Transformations for Linearity

📈 Scatterplots and Correlation

Part 1 of 7 — Exploring Bivariate Relationships

Describing Scatterplots

When examining a scatterplot, describe:

Correlation Coefficient $r$

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

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

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 $r \approx 0$
• $r$ is sensitive to outliers

🔑 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 $r^2$ from #1, express as a whole number)

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

📊 Least-Squares Regression Line

Part 2 of 7 — The LSRL

Topics in This Part

🔑 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

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

where:

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

What LSRL Minimizes

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

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

Interpreting Slope

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

⚠️ 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 ✅

📊 Residuals and Residual Plots

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

Topics in This Part

🔑 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

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

Properties of Residuals

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

Residual Plots

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

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

Worked Example

$\hat{y} = 10 + 2x$. 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 ✅

📊 Coefficient of Determination

Part 4 of 7 — Understanding $r^2$

Topics in This Part

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

The Definition

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

where:

• 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."

$r$ vs. $r^2$

🔑 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 — $r^2$ ✅

📊 Influential Points and Outliers

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

Topics in This Part

🔑 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)
• 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

Worked Example

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

1. Investigate — is there a data-entry error or special circumstance?
2. Report both analyses — with and without the point
3. Never silently delete data — explain your reasoning
4. 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 ✅

📊 Problem-Solving Workshop

Part 6 of 7 — Full Regression Analysis Problems

Workshop Goals

🔑 AP Tip: Free-response regression questions typically ask you to interpret slope/$r^2$ in context, describe the residual plot, and discuss whether the model is appropriate.

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:

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

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.

Analysis:

1. Slope interpretation: "For each additional hour of weekly studying, GPA is predicted to increase by 0.052 points."

2. $r^2$ interpretation: "46.2% of the variability in GPA is explained by the linear relationship with weekly study hours."

3. 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)

4. 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

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 ✅

📊 Review & Applications

Part 7 of 7 — Comprehensive Linear Regression Review

Complete Formula Reference

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

Decision Guide

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

🔑 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 ✅