Question

which type of model is most appropriate for the following data?
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$$\begin{array}{|c|c|} \\hline x&y\\ \\hline 1&2.63\\ \\hline 2&5.85\\ \\hline 3&7.74\\ \\hline 4&8.92\\ \\hline 5&10\\ \\hline 6&10.95\\ \\hline 7&11.49\\ \\hline 8&12.09\\ \\hline \\end{array}$$

Explanation:

Step1: Analyze first - differences

Calculate the differences in \(y\) values. \(\Delta y_1=5.85 - 2.63 = 3.22\), \(\Delta y_2=7.74 - 5.85 = 1.89\), \(\Delta y_3=8.92 - 7.74 = 1.18\), \(\Delta y_4=10 - 8.92 = 1.08\), \(\Delta y_5=10.95 - 10 = 0.95\), \(\Delta y_6=11.49 - 10.95 = 0.54\), \(\Delta y_7=12.09 - 11.49 = 0.6\). The first - differences are decreasing.

Step2: Consider model types

Linear models have constant first - differences. Quadratic models have non - constant first - differences but constant second - differences. Since the first - differences are decreasing and approaching zero as \(x\) increases, a logarithmic model \(y = a + b\ln(x)\) is a good fit as logarithmic functions have a decreasing rate of change as \(x\) increases.

Answer:

A logarithmic model is most appropriate.