Finding the Best Model
In Exercises 5–16, construct a scatterplot and identify the mathematical model that best fits the given data. Assume that the model is to be used only for the scope of the given data, and consider only linear, quadratic, logarithmic, exponential, and power models.
Global Warming Listed below are mean annual temperatures (°C) of the earth for each decade, beginning with the decade of the 1880s. Find the best model and then predict the value for 2090–2099. Comment on the result.

Verified step by step guidance

1. Step 1: Organize the data by pairing each decade with its corresponding mean annual temperature. For example, assign x-values as 1880, 1890, ..., 1970 and y-values as the temperatures given: 13.819, 13.692, ..., 14.636.
2. Step 2: Construct a scatterplot by plotting the decades on the x-axis and the mean annual temperatures on the y-axis. This visual will help identify the trend and the type of model that fits best.
3. Step 3: Consider the five types of models: linear, quadratic, logarithmic, exponential, and power. Fit each model to the data using appropriate regression techniques or software to find the best fit.
4. Step 4: Compare the goodness of fit for each model, such as by looking at the coefficient of determination (\[R^2\]) or residual plots, to determine which model best represents the data within the given range.
5. Step 5: Use the best-fitting model to predict the mean annual temperature for the decade 2090–2099 by substituting the corresponding x-value (e.g., 2090) into the model's equation. Then, interpret the prediction in the context of global warming.

Key Concepts

Scatterplot Construction

A scatterplot is a graphical representation of data points on a coordinate plane, showing the relationship between two variables. It helps visualize patterns, trends, or correlations, which is essential for selecting an appropriate mathematical model.

Model Types and Selection

Different models like linear, quadratic, logarithmic, exponential, and power describe various types of relationships between variables. Choosing the best model involves comparing how well each fits the data, often using visual fit and statistical measures like residuals or R-squared.

Prediction and Extrapolation

Once a model is selected, it can be used to predict values within the data range or slightly beyond (extrapolation). Predictions should be interpreted cautiously, especially when extending far beyond the observed data, as model accuracy may decrease.