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Algebra 2 Modelling with functions

Regression with technology

20 practice questions 0 video lessons Theory + worked examples

Regression with Technology

Texas Algebra II (TEKS) • Standard 2A.8(B) • Modelling with Functions

Regression with Technology is a topic in Modelling with Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.8(B), which requires students to use regression methods available through technology to write a linear, quadratic, and exponential function from a given set of data.

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

Texas Algebra II (TEKS) › Modelling with Functions › Regression with Technology  —  Standard 2A.8(B)

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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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