Exponential Regression Parameter Estimation

00:00 4 marks

Given the following data set:

x: 1, 2, 3, 4

y: 3.0, 4.1, 5.5, 7.2

Assuming that the relationship between x and y can be modeled by the exponential function \(y = a\,e^{b x}\), transform the model by taking the natural logarithm of both sides to obtain a linear relationship. Using the transformed data and any two suitable points, estimate the parameters a and b. Show your working steps.

Hint Hint

Hint

Remember, taking the natural log of the model gives \(\ln(y) = \ln(a) + bx\). This is in the form \(Y = C + bX\), where \(Y = \ln(y)\), \(C = \ln(a)\), and \(X = x\). Use any two points from the transformed data to estimate the slope (b) and the intercept (\(\ln(a)\)). For Paper 1, ensure you can perform these calculations without a calculator.

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Answer Answer

Complete Answer

Estimated parameters: a \(\approx 2.24\) and b \(\approx 0.292\).

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