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Introduction
Linear Regression helps you connect a mathematical rule to a result you can explain and check. Use the centered-sum formula or calculator regression to find the least-squares line. Interpret the resulting slope and intercept. This lesson belongs to linear functions and modeling and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in linear regression.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Read coordinate pairs, calculate means, and interpret slope.
Previous lesson: Line of Best Fit
Key mathematical vocabulary
- variable
- A letter representing an unknown or changing quantity.
- slope
- Vertical change divided by nonzero horizontal change.
- intercept
- A point where a graph meets an axis.
- correlation
- A measure of association, not proof of cause.
- residual
- Observed output minus predicted output.
Formulas and rules
- a(b+c)=ab+ac
- Equivalent expressions agree for all allowed inputs.
- A solution satisfies every original condition.
Data, association, and linear models
Use the centered-sum formula or calculator regression to find the least-squares line. Interpret the resulting slope and intercept.
A scatterplot records paired quantitative observations. A fitted line describes a trend and usually does not pass through every point. Pearson correlation r measures the direction and strength of a linear association, with −1≤r≤1 when both variables have nonzero variation.
Technology can calculate m=Sxy/Sxx and b=ȳ−m x̄ for a least-squares line, and r=Sxy/√(Sxx Syy). A residual is observed y minus predicted y. A small linear correlation does not rule out a strong nonlinear pattern. Association alone cannot prove causation; confounding variables and study design matter.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. The table has constant change Δy=12 for Δx=1, so slope m=12.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. The fitted line passes through the mean point and, for these exact data, every observation.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
Fit a linear model to (1, 33), (2, 45), (3, 57), (4, 69) and predict y at x=12.
| x | y |
|---|---|
| 1 | 33 |
| 2 | 45 |
| 3 | 57 |
| 4 | 69 |
- The table has constant change Δy=12 for Δx=1, so slope m=12.
- Using the mean point, b=ȳ−m x̄=51−12(2.5)=21. Least-squares technology gives the same result.
- The model is y=12x+21; at x=12, predict 165. Since this is beyond x=4, label the prediction extrapolation.
Answer: y = 12x + 21; prediction 165
Verify: The fitted line passes through the mean point and, for these exact data, every observation.
Avoid this mistake: Interpreting correlation as proof of cause.
Example 2 · Basic
Fit a linear model to (1, 34), (2, 47), (3, 60), (4, 73) and predict y at x=12.
| x | y |
|---|---|
| 1 | 34 |
| 2 | 47 |
| 3 | 60 |
| 4 | 73 |
- The table has constant change Δy=13 for Δx=1, so slope m=13.
- Using the mean point, b=ȳ−m x̄=53.5−13(2.5)=21. Least-squares technology gives the same result.
- The model is y=13x+21; at x=12, predict 177. Since this is beyond x=4, label the prediction extrapolation.
Answer: y = 13x + 21; prediction 177
Verify: The fitted line passes through the mean point and, for these exact data, every observation.
Avoid this mistake: Reversing observed minus predicted when finding a residual.
Example 3 · Intermediate
Fit a linear model to (1, 35), (2, 49), (3, 63), (4, 77) and predict y at x=12.
| x | y |
|---|---|
| 1 | 35 |
| 2 | 49 |
| 3 | 63 |
| 4 | 77 |
- The table has constant change Δy=14 for Δx=1, so slope m=14.
- Using the mean point, b=ȳ−m x̄=56−14(2.5)=21. Least-squares technology gives the same result.
- The model is y=14x+21; at x=12, predict 189. Since this is beyond x=4, label the prediction extrapolation.
Answer: y = 14x + 21; prediction 189
Verify: The fitted line passes through the mean point and, for these exact data, every observation.
Avoid this mistake: Trusting distant extrapolation without checking the context.
Example 4 · Advanced
Fit a linear model to (1, 24), (2, 26), (3, 28), (4, 30) and predict y at x=12.
| x | y |
|---|---|
| 1 | 24 |
| 2 | 26 |
| 3 | 28 |
| 4 | 30 |
- The table has constant change Δy=2 for Δx=1, so slope m=2.
- Using the mean point, b=ȳ−m x̄=27−2(2.5)=22. Least-squares technology gives the same result.
- The model is y=2x+22; at x=12, predict 46. Since this is beyond x=4, label the prediction extrapolation.
Answer: y = 2x + 22; prediction 46
Verify: The fitted line passes through the mean point and, for these exact data, every observation.
Avoid this mistake: Interpreting correlation as proof of cause.
Example 5 · Challenge
Fit a linear model to (1, 25), (2, 28), (3, 31), (4, 34) and predict y at x=12.
| x | y |
|---|---|
| 1 | 25 |
| 2 | 28 |
| 3 | 31 |
| 4 | 34 |
- The table has constant change Δy=3 for Δx=1, so slope m=3.
- Using the mean point, b=ȳ−m x̄=29.5−3(2.5)=22. Least-squares technology gives the same result.
- The model is y=3x+22; at x=12, predict 58. Since this is beyond x=4, label the prediction extrapolation.
Answer: y = 3x + 22; prediction 58
Verify: The fitted line passes through the mean point and, for these exact data, every observation.
Avoid this mistake: Reversing observed minus predicted when finding a residual.
Technology lab: fit and inspect your data
Enter paired measurements to calculate correlation and fit linear, quadratic, and positive exponential models. Compare the rule with the actual context before choosing a model.
Common mistakes and corrections
- Interpreting correlation as proof of cause.
- Reversing observed minus predicted when finding a residual.
- Trusting distant extrapolation without checking the context.
Real-world applications
- Study the association between practice time and performance. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Predict costs from observed production data and inspect prediction errors. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Use the centered-sum formula or calculator regression to find the least-squares line. Interpret the resulting slope and intercept.
A scatterplot records paired quantitative observations. A fitted line describes a trend and usually does not pass through every point. Pearson correlation r measures the direction and strength of a linear association, with −1≤r≤1 when both variables have nonzero variation.
Frequently asked questions
What should I know before linear regression?
Read coordinate pairs, calculate means, and interpret slope.
How can I check my work?
The fitted line passes through the mean point and, for these exact data, every observation. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Interpreting correlation as proof of cause. Reversing observed minus predicted when finding a residual. Trusting distant extrapolation without checking the context.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Study the association between practice time and performance.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Use the centered-sum formula or calculator regression to find the least-squares line. Interpret the resulting slope and intercept.
- Interpreting correlation as proof of cause.
- Reversing observed minus predicted when finding a residual.
- Trusting distant extrapolation without checking the context.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials