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Modeling Linear Functions with Scattergrams in Data Analysis

Learn to create linear models from scattergrams with real-world examples. Understand linear relationships and make predictions using data points and linear functions.

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Modeling Linear Functions with Scattergrams in Data Analysis

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  1. Chapter 2 Modeling with Linear Functions

  2. Section 2.1 Using Lines to Model Data

  3. Section 2.1 Slide 3 Using Lines to Model Data Scattergrams Example The number of Grand Canyon visitors is listed in the table for various years. Describe the data. Solution • Let v be the number (in millions) of visitors • Let t be the number of years since 1960

  4. Section 2.1 Slide 4 Using Lines to Model Data Scattergrams Example Continued Sketch a line that comes close to (or on) the data points. The graph on the left does the best job of this.

  5. Section 2.1 Slide 5 Definitions Linear Models If the points in a scattergram of data lie close to (or on) a line, then we say that the relevant variables are approximately linearly related. For the Grand Canyon situation, variables t and v are approximately linearly related. A model is a mathematical description of an authentic situation. We say that the description models the situation. Definition

  6. Section 2.1 Slide 6 Definitions Linear Models Definition A linear model is a linear function, or its graph, that describes the relationship between two quantities for an authentic situation. The Grand Canyon model is a linear model Every linear model is a linear function Functions are used to describe situations and to describe certain mathematical relationships Property

  7. Section 2.1 Slide 7 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions Example Use a linear model to predict the number of visitors in 2010. Solution • Year 2010 corresponds to t = 50: 2010 – 1960 = 50 • Locate point on linear model for t = 50 • The v-coordinate is approximately 5.6 • The model estimates 5.6 million visitors in 2010

  8. Section 2.1 Slide 8 Using a Linear Model to Make a Prediction and an Estimate Using a Linear Model to Make Estimates and Predictions Example Use a linear model to estimate the year there ware 4 million visitors. Solution • 4 million visitors corresponds to v = 4 • The corresponding v-coordinate is approx. t = 32 • According to the linear model, there were 4 million visitors in the year 1960 + 32 = 1992

  9. Section 2.1 Slide 9 Deciding Whether to Use a Linear Function to Model Data When to Use a Linear Function to Model Data Example Consider the scattergrams. Determine Situation 1 Situation 2 Situation 3 whether a linear function would model it well. • Situation 1 Close to line-describes a linear function • Situation 2 & 3 Points do not lie close to one line • A linear model would not describe these situations Solution

  10. Section 2.1 Slide 10 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown Example The wild Pacific Northwest salmon populations are listed in the table for various years. 1. Let P be the salmon population (in millions) at t years since 1950. Find a linear model that describes the situation. • Data is described in terms of P and t in a table • Sketch a scattergram (see the next slide) Solution

  11. Section 2.1 Slide 11 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown Example Continued 2. Find the P-intercept of the model. What does it mean? 3. Use the model to predict when the salmon will become extinct.

  12. Section 2.1 Slide 12 Intercepts of a Model; Model Breakdown Intercepts of a Model and Model Breakdown Solution • P- intercept is (0, 13) • When P = 13, t = 0 (the year 1950) • According to the model, there were 13 million salmon in 1950 • T-intercept is (45, 0) • When P = 0, t = 45 (the year 1950 + 45 = 1995 • Salomon are still alive today • Our model is a false prediction

  13. Section 2.1 Slide 13 Definition Intercepts of a Model and Model Breakdown Definition For situations that can be modeled by a function whose independent variable is t: We perform interpolation when we part of the model whose t-coordinates are not between the t-coordinates of any two data points.

  14. Section 2.1 Slide 14 Definition Intercepts of a Model and Model Breakdown Definition We perform extrapolation when we use a part of the model whose t-coordinates are not between the t-coordinates of any two data points. Definition When a model gives a prediction that does not make sense or an estimate that is not a good approximation, we say that model breakdown has occurred.

  15. Section 2.1 Slide 15 Modifying a Model Intercepts of a Model and Model Breakdown Example In 2002, there were 3 million wild Pacific Northwest salmon. For each of the following scenarios that follow, use the data for 2002 and the data in the table to sketch a model. Let P be the wild Pacific Northwest salmon population (in millions) at t years since 1950. 1. The salmon population levels off at 10 million. 2. The salmon become extinct.

  16. Section 2.1 Slide 16 Modifying a Model Intercepts of a Model and Model Breakdown Solution

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