Fitting Linear Models to Data: Fresh Take

  • Use linear functions to model and draw conclusions from real-world problems
  • Sketch scatter plots to see patterns and tell apart straight-line relationships from curves
  • Find the best straight line that goes through a set of data points
  • Identify the differences between linear and nonlinear relations

Building Linear Models

 

The Main Idea 

Modeling Linear Functions Problem-Solving Strategy

  1. Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
  2. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
  3. Determine what we are trying to find, identify, solve, or interpret.
  4. Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
  5. When needed, write a formula for the function.
  6. Solve or evaluate the function using the formula.
  7. Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
  8. Clearly convey your result using appropriate units, and answer in full sentences when necessary.

Helpful tips:

  • You’ve probably heard the phrase “starting point” a lot, right? The [latex]y[/latex]-intercept is your starting point, and the slope guides you from there. Always remember, slope is your “rate of change,” and the [latex]y[/latex]-intercept is your “initial value.”
  • When given two points, use them to find your slope.
  • Diagrams are not just doodles; they’re visual aids. Use them to map out the problem and see the relationships between variables.
A company sells doughnuts. They incur a fixed cost of [latex]$25,000[/latex] for rent, insurance, and other expenses. It costs [latex]$0.25[/latex] to produce each doughnut.

  1. Write a linear model to represent the cost [latex]C[/latex] of the company as a function of [latex]x[/latex], the number of doughnuts produced.
  2. Find and interpret the [latex]y[/latex]-intercept.

You can view the transcript for “Linear equation word problem | Linear equations | Algebra I | Khan Academy” here (opens in new window).

A city’s population has been growing linearly. In 2008, the population was [latex]28,200[/latex]. By 2012, the population was [latex]36,800[/latex]. Assume this trend continues.

  1. Predict the population in 2014.
  2. Identify the year in which the population will reach [latex]54,000[/latex].

You can view the transcript for “Linear Modeling” here (opens in new window).

There is a straight road leading from the town of Timpson to Ashburn [latex]60[/latex] miles east and [latex]12[/latex] miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located [latex]22[/latex] miles directly east of the town of Timpson, how far is the road junction from Timpson?

Drawing and Interpreting Scatterplots

The Main Idea

  • Scatterplots visually represent relationships between two variables.
  • Each point on a scatterplot represents a pair of values [latex](x, y)[/latex].
  • The pattern of points can indicate the type and strength of a relationship.
  • Linear relationships in scatterplots suggest a constant rate of change.
  • Not all scatterplots show clear relationships; some may show no pattern at all. 

Finding the Line of Best Fit

The Main Idea

  • The line of best fit represents the overall trend in a scatterplot.
  • It minimizes the overall distance between itself and all data points.
  • The line can be estimated visually or calculated mathematically.
  • Slope of the line indicates the rate of change between variables.
  • The line of best fit is used for making predictions within the data range. 

Understanding Interpolation and Extrapolation

The Main Idea

  • Interpolation predicts values within the range of observed data.
  • Extrapolation estimates values outside the range of observed data.
  • Interpolation is generally more reliable than extrapolation.
  • Both methods use the line of best fit or other trend models.
  • Understanding data limits is crucial for accurate predictions.
  • Model breakdown can occur, especially with extrapolation.
According to the data from the table in the cricket-chirp example, what temperature can we predict if we counted 20 chirps in 15 seconds?

Chirps 44 35 20.4 33 31 35 18.5 37 26
Temperature 80.5 70.5 57 66 68 72 52 73.5 53

Scatter plot, showing the line of best fit and where interpolation and extrapolation occurs. It is titled 'Cricket Chirps Vs Air Temperature'. The x-axis is 'c, Number of Chirps', and the y-axis is 'T(c), Temperature (F)'.

You can view the transcript for “What is Interpolation and Extrapolation?” here (opens in new window).

Distinguishing Between Linear and Nonlinear Models

The Main Idea

  • Data relationships can be linear or nonlinear.
  • The correlation coefficient (r) measures the strength and direction of linear relationships.
  • r ranges from -1 to 1, with values closer to ±1 indicating stronger linear relationships.
  • Correlation does not imply causation.
  • Visual inspection of scatterplots is crucial alongside numerical measures.
  • Nonlinear relationships require different modeling approaches.

Interpreting Correlation

  • Strong Positive ([latex]0.7 < r \leq 1[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to increase strongly.
  • Moderate Positive ([latex]0.3 < r \leq 0.7[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to increase moderately.
  • Weak Positive ([latex]0 < r \leq 0.3[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to increase weakly.
  • No Linear Correlation ([latex]r \approx 0[/latex]): No clear linear trend between [latex]x[/latex] and [latex]y[/latex].
  • Weak Negative ([latex]-0.3 \leq r < 0[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to decrease weakly.
  • Moderate Negative ([latex]-0.7 \leq r < -0.3[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to decrease moderately.
  • Strong Negative ([latex]-1 \leq r < -0.7[/latex]): As [latex]x[/latex] increases, [latex]y[/latex] tends to decrease strongly.