It is sometimes useful to find a function to model data, then use it to make predictions. Think about... How can you use data on the number of cars with large engines produced in previous years to predict the number of cars that will be produced in future years?
Year Thousands of cars (engine size 2 litres or more) Are there more cars with larger engines on the roads today? Source:
Think about... What type of function might provide a good model?
Finding an exponential model N = N 0 e kt Taking logs base e ln N = ln ( N 0 e kt ) ln N = ln N 0 + ln e kt Using the laws of logs ln N = ln N 0 + kt ln e ln N = ln N 0 + kt y = c + mx Compare with: Drawing a graph of ln N against t should give a straight line. If so, its gradient will give k and its intercept will give ln N 0.
Years after 1994 ( t )Thousands of cars ( N )ln N Cars registered with engine size 2 litres or more
k = gradient = ln N 0 = intercept = 7.318
ln N 0 = N 0 = e = 1507 N = 1507 e t From the graph Calculate N 0 Exponential model: Think about... How good is this model? e.g in 2000 i.e when t = 6 N = 1507 e × 6 = 2299 % error = predicted value – actual value actual value 100 % error = = 1.6 % 2299 –
Years after 1994 ( t )Data (000s)Model (000s)% Error % 0.0% – 0.1% – 3.3% 1.1% 1.2% 0.7% – 1.0% – 2.3% – 1.4% – 0.1% – 2.1% 0.9% 1.6% Comparison using percentage errors
Comparison using graph
N = 1507e t Exponential model: Prediction for 2008 t = 14 N = 1507 e × 14 = 1507 e = 4037 The actual values were 3731 and 3768 (thousand). Compare the predicted and actual values. Using the model to make predictions beyond 2007 Prediction for 2009 t = 15 N = 1507 e × 15 = 1507 e = 4332 Think about... Is there a better model?
Year Thousands of cars (engine size 2 litres or more) Source: Finding a new model Draw a graph to show the data for 2000 to Find a new model.
Reflect on your work Why is the percentage error a better measure for the accuracy of a model than the difference between the actual value and the value predicted by the model? What is indicated by a negative percentage error? Gas guzzlers Is your model valid for all values of t ?