Professor Ruddy Redback checks the chlorine residual in his...

Professor Ruddy Redback checks the chlorine residual in his swimming pool at the times shown in the following table, after his pool service has treated the pool with a water purifier. $$ \begin{array}{cc} \begin{array}{c} \text { Hours } \\ \text { (after treatment) } \end{array} & \begin{array}{c} \text { Chlorine Residual } \\ \text { (parts per million) } \end{array} \\ x & y \\ \hline & \\ 06 & 1.86 \\ 12 & 1.73 \\ 18 & 1.61 \\ 24 & 1.50 \\ 30 & 1.39 \\ 36 & 1.30 \\ & \\ \hline \end{array} $$ Use the method of least squares to fit an exponential curve of the form $y=\alpha \exp (\beta x)$ to the data for Professor Redback's pool.

Professor Ruddy Redback checks the chlorine residual in his swimming pool at the times shown in the following table, after his pool service has treated the pool with a water purifier.
$$
\begin{array}{cc}
\begin{array}{c}
\text { Hours } \\
\text { (after treatment) }
\end{array} & \begin{array}{c}
\text { Chlorine Residual } \\
\text { (parts per million) }
\end{array} \\
x & y \\
\hline & \\
06 & 1.86 \\
12 & 1.73 \\
18 & 1.61 \\
24 & 1.50 \\
30 & 1.39 \\
36 & 1.30 \\
& \\
\hline
\end{array}
$$
Use the method of least squares to fit an exponential curve of the form $y=\alpha \exp (\beta x)$ to the data for Professor Redback's pool.

Key Concepts

Parameter Estimation

Parameter estimation in the context of exponential curve fitting involves using the results from the linear regression on the transformed data to derive estimates for ln(?) and ?, and then converting ln(?) back to ? via exponentiation. This process yields the specific parameters for the exponential model that best fit the observed data according to the least squares criterion.

Linear Regression

Linear regression is a fundamental statistical method used for modeling the relationship between a dependent variable and one or more independent variables by fitting a linear equation to observed data. In the context of exponential regression, after applying a log transformation to the data, linear regression techniques are used to find the best-fit line for ln(y) as a function of x, which in turn facilitates the estimation of the parameters of the exponential model.

Log Transformation

Log transformation is a key technique used to linearize exponential relationships. By taking the natural logarithm of both sides of the exponential model y = ? exp(? x), the model is transformed into a linear form ln(y) = ln(?) + ? x. This linear form allows the use of linear regression methods to estimate the parameters ln(?) and ?, thereby enabling the fitting of the original exponential model.

Exponential Curve Fitting

Exponential curve fitting involves modeling data with functions of the form y = ? exp(? x), where the relationship between the independent variable x and the dependent variable y is exponential. This type of modeling is particularly useful when the rate of change of the dependent variable is proportional to its current value, often observed in growth and decay processes.

Least Squares Method

The least squares method is a standard approach in regression analysis used to determine the best-fitting curve or line that minimizes the sum of the squared residuals, which are the differences between the observed values and the values predicted by the model. This technique is widely used because it provides a mathematically tractable way of fitting models to data, ensuring that the overall error is minimized in a squared sense.

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