Question

Describe the steps you would follow to decide which of the three basic models best fits a collection of data.

Short answer

The right model can be decided by first plotting the data and observing the underlying pattern. Then, perform a line of best fit and calculate residuals for each model. The model with the smallest residuals will be a better fit. Additionally, using statistical measures like R Squared, AIC, or BIC, can also provide a clear indication of the better performing model.

Step-by-step solution

Data Visualization

Start by plotting the data given. You can either scatter plot it or create a line chart. This will help you visualize the distribution and pattern of the data.

Identifying the Pattern

By observing the plotted data, try to draw insights about the underlying pattern. If the data shows a straight-line pattern, it's likely a linear model. A curved pattern (either upwards or downwards facing) suggests a quadratic model. For a pattern where the data starts low, increases, and then levels off, or starts high and decreases before leveling off, it's likely to be an exponential model.

Comparative Analysis

Next, perform the line of best fit for each model type (linear, quadratic, exponential) and calculate the respective residuals. Residuals are the difference between the observed and predicted values. The model with the smallest residuals is considered the best fit.

Application of Goodness of Fit Test

Another precise way to decide the goodness of fit is to use statistical measures. Use the coefficient of determination (R Squared), Akaike information criterion (AIC), or Bayesian information criterion (BIC). The model having higher R Squared (closer to 1), or lower AIC and BIC is considered to be a better fit.