The following table shows the average Valentine's Day...

The following table shows the average Valentine's Day spending in dollars per consumer for various years. $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2011 & 2012 & 2013 \\ \hline \text { Spending ( } \$ \text { ) } & 103 & 116 & 126 & 131 \\\hline\end{array}$$ (a) Use exponential regression to approximate values for $a$ and $b$ so that $f(x)=a+b \ln x$ models the data, where $x=1$ corresponds to $2010, x=2$ to 2011 and so on. (b) Use your function to estimate average spending in 2012 and compare to the value in the table.

The following table shows the average Valentine's Day spending in dollars per consumer for various years.
$$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2010 & 2011 & 2012 & 2013 \\ \hline \text { Spending ( } \$ \text { ) } & 103 & 116 & 126 & 131 \\\hline\end{array}$$
(a) Use exponential regression to approximate values for $a$ and $b$ so that $f(x)=a+b \ln x$ models the data, where $x=1$ corresponds to $2010, x=2$ to 2011 and so on.
(b) Use your function to estimate average spending in 2012 and compare to the value in the table.

Key Concepts

Model Prediction and Evaluation

Once a regression model is constructed by estimating its parameters, it can be used to predict new or unseen data points. The predicted values can then be compared with actual observed values to assess the model's accuracy and reliability, which is essential for validating the model's practical use in forecasting.

Logarithmic Transformation

A logarithmic transformation involves applying the natural logarithm to one or more variables in a dataset. This transformation can help linearize data that follow a non-linear pattern, making it easier to use linear regression techniques on models that initially appear exponential or multiplicative in nature.

Regression Analysis

Regression analysis is a statistical method used to model and analyze the relationship between a dependent variable and one or more independent variables. It involves fitting a mathematical equation to observed data to understand how the variables are related and to make predictions about future values.

Least Squares Estimation

Least squares estimation is a foundational method in regression analysis for finding the best-fitting line through a set of data points. By minimizing the sum of the squares of the differences between the observed values and the values predicted by the model, this technique determines the most appropriate parameters for the chosen regression model.

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