An ecologist wished to know whether certain species of...

An ecologist wished to know whether certain species of aquatic insects have their ecological range limited by temperature. He collected the data in Table $8,$ relating the average daily temperature at different portions of a creek with the elevation (above sea level) of that portion of the creek. (The authors express their thanks to Dr. J. David Allen, formerly of the Department of Zoology at the University of Maryland, for providing the data for this exercise.) (a) Find the straight line that provides the best least-squares fit these data. (b) Use the linear function to estimate the average daily temperature for this creek at altitude 3.2 kilometers. $$\begin{array}{cc} \text {Table 8 Relationship between Elevation and} \\ \text {Temperature in a Creek} \\ \hline & \text { Average Temperature } \\ \text { Elevation (kilometers) } & \text { (degrees Celsius) } \\ \hline 2.7 & 11.2 \\ 2.8 & 10 \\ 3.0 & 8.5 \\ 3.5 & 7.5 \\ \hline \end{array}$$

An ecologist wished to know whether certain species of aquatic insects have their ecological range limited by temperature. He collected the data in Table $8,$ relating the average daily temperature at different portions of a creek with the elevation (above sea level) of that portion of the creek. (The authors express their thanks to Dr. J. David Allen, formerly of the Department of Zoology at the University of Maryland, for providing the data for this exercise.)
(a) Find the straight line that provides the best least-squares fit these data.
(b) Use the linear function to estimate the average daily temperature for this creek at altitude 3.2 kilometers.
$$\begin{array}{cc} \text {Table 8 Relationship between Elevation and} \\ \text {Temperature in a Creek} \\
\hline & \text { Average Temperature } \\
\text { Elevation (kilometers) } & \text { (degrees Celsius) } \\
\hline 2.7 & 11.2 \\
2.8 & 10 \\
3.0 & 8.5 \\
3.5 & 7.5 \\
\hline
\end{array}$$

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Key Concepts

Least Squares Regression

Least squares regression is a method used to determine the best-fitting straight line through a set of data points by minimizing the sum of the squares of the vertical deviations (errors) of the points from the line. This approach helps in finding the line that most accurately represents the trend in the data, and its slope and intercept are calculated in a way that minimizes the overall error.

Linear Function

A linear function is a mathematical relationship between two variables that can be graphed as a straight line. It is typically expressed in the form y = mx + b, where m represents the slope (the rate of change) and b represents the y-intercept (the value of y when x equals zero). Linear functions are fundamental in data modeling as they provide a simple way to describe linear relationships between variables.

Prediction using Regression

Prediction using regression involves using a regression model, such as a best-fit line, to estimate the value of the dependent variable for a given independent variable. Once the regression equation is established using methods like least squares, it can be used to interpolate or extrapolate data, providing estimated outcomes based on the observed trend in the existing data.

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