When deciding whether a model is reliable, it is important to consider the source of the data. Data generated by a law of physics or a geometric formula will have a mathematical model that fits the data and yields accurate predictions. With other data, such as sales records, you can approximate the data only within, or close to, the domain over which it was generated.
Using a model to predict a y-value “outside” the domain of a data set is extrapolation. Estimating within the domain is interpolation. Interpolation usually yields reliable estimates. Extrapolation becomes less reliable as you move farther away from the data.
Problem 4 Using Interpolation and Extrapolation
Cheese Consumption The table shows average annual consumption of cheese per person in the U. S. for selected years from 1910 to 2001.
| Year | Pounds Consumed |
|---|---|
| 1910 | 4 |
| 1940 | 5 |
| 1970 | 8 |
| 1975 | 10 |
| 1995 | 25 |
| 2001 | 30 |
Source: U.S. Department of Agriculture
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Use CUBICREG. Model the data with a cubic function. Graph the function with a scatter plot of the data.
Since
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Use the model to estimate cheese consumption for 1980, 2000, and 2012. In which estimate do you have the most confidence? The least confidence? Explain.
Use the cubic model from part A to estimate the cheese consumption for each year:
1980: 12.6 lb of cheese per person
2000: 29.4 lb of cheese per person
2012: 46.2 lb of cheese per person
Think
What affects your confidence in drawing conclusions from the model?
Your confidence can waver where model behavior is extreme or where there are large gaps in the data.
You can be confident in interpolating the estimates for 1980 (Y1(80)) and 2000 (Y1(100)) because the cheese consumption fits the pattern of increase shown in the table. You should have the least confidence in the extrapolated 2012 (Y1(112)) estimate because the cubic model increases so quickly beyond 2001.
✓ Got It?
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- Use LINREG to find a linear model for cheese consumption. Graph it with a scatter plot.
- Reasoning Use the model to estimate consumption for 1980, 2000, and 2012. In which of these estimates do you have the most confidence? The least confidence? Explain.
Table of Contents
- 5-1 Polynomial Functions
- 5-2 Polynomials, Linear Factors, and Zeros
- 5-3 Solving Polynomial Equations
- 5-4 Dividing Polynomials
- 5-5 Theorems About Roots of Polynomial Equations
- 5-6 The Fundamental Theorem of Algebra
- 5-7 The Binomial Theorem
- 5-8 Polynomial Models in the Real World
- 5-9 Transforming Polynomial Functions