Using a Line of Best Fit
A line of best fit turns data into a rule you can use. Its numbers tell a story about the data, but only near the data.
What you'll be able to do
- Say what a fit line's slope means, with units.
- Say what its y-intercept means, and if it makes sense.
- Predict from a fit line, and compare with real values.
- Know when a prediction can't be trusted.
The big idea
A fit line is a trend. It shows how the data tend to go. Single points can be above or below it.
The slope. It is the predicted change in y for 1 more unit of x. Say it with units.
An example. y = 0.25x + 40 for chirps a minute and °F. Each extra chirp a minute means about 0.25 °F warmer.
The y-intercept. It is the predicted y when x = 0. Ask: is x = 0 inside the data?
Predict. Put x into the rule. Say 'about', because it is a trend.
Real against predicted. Subtract: real y minus predicted y. Plus means above the trend.
Stay near the data. Far outside the data, a trend can give answers that make no sense.
A link is not a cause. Two things can rise together because a third thing drives both.
Words you need
- Line of best fit
- A straight line that follows the trend of a scatter plot.
- Slope
- How steep a line is: the change in y for 1 more x.
- y-intercept
- Where the line crosses the y-axis: the value of y when x = 0.
- Predict
- Use the rule to find a likely value.
- Trend
- The overall way the data go.
- Association
- A link between two sets of data.
Reading a fit line
| Part | What it tells you |
|---|---|
| Slope | About how much y changes for 1 more x |
| y-intercept | The predicted y at x = 0, if 0 is near the data |
| A prediction | A likely value, not an exact one |
| Far from the data | Don't trust it |
Worked examples
Example 1
x is a book's pages. y is its price in dollars. The data run from 100 to 600 pages. The fit line is y = 0.02x + 6. What do 0.02 and 6 mean?
- Slope: each extra page adds about $0.02 to the price.
- Start: a 0-page book is predicted at $6.
- No book has 0 pages. 0 is far outside the data, so the start has no real meaning.
Answer: A 0-page book is predicted at about 6 dollars. It has no real meaning: 0 is far outside the data
Example 2
The laptop fit line is y = −35x + 300. Predict the price of a 4-year-old laptop. One sold for $150.
- 300 − 35 × 4 = 160. The line predicts about $160.
- The laptop sold for $150.
- 150 − 160 = −10. It sold for $10 less than predicted.
Answer: $10 less than predicted
Example 3
Cricket data run from 80 to 200 chirps a minute. The fit line is y = 0.25x + 40. Should you use it for 400 chirps a minute?
- 400 is far outside 80 to 200.
- The line predicts 0.25 × 400 + 40 = 140 °F.
- That is hotter than any cricket could live in. Use the line only near 80 to 200.
Answer: At 400: 140 (far outside the data)
Common mistakes
Treating a prediction as exact. A fit line shows a trend. One person can be above or below it.
Giving the slope with no units. Say 'about 0.25 °F per chirp', not just '0.25'.
Trusting predictions far from the data. A trend can break. Stay near the data.
Saying a link proves a cause. Hot days bring ice cream and sunburns. Ice cream doesn't cause sunburns.
Practice: 20 questions
17 of 20 are word problems. For a word problem, sketch it first and mark what you know.
Tier 1 Foundations Use the rule directly
- The plot shows crickets. x is chirps a minute. y is the air temperature in °F. The fit line is y = 0.25x + 40. What does 0.25 mean? Say it with units. Word problem
- x is foot length in cm. y is height in cm. The fit line is y = 4.5x + 52. Predict the height of a person with a 26 cm foot. Word problem
- x is hours of typing practice. y is typing speed in words a minute. The data run from 0 to 20 hours. The fit line is y = 3x + 20. What does 20 mean? Does it make sense? Word problem
- x is a used laptop's age in years. y is its price in dollars. The fit line is y = −35x + 300. A 3-year-old laptop sold for $230. Is that more or less than predicted? By how much? Word problem
- x is hours of study. y is a test score. The fit line is y = 4x + 60. Lu studied 5 hours and scored 74. Her friend says the line must be wrong, because it predicts 80. Find the mistake. Word problem
Tier 2 Intermediate One more step to think through
- Data on kids aged 6 to 12 give the fit line y = 6x + 80. x is age. y is height in cm. Predict the height at 9 and at 30. Which prediction can you trust? Word problem
- The cricket fit line is y = 0.25x + 40. x is chirps a minute. y is °F. It is 70 °F. About how many chirps a minute does the line predict? Word problem
- x is minutes a tap runs. y is liters in a tub. The fit line's slope is 0.4. About how many liters does the tub gain per hour (60 minutes)? Word problem
- Two students fit lines to the same data, with x from 0 to 10. Ari's line is y = 3x + 20. Bea's is y = 3.4x + 17. How far apart are their predictions at x = 5? At x = 30?
- The plot shows runners. x is weeks of training. y is 5 km time in minutes. The fit line passes through (0, 32) and (10, 27). Write it. What does the slope mean? Word problem
Tier 3 Advanced Several steps - find the hidden structure
- The laptop fit line is y = −35x + 300. x is age in years. y is price in dollars. From age 2 to age 7, about how much does the line say the price falls? Word problem
- The cricket fit line is y = 0.25x + 40. Kai says 10 more chirps a minute means 10 °F warmer. Find his mistake. Word problem
- The typing fit line is y = 3x + 20. Ana practiced 6 hours. She types 5 words a minute faster than predicted. How fast does she type? Word problem
- x is hours of practice a week. y is points a game. Team A's fit line is y = 0.8x + 10. Team B's is y = 0.8x + 18. What does the gap between the lines mean? Word problem
- The foot fit line is y = 4.5x + 52. x is foot length in cm. y is height in cm. Four people: (24, 170), (25, 160), (27, 175), (28, 182). Who is farthest from the prediction? Word problem
Tier 4 Challenge Stretch problems, some above grade level
- Winning 100 m times since 1900 give the fit line y = 11 − 0.01x. x is years since 1900. y is seconds. When does the line predict 0 seconds? What does that tell you? Word problem
- x is a student's birth month, 1 to 12. y is a test score. The fit line is y = 0.1x + 74. What does the line tell you? Word problem
- A fit line is y = 2.5x + 40, with x in minutes and y in dollars. Which makes the slope bigger: y in cents, or x in hours?
- On 10 summer days, a beach town counted ice creams sold and sunburns. The fit line is y = 0.15x + 4. Dev says ice cream causes sunburns. He wants to ban ice cream. Find his mistake. Word problem
- Least-squares regression picks the line with the smallest sum of squared gaps. For (1, 2), (2, 3), (3, 7), find this sum for y = 2.5x − 1 and for y = 2x. Which line wins? Above grade level
Answers
Tier 1 Foundations
- For each extra chirp a minute, the temperature is about 0.25 °F higher. The slope is the predicted change in y for 1 more chirp a minute. It is a trend, so say 'about'.
- About 169 cm. 4.5 × 26 + 52 = 169. A fit line predicts the trend, so say 'about'.
- A person with no practice types about 20 words a minute. It makes sense: 0 is inside the data. The y-intercept is the predicted y when x = 0. Trust it only if x = 0 is inside the data.
- $35 more than predicted. The line predicts $195. $230 − $195 = $35.
- 6 points less than predicted. A fit line shows the trend, not each person. Lu is just below the trend. The line predicts 80 points. 74 − 80 = −6.
Tier 2 Intermediate
- At 9: about 134 cm, which you can trust. At 30: 260 cm, far outside the data. Nobody is that tall. The data run from 6 to 12. Trust predictions inside that range only.
- About 120 chirps a minute. Set the line equal to 70: 0.25x + 40 = 70. Then 0.25x = 30, so x = 120.
- About 24 liters an hour. The slope is the change for 1 unit of x. 60 × 0.4 = 24.
- 1 apart at 5; 9 apart at 30. Predictions at 5: 35 and 34, at 30: 110 and 119. The slopes differ, so the gap grows away from the data.
- y = −0.5x + 32. Each week of training, the time drops about half a minute. y = −0.5x + 32 misses by 0.7 in all.
Tier 3 Advanced
- About $175 less. The slope is the change for 1 unit of x. 5 × 35 = 175.
- About 2.5 °F warmer. The slope is the change for 1 chirp, not for 10. The slope is the change for 1 unit of x. 10 × 0.25 = 2.5.
- 43 words a minute. The line predicts 3 × 6 + 20 = 38. 38 + 5 = 43.
- For the same practice, Team B players score about 8 more points. Both gain about 0.8 points per hour. Predictions at 5: 14 and 22, at 10: 18 and 26. The slopes match, so the gap stays the same.
- (24, 170): 10 above. Predictions: 160, 164.5, 173.5, 178. Gaps: 10, −4.5, 1.5, 4.
Tier 4 Challenge
- In the year 3000 (x = 1100). No one can run 100 m in 0 seconds, so the trend can't go on forever. 11 − 0.01x = 0 gives x = 1100, and 1900 + 1100 = 3000. The data stop long before that.
- Almost nothing. December predicts only 1.1 points more than January. A slope near 0 means x barely changes the prediction. There is almost no association.
- y in cents: 250 cents a minute, against $150 an hour. Cents: 2.5 × 100 = 250 cents a minute. Hours: 2.5 × 60 = $150 an hour. 250 is bigger.
- A link is not a cause. Dev reads the fit line as a cause. Both numbers rise on hot, sunny days. The sun causes the sunburns, not the ice cream.
- y = 2.5x − 1. y = 2.5x − 1: gaps 0.5, −1, 0.5, so squares add to 1.5. y = 2x: gaps 0, −1, 1, so squares add to 2.
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