Practice Least Squares Regression Line in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

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 3

medium

For the LSRL passing through

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

with slope

1.7

, write the equation.

Example 4

hard

A regression has slope

b=3

. If

y

is rescaled to

y' = 2y

, what is the new slope?

Example 5

medium

Given

\bar{x}=4

\bar{y}=20

r=0.8

s_x=2

s_y=5

, find the LSRL.

Example 6

medium

A line passes through

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

with slope

b = 2.5

. Find its equation.

Example 7

easy

\hat{y} = 7 - 2x

, what is the y-intercept?

Example 8

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 9

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 10

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 11

medium

Two data points lie exactly on

\hat{y} = 2 + 3x

(1, ?)

and

(4, ?)

. Find both predicted values.

Example 12

medium

What does it mean if

r^2 = 1

for a regression?

Example 13

easy

\hat{y} = 10 + 4x

where

y

is cost in dollars and

x

is hours, interpret the slope.

Example 14

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 15

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}

Example 16

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 17

medium

Using

\hat{y} = 26 + 1.2x

, predict

y

x = 30

Example 18

medium

Compute slope:

r = -0.6

s_y = 12

s_x = 4

Example 19

medium

Why is predicting

y

at an

x

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

Example 20

medium

Given five data points

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

, compute

\bar{x}

and

\bar{y}

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Related Concepts

Correlation Scatter Plot Mean Standard Deviation Residuals Coefficient of Determination