The amount of waste (in millions of tons a day) generated in...

The amount of waste (in millions of tons a day) generated in a certain city from 1960 to 1995 was \begin{tabular}{c|cccccccc} year & 1960 & 1965 & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 \\ \hline amount & 86 & 99.8 & 115.8 & 125 & 132.6 & 143.1 & 156.3 & 169.5 \end{tabular} (a) Find the equation for the least squares line that best fits these data. (b) Use the result to estimate the amount of waste in the year 2000, and in the year 2005. (c) Redo your calculations using an exponential growth model $y=c e^{\alpha t}$. (d) Which model do you think most accurately reflects the data? Why?

The amount of waste (in millions of tons a day) generated in a certain city from 1960 to 1995 was
\begin{tabular}{c|cccccccc}
year & 1960 & 1965 & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 \\
\hline amount & 86 & 99.8 & 115.8 & 125 & 132.6 & 143.1 & 156.3 & 169.5
\end{tabular}
(a) Find the equation for the least squares line that best fits these data.
(b) Use the result to estimate the amount of waste in the year 2000, and in the year 2005.
(c) Redo your calculations using an exponential growth model $y=c e^{\alpha t}$.
(d) Which model do you think most accurately reflects the data? Why?

Key Concepts

Model Assessment and Comparison

Model assessment and comparison involve evaluating multiple models to determine which one best represents the underlying data trend. This process may include checking the model's fit by comparing predicted values to actual data points, analyzing residuals, or considering the theoretical appropriateness of the model for the phenomenon being studied. It helps to ensure that predictions and interpretations based on the model are reliable and meaningful.

Exponential Growth Model

An exponential growth model describes processes in which the rate of change of a quantity is proportional to the current amount of that quantity. In this model, values are expressed as the product of an initial amount and an exponential function of time. Often, by taking the logarithm of the data, the model can be linearized to facilitate parameter estimation, which is particularly useful for data that may grow at an accelerating rate.

Data Extrapolation

Data extrapolation involves using a model developed from observed data to estimate values outside the range of the data. After determining a suitable model, such as the least squares regression line or an exponential curve, predictions for future or past values can be made by substituting these values into the equation. Extrapolation assumes that the established trends continue beyond the original dataset.

Least Squares Regression

Least squares regression is a statistical method used to determine the best-fitting line through a set of data points. This method minimizes the sum of the squares of the differences between the observed values and the values predicted by the linear model. It is particularly useful for identifying a trend in a dataset and for making predictions about values that were not observed directly.

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