Practice Coefficient of Determination 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 proportion of the total variation in the response variable $y$ that is explained by the linear relationship with the explanatory variable $x$ . It equals the square of the correlation coefficient: $r^2$ .

Total variation in $y$ has two parts: what the regression line explains and what's left over (residual variation). If $r^2 = 0.85$ , the regression line accounts for $85\%$ of why $y$ values differ from each other, and $15\%$ is unexplained. Think of $r^2$ as a report card for how well $x$ predicts $y$ .

Showing a random 20 of 50 problems.

Example 1

challenge

Total variation SST

= 1000

. Adding a predictor reduces residual variation from SSE

= 400

to SSE

= 250

. By how much does

r^2

increase?

Example 2

medium

SSE=120

and

r^2=0.4

. Find

SST

Example 3

hard

Explain why

r^2

is identical whether the regression line is

y

x

x

y

Example 4

hard

Five

y

-values are

4,5,7,8,11

with

\bar y=7

. The residuals from a fitted line are

1,-1,0,-1,1

. Compute

r^2

Example 5

medium

Model A has

r^2 = 0.9

but a strongly curved residual plot. Model B has

r^2 = 0.8

with random residuals. Which has the better-justified linear fit?

Example 6

easy

The correlation between study hours and test score is

r=0.8

. Calculate

R^2

and interpret it.

Example 7

medium

r^2=0.04

. A student calls the relationship 'strong.' Correct them.

Example 8

medium

A regression model has

SST = 500

(total variation) and

SSE = 125

(unexplained variation). Calculate

R^2

and interpret its meaning.

Example 9

hard

SST=900

. The fitted line gives residual SS

SSE=144

. Compute

r^2

to two decimals.

Example 10

medium

r^2 = 0.81

. A student says '

81\%

of the data points fall on the line.' What is wrong, and what is correct?

Example 11

medium

r^2=0.36

and slope is positive, give both

r

and the percentage of explained variation.

Example 12

medium

A regression has SST

= 500

and

r^2 = 0.6

. Find the residual sum of squares SSE.

Example 13

easy

The correlation is

r = 0.7

. Compute

r^2

Example 14

medium

r^2

rises from

0.49

0.64

, how does the magnitude of

r

change (slope stays positive)?

Example 15

medium

A model has

SST=1200

and

r^2=0.7

. Find the explained sum of squares

SSR

and the residual

SSE

Example 16

easy

True or false:

r^2

can be negative.

Example 17

easy

Interpret

r^2=0.5

in one sentence.

Example 18

medium

A model on

n=20

points has

SST=500

and

r^2=0.84

. Compute

SSE

and the residual standard deviation

s=\sqrt{SSE/(n-2)}

Example 19

medium

A report states

r = 0.5

. A student claims '

50\%

of the variation is explained.' Correct them.

Example 20

easy

r^2 = 0

, how much of the variation in

y

does the line explain?

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Related Concepts

Correlation Least Squares Regression Line Residuals Inference for Regression