How far outside the data can I predict?

There is no safe distance. A little outside the range may still be fine if the pattern clearly continues, but any extrapolation should be flagged as uncertain, and predictions far away (or impossible values) should not be used.

What is the difference between r and R²?

$r$ is between $-1$ and $1$ and shows the direction (sign) and strength of a linear relationship. $R^2 = r^2$ is between $0$ and $1$ and tells you what share of the variation in $y$ the line explains. $R^2$ loses the sign, so check the slope for the direction.

If R² is high, does x cause y?

Not necessarily. A high $R^2$ only says the line fits the data well. A third variable or chance can create the pattern; only a controlled experiment can show cause and effect.

Teacher Resources

Regression Word Problems

Q: Does the intercept always have a meaning?

No. It is the predicted $y$ at $x = 0$. If $x = 0$ is impossible (a person 0 cm tall) or far from the data (a house with 0 m²), the intercept is only a number that positions the line, not something to interpret.

Solve complete regression problems: interpret slope and intercept in context, predict, spot extrapolation, compute residuals and read r and R².

Advanced30 minLesson

Printable worksheet

Definition

A regression word problem gives you real data (or a regression equation already fitted to it) and asks what the model says about the situation. Every such problem uses the same toolkit:

\hat{y} = mx + b

**Slope $m$ in context**: for each increase of 1 unit in $x$ , the predicted $y$ changes by $m$ units, on average.
**Intercept $b$ in context**: the predicted $y$ when $x = 0$ . It only has a real meaning if $x = 0$ makes sense and lies near the data.
Prediction: substitute $x$ into the equation to get the predicted value $\hat{y}$ .
Interpolation vs extrapolation: a prediction for an $x$ inside the range of the data is interpolation (usually reliable); outside that range it is extrapolation (risky).
Residual: $e = y - \hat{y}$ (actual minus predicted). Positive means the point lies above the line.
r and R²: $r$ gives the direction and strength of the linear relationship; $R^2 = r^2$ is the share of the variation in $y$ that the linear relationship with $x$ explains.

A good answer always uses the units and words of the problem, not just numbers.

Try it now

The model

\hat{y} = 6x + 48

predicts a test score

\hat{y}

(points) from weekly study hours

x

. What does the slope

6

mean?

Worked Examples

A café recorded the outdoor temperature

x

(in °C, from

0

20

) and the number of hot drinks

y

sold on five mornings. The regression line is

\hat{y} = 110 - 3x

. Interpret the slope and the intercept.

Identify the slope and the intercept

\hat{y} = -3x + 110

, so

m = -3

and

b = 110

→

m = -3

b = 110

Interpret the slope with units

For each 1 °C warmer, the predicted number of hot drinks changes by

-3

→ About 3 fewer hot drinks per extra degree, on average

Interpret the intercept with units

x = 0

°C the model predicts

\hat{y} = 110 - 3 \times 0 = 110

→ About 110 hot drinks on a 0 °C morning

Check that the intercept makes sense

0

°C is a real temperature and it is inside the data range

0

20

→ The intercept has a real meaning here

Answer: Slope: for every 1 °C rise in temperature, the café is predicted to sell about 3 fewer hot drinks. Intercept: on a 0 °C morning the model predicts about 110 hot drinks, which makes sense because 0 °C is part of the data.

Common Mistakes

Stating the slope as a bare number: "the slope is 6"

Why it's wrong: A number without context does not answer the question. The slope is a rate: it needs both units and the word "predicted" or "on average", because real points scatter around the line.

Correct: Say: "For each extra hour of study per week, the predicted test score rises by about 6 points."

Trusting a prediction far outside the data (extrapolation)

Why it's wrong: The line was fitted only to the observed range. Outside it the pattern may change, and the line can give impossible values like a score of 120 out of 100 or a negative number of sales.

Correct: Check the range of $x$ first. Inside it: interpolation. Outside it: extrapolation, so say the prediction is unreliable.

Computing the residual the wrong way round: $\hat{y} - y$

Why it's wrong: The sign then flips, and "above the line" becomes "below the line". By convention the residual is actual minus predicted.

Correct: Always use $e = y - \hat{y}$ : positive means the model underpredicted, negative means it overpredicted.

Reading $R^2 = 0.9$ as "90% of the points lie on the line" or as "the model is right 90% of the time"

Why it's wrong: $R^2$ measures how much of the variation in $y$ the linear relationship explains. Points can all be off the line and $R^2$ can still be 0.9.

Correct: Say: "90% of the variation in $y$ is explained by the linear relationship with $x$ ."

Interactive Visual

Scatter Plot

Weekly study hours and test score

ŷ = 6x + 48

r = 0.949 · strong positive correlation

Tap a point to see its coordinates.

Scatter Plot

Outdoor temperature and hot drinks sold

ŷ = −3x + 110

r = -0.949 · strong negative correlation

Tap a point to see its coordinates.

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

The model

\hat{y} = 6x + 48

predicts a test score

\hat{y}

(points) from weekly study hours

x

. What does the slope

6

mean?

Practice Problems: Regression Word Problems

1. The model \hat{y} = 6x + 48 predicts a test score \hat{y} (points) from weekly study hours x. What does the slope 6 mean?
2. Using \hat{y} = 6x + 48 (test score from weekly study hours x), predict the score of a student who studies 4 hours per week.
3. A regression line was fitted to data with x values from 2 to 10. For which value of x is a prediction an extrapolation?
4. A data point has actual value y = 80, and the regression line predicts \hat{y} = 74 for it. What is its residual?
5. A regression of exam score on study time has R² = 0.64. Which statement is correct?
6. The correlation between daily steps and calories burned is r = 0.7. What is R²? Give it as a decimal.
7. A dealer's model for used cars is \hat{y} = 18000 - 1500x, where \hat{y} is the price in dollars and x the age in years (data for cars from 0 to 8 years old). What does the intercept 18000 mean?
8. A café uses \hat{y} = 110 - 3x to predict the number of hot drinks sold from the morning temperature x in °C (data from 0 to 20 °C). How many hot drinks does it predict for 12 °C?
9. In a model of delivery times, one delivery has a residual of -5 minutes. What does this mean?
10. A plant's height is modelled by \hat{y} = 2.5x + 4 (cm, x in weeks). By how many cm does the predicted height increase over 4 more weeks?
11. The café model is \hat{y} = 110 - 3x (hot drinks, x in °C). On a 15 °C morning the café sold 75 hot drinks. Find the residual of this morning.
12. For the study-hours data, the sum of squared residuals is SS_{\text{res}} = 40 and the total variation of the scores is SS_{\text{tot}} = 400. Find R^2 and say what percentage of the variation in scores the line explains.
13. The model \hat{y} = 6x + 48 was fitted to students who study from 1 to 5 hours per week (tests out of 100 points). Predict the score for 12 hours and decide what kind of prediction this is.
14. The dealer's model is \hat{y} = 18000 - 1500x (price in dollars, x = age in years, data from 0 to 8 years). A 4-year-old car was sold for 13000 dollars. Find the predicted price and the residual. Was the car sold above or below the predicted price?
15. A student counts the push-ups y she can do after x weeks of training: | x | 1 | 2 | 3 | 4 | |---|---|---|---|---| | y | 3 | 5 | 6 | 10 | Find the least-squares regression line \hat{y} = mx + b.
16. The café model is \hat{y} = 110 - 3x (hot drinks, x in °C, data from 0 to 20 °C), with R^2 = 0.9. On an 8 °C morning the café sold 90 hot drinks. Find the prediction, the residual, the correlation coefficient r and the percentage of variation explained.
17. An ice-cream stand collected data for temperatures from 10 to 30 °C. The regression line of ice creams sold per hour on temperature has slope 1.5, and the means are \bar{x} = 20 °C and \bar{y} = 50 ice creams. Find the intercept, predict sales at 30 °C, and find the residual of an hour at 30 °C with 61 ice creams sold.
18. With \hat{y} = 6x + 48 (test score from weekly study hours, data from 1 to 5 hours), how many hours of study does the model say are needed for a predicted score of 84? Can this answer be trusted?

Why It Matters

Regression is how people turn data into decisions:

Shops and cafés predict how much stock to order from the weather forecast.
Schools look at how study time relates to results, and which students do better or worse than predicted.
Doctors and scientists use regression to estimate doses, growth rates and risks.
Car dealers and banks estimate how fast the value of a car drops each year.

In all of these, the hard part is not the arithmetic but the interpretation: saying what the slope means, knowing when a prediction can be trusted, and explaining how well the line fits. That is exactly what exams and real jobs ask for, and what this lesson trains.

Real World Applications

Biology Lab: Plant Growth

Biology students measure a seedling every week and fit a regression line to height against time to describe its growth rate.

Example:

For $\hat{y} = 2.5x + 4$ (height in cm, $x$ in weeks, data from week 0 to week 8), the slope says the plant grows about 2.5 cm per week and the intercept says it was about 4 cm tall when measuring started.

1Try It Yourself

The class model is $\hat{y} = 2.5x + 4$ , with height $\hat{y}$ in cm and time $x$ in weeks (data from week 0 to week 8).

What height does the model predict at week 6?

Step 1: Write the mathematical expression

Substitute $x = 6$ into the equation:

Delivery Times for an Online Shop

Delivery companies model delivery time from distance to give customers an estimated arrival time and to spot routes that are slower than expected.

Example:

With $\hat{y} = 12 + 3x$ (minutes, $x$ = distance in km, data from 1 to 10 km), a 4 km delivery is predicted to take $12 + 12 = 24$ minutes.

2Try It Yourself

The model is $\hat{y} = 12 + 3x$ (time in minutes, $x$ = distance in km). A delivery of 6 km actually took 33 minutes.

What is the residual of this delivery?

Step 1: Write the mathematical expression

Residual = actual time minus predicted time:

Phone Battery Testing

Tech reviewers relate screen-on time to battery drain and report how strongly the two are linked, so readers know how much the regression line can be trusted.

Example:

If screen time and battery used have $r = 0.8$ , then $R^2 = 0.64$ : 64% of the variation in battery use is explained by screen time.

3Try It Yourself

In a battery test the correlation between temperature of the room and battery life is $r = -0.9$ .

What percentage of the variation in battery life is explained by the linear relationship with room temperature?

Step 1: Write the mathematical expression

Square r and turn it into a percentage:

Key Takeaways

1Interpret the slope as a rate with units: for each 1-unit increase in $x$ , the predicted $y$ changes by $m$ , on average
2The intercept is the predicted $y$ at $x = 0$ ; it only means something if $x = 0$ makes sense and is near the data
3Predict by substituting $x$ into $\hat{y} = mx + b$ ; inside the data range this is interpolation, outside it is risky extrapolation
4A residual is actual minus predicted, $e = y - \hat{y}$ : positive above the line, negative below it
5 $r$ shows direction and strength; $R^2 = r^2$ is the share of the variation in $y$ explained by the line
6A strong fit shows a link, not a cause: always answer in the words and units of the problem

Frequently Asked Questions

No. It is the predicted

y

x = 0

. If

x = 0

is impossible (a person 0 cm tall) or far from the data (a house with 0 m²), the intercept is only a number that positions the line, not something to interpret.

Glossary

Predicted value $\hat{y}$: The value of $y$ that the regression line gives for a chosen $x$
Slope in context: The predicted change in $y$ , in the units of $y$ , for each 1-unit increase in $x$
Intercept in context: The predicted value of $y$ when $x = 0$ , meaningful only if $x = 0$ is realistic and near the data
Interpolation: Predicting $y$ for an $x$ inside the range of the data
Extrapolation: Predicting $y$ for an $x$ outside the range of the data; often unreliable
Coefficient of determination R²: The share of the variation in $y$ explained by the linear relationship with $x$ ; equals $r^2$ for a line

Formula Card

Regression line

\hat{y} = mx + b

Substitute x to get the predicted value

Least-squares slope

m = \frac{\sum(x-\bar{x})(y-\bar{y})}{\sum(x-\bar{x})^2}

Computed from the deviations from the means

Least-squares intercept

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

The line always passes through the point of means

Residual

e = y - \hat{y}

Actual minus predicted; positive above the line

Coefficient of determination

R^2 = r^2 = 1 - \frac{SS_{\text{res}}}{SS_{\text{tot}}}

$SS_{\text{res}}$ is the sum of squared residuals, $SS_{\text{tot}}$ the sum of squared deviations of y from its mean

Today's challengeOne problem, three tries

Regression Word Problems

Definition

Worked Examples

Common Mistakes

Interactive Visual

Scatter Plot

Weekly study hours and test score

Scatter Plot

Outdoor temperature and hot drinks sold

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Biology Lab: Plant Growth

Delivery Times for an Online Shop

Phone Battery Testing

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Regression line

Least-squares slope

Least-squares intercept

Residual

Coefficient of determination

More in This Topic

Introduction to Regression

Linear Regression

Scatter Plots

Line of Best Fit