Least Squares Regression Line Examples in Math

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

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

Concept Recap

The unique straight line $\hat{y} = a + bx$ that minimizes the sum of squared vertical distances (residuals) between the observed data points and the line.

You have a scatter plot with points scattered around a general trend. The LSRL is the line that gets as close as possible to all the points simultaneously—it's the 'best' straight line through the cloud. 'Best' means it minimizes the total squared prediction error.

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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: The least-squares regression line is the unique line $\hat{y}=a+bx$ minimizing the total squared vertical distance to the data.

Common stuck point: The procedure for least squares regression line is the easy part; the trap is minimizing perpendicular or horizontal distances. Asking "Am I fitting a single straight line to two-variable numeric data by minimizing squared vertical distances?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I fitting a single straight line to two-variable numeric data by minimizing squared vertical distances?

Worked Examples

Example 1

medium

Find the least-squares regression line for:

(x,y)

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

. Use

b = r \frac{s_y}{s_x}

and

a = \bar{y} - b\bar{x}

Answer

\hat{y} = 2.2 + 0.60x

First step

\bar{x} = 3

\bar{y} = 4

;

s_x = \sqrt{2.5} \approx 1.58

;

s_y = \sqrt{1.5} \approx 1.22

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Example 2

hard

The LSRL for predicting weight (

y

, kg) from height (

x

, cm) is

\hat{y} = -100 + 0.8x

. Interpret the slope and intercept, predict weight for height=175 cm, and explain why extrapolating to height=50 cm is problematic.

Example 3

medium

Given

\bar{x}=4

\bar{y}=20

r=0.8

s_x=2

s_y=5

, find the LSRL.

Example 4

medium

Why is predicting

y

at an

x

-value far outside the observed range dangerous? Give one example.

Example 5

hard

y

is shifted by adding

5

to each observation, how does the LSRL change?

Practice Problems

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

Example 1

easy

Given

\hat{y} = 5 + 3x

: (a) predict

y

when

x=4

, (b) interpret the slope, (c) does the line pass through the origin?

Example 2

hard

The LSRL has the property of minimizing

\sum e_i^2 = \sum (y_i - \hat{y}_i)^2

. Explain why minimizing squared residuals (rather than absolute residuals) is preferred, and name two consequences of this choice.

Example 3

easy

A regression line is

\hat{y} = 3 + 2x

. Predict

y

when

x = 4

Example 4

easy

\hat{y} = 5 + 3x

, what is the slope?

Example 5

easy

\hat{y} = 7 - 2x

, what is the y-intercept?

Example 6

easy

The LSRL minimizes the sum of squared what?

Example 7

easy

Compute the slope:

r = 0.8

s_y = 4

s_x = 2

. Use

b = r \frac{s_y}{s_x}

Example 8

easy

A line passes through the point of averages

(\bar{x}, \bar{y}) = (5, 12)

with slope

b = 2

. Find the intercept using

a = \bar{y} - b\bar{x}

Example 9

easy

\hat{y} = 10 + 4x

where

y

is cost in dollars and

x

is hours, interpret the slope.

Example 10

easy

Does the regression of

y

x

generally give the same line as the regression of

x

y

Example 11

medium

Given

r = 0.6

s_x = 5

s_y = 10

\bar{x} = 20

\bar{y} = 50

, find the LSRL.

Example 12

medium

Using

\hat{y} = 26 + 1.2x

, predict

y

x = 30

Example 13

medium

Two data points lie exactly on

\hat{y} = 2 + 3x

(1, ?)

and

(4, ?)

. Find both predicted values.

Example 14

medium

A model

\hat{y} = 100 + 5x

predicts plant height (cm) from days

x

. Why is predicting height at

x = 10{,}000

days unwise?

Example 15

medium

Find the slope of the LSRL through summary stats

r = -0.5

s_y = 8

s_x = 4

Example 16

medium

\hat{y} = 200 - 0.5x

y

is weight (lb) and

x

is age in days for a dieting program. Interpret the intercept and say whether it is meaningful.

Example 17

medium

A slope is computed as

b = r \frac{s_y}{s_x}

with

b = 3

and

\frac{s_y}{s_x} = 5

. Find

r

Example 18

medium

The LSRL is

\hat{y} = 4 + 2x

. A new point

(3, 12)

is observed. Is the observed value above or below the line?

Example 19

medium

A line passes through

(\bar{x},\bar{y}) = (8, 30)

with slope

b = 2.5

. Find its equation.

Example 20

challenge

A dataset has

\bar{x}=10, \bar{y}=50, s_x=2, s_y=6, r=0.9

. Find the LSRL and predict

y

x=12

Example 21

challenge

Show that if

r = 1

, the regression of

y

x

and of

x

y

describe the same line. (Use slopes

b_{yx} = r\frac{s_y}{s_x}

and

b_{xy} = r\frac{s_x}{s_y}

Example 22

challenge

A regression on temperature (

x

, in

^\circ

C) gives

\hat{y} = 2 + 0.5x

. If temperature is re-expressed in tenths of a degree (

x' = 10x

), what is the new slope?

Example 23

easy

Given

\hat{y} = 8 + 1.5x

, predict

y

when

x = 6

Example 24

easy

Given

r = 0.5

s_y = 6

s_x = 3

, compute the slope of the LSRL.

Example 25

easy

For LSRL

\hat{y} = 100 - 2x

predicting weight (lb) from days

x

in a diet program, interpret the slope.

Example 26

easy

Residual = observed

-

predicted. If

y_{\text{obs}}=15

and

\hat{y}=12

, find the residual.

Example 27

medium

A regression

\hat{y} = 50 - 0.2x

predicts time-on-test (min) from age

x

(yr). Predict for

x=25

and compute residual if observed time is

42

Example 28

medium

Given the LSRL has slope

b = 0.6

and passes through

(\bar{x}, \bar{y}) = (12, 25)

, find the equation.

Example 29

medium

In a regression with

r=0.9

r^2

describes what?

Example 30

medium

Compute slope:

r = -0.6

s_y = 12

s_x = 4

Example 31

medium

A residual plot shows a clear curved pattern. What does this indicate?

Example 32

medium

For

\hat{y} = 30 + 4x

, predict the change in

y

x

increases by

5

Example 33

medium

Given five data points

(1,3), (2,5), (3,7), (4,8), (5,10)

, compute

\bar{x}

and

\bar{y}

Example 34

medium

For the LSRL passing through

(\bar{x},\bar{y}) = (3, 6.6)

with slope

1.7

, write the equation.

Example 35

hard

For

\hat{y} = 20 + 5x

in dollars per item (

y

) vs. quantity (

x

), interpret slope and intercept and judge if intercept is meaningful when

x=0

Example 36

hard

A regression equation rescales

x

x' = x/10

. If original LSRL is

\hat{y} = 5 + 2x

, what is the new equation in

x'

Example 37

hard

For the LSRL

\hat{y}=4+2x

, the sum of residuals over all data points is always what?

Example 38

hard

A regression has slope

b=3

. If

y

is rescaled to

y' = 2y

, what is the new slope?

Example 39

hard

For

r=0.6

, what percent of variation in

y

is NOT explained by linear regression on

x

Example 40

challenge

Given that the LSRL of

y

x

has slope

b_{yx}

and the LSRL of

x

y

has slope

b_{xy}

, show

b_{yx} \cdot b_{xy} = r^2

Example 41

challenge

A regression line

\hat{y} = 3 + 2x

has

r^2 = 0.81

and

s_x = 4

. Find

s_y

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Related Concepts

Correlation Scatter Plot Mean Standard Deviation Residuals Coefficient of Determination

Background Knowledge

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

correlationscatter plotmeanstandard deviation