the residual?

Interpretation Decisions 🔍

Exit Quiz — Regression Workshop ✅

b = r \cdot \frac{s _{y}}{s _{x}}

Interpretation Templates (AP Exam Ready)

Slope: "For each additional [1 unit of x], the predicted [y in context] [increases/decreases] by [|b|] [units of y]."

Intercept: "When [x in context] is 0, the predicted [y in context] is [a] [units of y]." (Only if $x = 0$ is in the data range.)

$r$ : "There is a [strong/moderate/weak], [positive/negative], linear association between [x] and [y]."

$r^2$ : "[ $r^2 \times 100$ ]% of the variability in [y in context] is explained by the linear relationship with [x in context]."

Residual: "The actual [y in context] was [e] [units] [above/below] the value predicted by the model."

Key Concepts Summary

Topic	Key Takeaway
Scatterplot	Always plot data first; describe direction, form, strength, unusual features
LSRL	Minimizes $\sum e_i^2$ ; passes through $(\bar{x}, \bar{y})$ ; $\sum e_i = 0$
Slope & Intercept	Slope = rate of change; intercept = starting value (if meaningful)
Residuals	$e = y - \hat{y}$ ; residual plot checks model appropriateness
$r$	Direction + strength; $-1 \leq r \leq 1$ ; only for linear relationships
$r^2$	Proportion of variability explained; $0 \leq r^2 \leq 1$
Outliers	Large residual; may or may not be influential
High Leverage	Extreme $x$ -value; potential to influence
Influential	Removing changes slope/ $r^2$ substantially
Extrapolation	Predicting outside data range — unreliable

$\text{Is the relationship linear?}$ $\downarrow$ $\text{Check scatterplot → Fit LSRL → Check residual plot}$ $\downarrow$ $\text{Random scatter → Linear model OK}$ $\text{Curved pattern → Nonlinear model needed}$ $\downarrow$ $\text{Interpret: slope, } r, r^2 \text{ in context}$ $\downarrow$ $\text{Check for unusual points}$ $\downarrow$ $\text{Make predictions (within data range only)}$

🔑 AP Exam Strategy: Regression appears on the exam every year. Master the interpretation templates — they earn you full credit on free-response questions.

Comprehensive Review 🎯

Mixed Calculations 🧮

Given: $\bar{x} = 10$ , $\bar{y} = 25$ , $s_x = 4$ , $s_y = 8$ , $r = 0.75$ .

1) Calculate the slope $b$ .

2) Calculate the intercept $a$ .

3) What is $r^2$ ? (two decimal places)

Concept Connections 🔍

Final Exam — Linear Regression ✅

Coefficient of Determination

What LSRL Minimizes

Interpreting Slope

Interpreting Intercept

Predictions & Extrapolation

Properties of Residuals

Residual Plots

Reading Residual Plots

Worked Example

The Definition

Interpretation Template

$r$ vs. $r^2$

Visualizing the Distinction

Worked Example

What to Do with Unusual Points

Worked Example 1 — Temperature and Ice Cream Sales

Worked Example 2 — Study Hours and GPA

Common Mistakes on the AP Exam

Interpretation Templates (AP Exam Ready)

Key Concepts Summary

Decision Guide

Coefficient of Determination

Coefficient of Determination - Complete Interactive Lesson

Part 1: Scatterplots and Correlation

📈 Scatterplots and Correlation

Describing Scatterplots

Correlation Coefficient rrr

Important Properties of rrr

Part 2: Least-Squares Regression Line

📊 Least-Squares Regression Line

Topics in This Part

The LSRL Equation

Part 3: Residuals and Residual Plots

📊 Residuals and Residual Plots

Topics in This Part

Residual Formula

Part 4: Coefficient of Determination

📊 Coefficient of Determination

Topics in This Part

Part 5: Influential Points and Outliers

📊 Influential Points and Outliers

Topics in This Part

Three Types of Unusual Points

Part 6: Problem-Solving Workshop

📊 Problem-Solving Workshop

Workshop Goals

Part 7: Review & Applications

📊 Review & Applications

Complete Formula Reference

What LSRL Minimizes

Interpreting Slope

Interpreting Intercept

Predictions & Extrapolation

Properties of Residuals

Residual Plots

Reading Residual Plots

Worked Example

The Definition

Interpretation Template

rrr vs. r2r^2r2

Visualizing the Distinction

Worked Example

What to Do with Unusual Points

Worked Example 1 — Temperature and Ice Cream Sales

Worked Example 2 — Study Hours and GPA

Common Mistakes on the AP Exam

Interpretation Templates (AP Exam Ready)

Key Concepts Summary

Decision Guide

Correlation Coefficient $r$

Important Properties of $r$

$r$ vs. $r^2$