Competing Function Model Validation

Competing Function Model Validation

  • Date:

  • Learning Target: Determine the best models for a data set.

Overview

  • Throughout this course, students will work with and create models based on given data sets. Three particularly important types of models are:

    • Linear

    • Quadratic

    • Exponential

  • These models are significant as they frequently occur in real-world scenarios.

  • On the AP Exam, students must determine the most appropriate model for a given data set.

Strategies for Model Selection

  • Several strategies and methods exist for determining whether a linear, quadratic, or exponential model best fits a data set.

Comparing Models

Type of Model

  • Linear Model

  • Example Graph/Data:
    \begin{array}{|c|c|} \hline x & y \ \hline 0 & 1 \ 2 & 2.6 \ 4 & 5 \ 6 & 6.4 \ 8 & 8.7 \ \hline \end{array}

  • Data Interpretation: Use a linear model when data shows a relatively constant rate of change.

  • Quadratic Model

  • Example Graph/Data:
    \begin{array}{|c|c|} \hline x & y \ \hline 0 & 9 \ 2 & 3.95 \ 4 & 2.1 \ 6 & 4.2 \ 8 & 8.8 \ \hline \end{array}

  • Data Interpretation: Use a quadratic model when rates of change are increasing/decreasing at a relatively constant rate. Data typically exhibits a "U"-shaped or inverted pattern.

  • Exponential Model

  • Example Graph/Data:
    \begin{array}{|c|c|} \hline x & y \ \hline 0 & 0.5 \ 2 & 1 \ 4 & 1.9 \ 6 & 4.2 \ 8 & 8.8 \ \hline \end{array}

  • Data Interpretation: Use an exponential model when output values are approximately proportional, meaning each successive output results from repeated multiplication.

Standards

  • APPC.2.6.A: Construct linear, quadratic, and exponential models based on a data set.

  • APPC.2.6.B: Validate a model constructed from a data set.

Example 1: Analyzing Functions

  • Selected values from several functions are provided. The task is to:

    • a) Sketch the scatterplot for each function and examine it to determine the best-fitting model (linear, quadratic, or exponential).

    • b) Data sets:

    • (i)
      \begin{array}{|c|c|} \hline x & f(x) \ \hline 0 & 11 \ 2 & 8.2 \ 4 & 5 \ 6 & 2.3 \ 8 & -1 \ \hline \end{array}

    • (ii)
      \begin{array}{|c|c|} \hline x & g(x) \ \hline -1 & 2 \ 1.5 & 5.5 \ 4 & 10.5 \ 6.5 & 5.75 \ 9 & 2.25 \ \hline \end{array}

    • (iii)
      \begin{array}{|c|c|} \hline x & h(x) \ \hline 1 & 10 \ 3 & 5.2 \ 5 & 2.4 \ 7 & 1.3 \ 9 & 0.7 \ \hline \end{array}

    • (iv)
      \begin{array}{|c|c|} \hline x & k(x) \ \hline -2 & 2 \ 1 & 3 \ 4 & 4.5 \ 7 & 6.75 \ 10 & 10.25 \ \hline \end{array}

Residuals

  • When creating a model, predictions can be made for the dependent variable (output) using the independent variable (input).

  • The residual, which is the difference between the actual output and predicted output, is crucial:

    • Formula:
      \text{Residual} = \text{Actual Output Value} - \text{Predicted Output Value}

Standards

  • APPC.2.6.A: Construct linear, quadratic, and exponential models based on a data set.

  • APPC.2.6.B: Validate a model constructed from a data set.

Example 2: Modeling Newborn Weight

  • Context: The weight of newborn babies is modeled using a linear function for the first four months after birth.

  • Data Set:

    • Table of the weight W(t), in kilograms, for a specific newborn baby, where t represents the months after birth:
      \begin{array}{|c|c|} \hline t & W(t) \ \hline 0 & 3.2 \ 1 & 4.2 \ 2 & 5.1 \ 3 & 5.8 \ 4 & 6.4 \ \hline \end{array}

  • Tasks:

    • a) Use regression to find a linear model of the form y = a + bx for the weight.

    • b) Predict the baby’s weight for t = 2.5 months.

    • c) Calculate the residual for this weight given that the actual weight was 5.5 kg. Determine if the model underestimated or overestimated.

Using a Residual Plot

  • A residual plot displays all the residuals for assessed data and can determine model appropriateness:

    • If the residual plot appears without any pattern, the model is appropriate.

    • A clear pattern indicates that the regression model was unsuitable.

Standards

  • APPC.2.6.A: Construct linear, quadratic, and exponential models based on a data set.

  • APPC.2.6.B: Validate a model constructed from a data set.

Example 3: Analyzing a Residual Plot

  • Context: An exponential regression was used for a data set. A residual plot is provided.

  • Task: Determine the appropriateness of the exponential regression model based on the residual plot:

    • (A) The exponential model is not appropriate because the residuals show no pattern.

    • (B) The exponential model is not appropriate because the residuals show a pattern.

    • (C) The exponential model is appropriate because the residuals show no pattern.

    • (D) The exponential model is appropriate because the residuals show a pattern.

Example 4: Analysis of Multiple Models

  • Context: Students created linear, quadratic, and exponential regression models and constructed residual plots for each.

  • Task: Determine which model is most appropriate based on the given residual plots.

Example 5: Paint Usage for a Mural

  • Context: Mr. Passwater plans to measure how much paint (in quarts) is needed for circles of varying sizes, based on radius r (in feet).

  • Tasks:

    • a) Determine whether a linear, quadratic, or exponential model is more suitable and provide reasoning.

    • b) Discuss whether it is more appropriate for the model used to underestimate or overestimate the paint needed and justify the reason.