Linear Regression Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Linear Regression.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a straight line that minimizes the sum of squared distances from data points to the line (least squares method).

Given scattered points, draw the 'best' line through them. 'Best' means the line that's closest to all points on average. This line lets you predict Y from X.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Linear Regression asks whether the same cases connect two variables or groups in a pattern that can be described carefully.

Common stuck point: Students often know a procedure related to linear regression but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I studying a relationship between variables, and have I separated association from causation?

Worked Examples

Example 1

medium

A regression model of plant height (cm) on water (mL) gives

\hat{y} = 0.1x + 4

. Interpret slope and intercept in context.

Plant height vs water: ŷ = 0.1x + 4; slope = 0.1 cm/mL, intercept = 4 cm

Answer

\text{slope }0.1\text{ cm/mL; intercept }4\text{ cm at }x=0

First step

Slope: each additional mL of water raises predicted height by

0.1

cm.

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Example 2

medium

Data: (

\bar{x},\bar{y}) = (5, 30)

s_x = 2

s_y = 8

r = 0.75

. Write the regression equation.

Regression line ŷ = 3x + 15 passes through the centroid (x̄=5, ȳ=30)

Example 3

hard

For data with

\sum x = 20

\sum y = 60

\sum xy = 250

\sum x^2 = 90

n = 5

, find the regression line.

Example 4

hard

You scale all

y

values by

2

but leave

x

unchanged. How do slope and intercept of the new regression compare to the old?

Example 5

hard

A regression line is

\hat{y} = 2.5 + 1.8x

, where

x

is hours studied and

\hat{y}

is predicted exam score. Interpret the slope and y-intercept.

Example 6

hard

Using

\hat{y} = 10 + 3x

, predict

y

when

x = 5

. Is it appropriate to predict for

x = 50

if the data ranged from

x = 1

x = 10

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A regression line is

\hat{y} = 3x + 2

. Predict y when

x = 5

Regression line ŷ = 3x + 2; find the predicted value at x = 5

Example 2

easy

In the regression equation

\hat{y} = 4x + 9

, what is the slope?

Example 3

easy

\hat{y} = 2x + 6

, interpret the slope in words.

Example 4

easy

Linear regression fits a line by minimizing what?

Example 5

easy

Regression line

\hat{y} = -2x + 30

. Predict y at

x = 4

Regression line ŷ = −2x + 30; find the predicted value at x = 4

Example 6

easy

\hat{y} = 5x + 8

, what is the intercept and what does it mean?

ŷ = 5x + 8; the intercept a = 8 is the predicted y at x = 0

Example 7

easy

Does the regression equation

\hat{y}=2x+1

tell us x causes y?

Example 8

easy

Regression line

\hat{y}=0.5x+10

. Predict y at

x=20

Regression line ŷ = 0.5x + 10; find the predicted value at x = 20

Example 9

medium

From summary stats

\bar{x}=10

\bar{y}=50

, slope

b=3

, find the regression intercept

a

Example 10

medium

Given slope

b=r\frac{s_y}{s_x}

with

r=0.6

s_y=10

s_x=4

, find the regression slope.

Example 11

medium

A regression of crop yield on rainfall is built from rainfall 10-40 mm. Why shouldn't it predict yield at 200 mm?

Example 12

medium

Regression line

\hat{y}=2.5x+4

. The number of hours studied is x and the test score is

\hat{y}

. How much does each extra hour raise the predicted score?

Slope of ŷ = 2.5x + 4: each additional hour raises predicted score by 2.5 points

Example 13

medium

A regression of ice-cream sales on temperature shows a strong positive slope. Why is concluding 'heat causes more sales' overreaching from regression alone?

Example 14

medium

Regression predicts

\hat{y}=2x+5

. At

x=10

the observed value is 22. Find the residual.

Residual at x=10: observed y=22 vs predicted ŷ=25; residual = 22−25 = −3

Example 15

medium

After fitting a regression line, the residual plot shows a clear curved pattern. What does this indicate?

Example 16

medium

A regression gives

\hat{y}=1.2x+3

with

r=0.9

. What is

R^2

, and what does it mean?

Example 17

medium

Regression line

\hat{y}=-1.5x+40

models defects vs machine speed. Predict

\hat{y}

x=10

and state whether higher speed predicts more or fewer defects.

Regression line ŷ = −1.5x + 40; predict defects at machine speed x = 10

Example 18

challenge

Data:

\bar{x}=5

\bar{y}=20

s_x=2

s_y=8

r=0.75

. Write the regression equation

\hat{y}=a+bx

Example 19

challenge

Regression

\hat{y}=4x+2

from data with x from 1 to 6. A student predicts

\hat{y}

x=6

as 26 and at

x=20

as 82. Which prediction is trustworthy and why?

Example 20

challenge

Two regression models for the same data: Model A has

R^2=0.95

using 1 predictor; Model B has

R^2=0.96

but adds 5 extra predictors. Why might Model A be preferred?

Example 21

easy

Regression line

\hat{y} = 6x - 1

. Predict

y

x = 4

Regression line ŷ = 6x − 1; predict ŷ at x = 4

Example 22

easy

Regression line

\hat{y} = -0.5x + 12

. Predict

y

x = 6

Regression line ŷ = −0.5x + 12; predict ŷ at x = 6

Example 23

easy

\hat{y} = 4x + 11

, by how much does predicted

y

change when

x

increases by

1

Slope of ŷ = 4x + 11: predicted y rises by 4 for each unit increase in x

Example 24

easy

Regression line

\hat{y} = 2.5x + 4

. Find

\hat{y}

x = 0

ŷ = 2.5x + 4; the intercept a = 4 is the predicted y at x = 0

Example 25

medium

From

\bar{x}=6

\bar{y}=20

, slope

b = 2.5

, find the regression intercept.

Example 26

medium

Given

r = -0.8

s_y = 5

s_x = 2

, find the regression slope.

Example 27

medium

A regression with slope

b = 0

produces what fitted line?

Example 28

medium

A regression slope is

b = 3

with

s_x = 4

and

s_y = 20

. Find the correlation

r

Example 29

medium

Regression line

\hat{y} = 7x + 50

. Solve for

x

when

\hat{y} = 134

Regression line ŷ = 7x + 50; solve for x when ŷ = 134

Example 30

medium

You shift all

x

values by

+10

without changing

y

. Does the regression slope change?

Example 31

hard

A regression on

x

[1,10]

gives

\hat{y} = 4x + 3

. Predict

\hat{y}

x = 100

and comment on reliability.

Regression fit ŷ = 4x + 3; data range is x∈[1,10] — predictions beyond x=10 are extrapolation

Example 32

hard

A perfect linear data set has

r = 1

. What is the relation between slope

b

and the standard deviations

s_x,s_y

Example 33

hard

Why might removing one influential outlier dramatically change the regression slope?

Example 34

hard

Regression residuals are plotted vs.

\hat{y}

and show a clear curved pattern. What does this suggest about the linear model?

Example 35

challenge

For data

(0,1),(1,3),(2,2),(3,5),(4,4)

, find the least-squares regression line.

Scatter plot with least-squares regression line ŷ = 0.8x + 1.4

Example 36

hard

A regression equation is

\hat{y} = 50 - 2.3x

. Interpret the slope. If

x = 8

, find

\hat{y}

Example 37

hard

A regression line is

\hat{y} = 4 + 1.5x

. Predict

y

when

x = 12

, and decide whether this is interpolation if the observed x-values ranged from 5 to 15.

← Back to Linear Regression Practice Mode

Related Concepts

Scatter Plot Correlation Line of Best Fit Residuals R-Squared (Coefficient of Determination)

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat scatter plotcorrelation introline of best fit