Statistics

Linear Regression

Lesson

When two variables show a roughly linear relationship, the line of best fit (also called the regression line) is the line that comes closest to all the points. Once you have it, you can predict a value of $y$ from any $x$ .

y = mx + b

The slope and intercept formulas:

m = \frac{n\,\sum xy - \sum x \sum y}{n\,\sum x^2 - (\sum x)^2}

b = \bar{y} - m\,\bar{x}

Where $\bar{x}$ and $\bar{y}$ are the means of the $x$ - and $y$ -values. Calculators and spreadsheets do this for you in practice; for small data sets it’s manageable by hand.

Once you have the line, prediction is just substitution — plug an $x$ value into $y = mx + b$ .

Worked example 1 — fit and predict

Data: (1, 5), (2, 8), (3, 11), (4, 14). The points jump by 3 each time, so the line is exactly $y = 3x + 2$ : slope $m = 3$ , intercept $b = 2$ .

To predict $y$ at $x = 5$ :

y = 3(5) + 2 = 17

Worked example 2 — non-perfect data

Data: (1, 3), (2, 5), (3, 4), (4, 6). Using the formulas: $\bar{x} = 2.5,\ \bar{y} = 4.5$ , $\sum xy = 49,\ \sum x^2 = 30$ .

m = \frac{4(49) - 10(18)}{4(30) - 100} = \frac{16}{20} = 0.8

b = 4.5 - (0.8)(2.5) = 2.5

Regression line: $y = 0.8x + 2.5$ .

How to type your answer

A single number — slope, intercept, or a predicted $y$ value, depending on the question. Use a decimal point if needed. Examples: 3, 2.5, 17, -8.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

\text{Slope of regression line for } (1,2),\ (2,4),\ (3,6),\ (4,8)

Problem 2

\text{y-intercept of regression line for } (1,2),\ (2,4),\ (3,6),\ (4,8)

Problem 3

\text{Slope of regression line for } (0,1),\ (1,3),\ (2,5),\ (3,7)

Problem 4

\text{y-intercept of regression line for } (0,1),\ (1,3),\ (2,5),\ (3,7)

Practice

Standard problems matching the lesson.

Problem 5

\text{Slope for } (1,5),\ (2,8),\ (3,11),\ (4,14)

Problem 6

\text{y-intercept for } (1,5),\ (2,8),\ (3,11),\ (4,14)

Problem 7

\text{Slope for } (1,10),\ (2,8),\ (3,6),\ (4,4)

Problem 8

\text{y-intercept for } (1,10),\ (2,8),\ (3,6),\ (4,4)

Problem 9

\text{Using } y = 3x + 2,\ \text{predict } y \text{ at } x = 5

Problem 10

\text{Using } y = -2x + 12,\ \text{predict } y \text{ at } x = 10

Problem 11

\text{Using } y = 0.5x + 3,\ \text{predict } y \text{ at } x = 8

Problem 12

\text{Using } y = -1.5x + 10,\ \text{predict } y \text{ at } x = 4

Problem 13

\text{Slope for } (2,5),\ (4,9),\ (6,13),\ (8,17)

Problem 14

\text{y-intercept for } (2,5),\ (4,9),\ (6,13),\ (8,17)

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\text{Using } y = 2.5x - 4,\ \text{predict at } x = 12

Problem 16

\text{Using } y = -3x + 25,\ \text{predict at } x = 7

Problem 17

\text{Given } m=4,\ b=-7.\ \text{Predict at } x = 6

Problem 18

\text{Given } m=-0.5,\ b=12.\ \text{Predict at } x = 8

Problem 19

\text{Slope for } (1,3),\ (2,5),\ (3,4),\ (4,6)

Problem 20

\text{y-intercept for } (1,3),\ (2,5),\ (3,4),\ (4,6)

Problem 21

\text{Using } y = 0.8x + 2.5,\ \text{predict at } x = 10

Problem 22

\text{Using } y = -0.8x + 7,\ \text{predict at } x = 5

Practice

Standard problems matching the lesson.

Problem 23

A study-hours-to-grade model: y = 5x + 60. Predict the grade for a student who studies 5 hours.

Problem 24

A car's value (thousands of dollars) after x years: y = -2x + 25. Predict the value at x = 4.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Ad spend predicts sales via y = 3.5x + 12 (both in thousands of dollars). Predict sales when ad spend is 8.

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