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Algebra 2 Modelling with functions

Regression with technology

20 practice questions 0 video lessons Theory + worked examples

Regression with Technology

Common Core Algebra 2 • Standard S-ID.6a • Modelling with Functions

Regression with Technology is a topic in Modelling with Functions in the Common Core State Standards. It is aligned to Standard S-ID.6a, which requires students to use technology to fit a linear, quadratic, or exponential function to data and interpret the correlation coefficient.

Regression uses technology to fit a model to data; the correlation coefficient \(r\) (from \(-1\) to \(1\)) measures how well a line fits.

Common Core Algebra 2 › Modelling with Functions › Regression with Technology  —  Standard S-ID.6a

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Theory

Regression fits a model to data using technology, minimizing the overall error:
  • Choose the model type (linear, quadratic, exponential).
  • The calculator returns the equation of best fit.
  • The correlation coefficient \(r\) (from \(-1\) to \(1\)) measures fit.
\(|r|\) near \(1\) means a strong fit; near \(0\) means weak.
Regression line of best fit Regression finds the line or curve that best fits a scatter of data. line of best fit
The line of best fit through the data.
Regression Regression Regression technology fits a model to data r: correlation coefficient (-1 to 1) |r| near 1 → strong fit r² : fraction of variation explained
Reading a regression.

Correlation:

\[-1\le r\le 1,\qquad r^2=\text{fraction of variation explained}\]
r ranges from negative 1 to 1; r squared is the fraction of variation explained
Sign of \(r\) shows direction; magnitude shows strength.

How to run a regression

  1. Enter the data.
  2. Choose the model type.
  3. Read the equation of best fit.
  4. Check \(r\) (or \(r^2\)) for the strength of fit.
Example 1 — Read the equation
Technology gives \(y=1.2x+0.8\). What is the slope?
Solution

The slope is the coefficient of \(x\).

\(\text{slope}\)\(=\)\(1.2\)
the slope is 1.2
Example 2 — Interpret r
A regression has \(r=0.97\). Describe the fit.
Solution

\(|r|\) near \(1\) is a strong positive fit.

\(|0.97|\)\(\approx\)\(1\ \text{(strong)}\)
a strong positive linear fit
Example 3 — Exponential regression
When would you fit an exponential regression?
Solution

When the data grows by a roughly constant percent, use an exponential model \(y=ab^x\).

use exponential regression for constant-percent growth
Example 4 — Use the model
Using \(y=1.2x+0.8\), predict \(y\) at \(x=10\).
Solution

Substitute.

\(y\)\(=\)\(1.2(10)+0.8\)
\(=\)\(12.8\)
the prediction is 12.8

Common pitfalls

Match the model type to the data's pattern.
\(r\) measures linear fit; a curved pattern may need another model.
Correlation is not causation.

Frequently asked questions

What is regression?

Fitting a model equation to data, usually with technology.

What is the correlation coefficient?

\(r\), a value from \(-1\) to \(1\) measuring how well a line fits.

What does \(r=0.97\) mean?

A strong positive linear relationship.

Does a strong correlation prove causation?

No — correlation does not imply causation.

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