Linear Regression

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Linear Regression: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Linear regression turns a cloud of paired measurements into one straight-line equation, giving a rule that estimates the second measurement from the first.

Algebra 1StatisticsS-ID.B.6a

What is Linear Regression?

When scatter-plot dots trail along a straight band, linear regression chooses the one line that hugs the band most tightly. The equation of that line becomes a prediction rule: feed it an \(x\) and it returns an estimate \(\hat{y}\).

The standard choice is the least-squares line \(\hat{y} = ax + b\), which makes the total of the squared vertical misses as small as possible. The number \(a\) is the predicted change in \(y\) for each extra unit of \(x\), and \(b\) is the predicted value at \(x = 0\).

How to spot it

  • A table pairs two measurements, such as hours studied and score, and the question asks for an equation that predicts one from the other.
  • A scatter plot looks like a thin band rising or falling, with no strong curve.
  • A calculator or spreadsheet is asked to run linear regression, or the problem hands you a fitted equation to interpret.
  • The task is to predict a value that is not in the table, such as next year or a larger size.

The method

  1. Sketch the points first and check that they follow a roughly straight trail; a curve calls for another model.
  2. Type both lists into a calculator or spreadsheet and run its line-fitting command to obtain \(a\) and \(b\).
  3. Write \(\hat{y} = ax + b\), then say what \(a\) and \(b\) mean using the units of the story.
  4. Substitute the new \(x\) to predict, and stay near the span of values you actually measured.
y = 2.8x + 1.2prediction (6, 18)123456
The fitted line passes through the cloud of blue points and predicts 18 at x = 6.

Linear Regression: solved example

Problem

At a camp in Indiana, five players log hours of shooting drills \(x\) and baskets sunk \(y\) in a 20-shot test: \((1, 4)\), \((2, 7)\), \((3, 9)\), \((4, 13)\), \((5, 15)\). Software returns \(\hat{y} = 2.8x + 1.2\). Interpret the slope, then predict the score after 6 hours.

Solution

A slope of 2.8 means one extra drill hour goes with about 2.8 more baskets. The intercept 1.2 is the predicted score for a player with no drills.

At \(x = 6\): \(\hat{y} = 2.8(6) + 1.2 = 16.8 + 1.2 = 18\). The model predicts 18 baskets, under the 20-shot limit, so the estimate is believable.

Try it

Question 1

A ferry’s fuel use follows \(\hat{y} = 14.5x + 60\), where \(x\) is nautical miles and \(\hat{y}\) is gallons. Predict the fuel for a 22-mile trip.

Show the answer

\(14.5(22) + 60 = 319 + 60 = 379\) gallons.

Question 2

In that same ferry equation, explain what 14.5 means.

Show the answer

Each additional nautical mile is predicted to use about 14.5 more gallons of fuel.

Linear Regression vs. Line of Best Fit

A line of best fit can be sketched by eye with a ruler, and two students may draw two different ones. Linear regression is the exact calculation that removes the guesswork: it uses least squares, so everyone gets the same equation from the same data and can measure how far each point strays from it.

Line of Best Fit

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