Introductory Statistical Methods - MODELLING ASSESSMENT TASK
EXERCISE 1: Answer this question using R. Present your R code and output in your answers.
The following data are from a long-term study of a dam in NSW.
|
Depth (m)
|
0
|
3
|
6
|
12
|
18
|
24
|
30
|
36
|
|
Temperature (°C)
|
25.0
|
24.7
|
24.8
|
21.7
|
17.2
|
15.5
|
14.6
|
14.0
|
(i) Use R and the nls function to model this data with a logistic growth curve where depth is equivalent to the time variable and temperature is the variable we are trying to model.
(ii) How well does the model fit the data? Present the residual SS.
(iii) Fit a cubic polynomial to the data. Based on the residual SS does the cubic polynomial fit the data better than a logistic growth model? Ignoring which model fits the data better, give two reasons why you would use a logistic growth model as compared to a cubic polynomial
(iv) Plot the data and the fitted logistic equation on the same graph.
(v) Interpret the estimates of the parameters of the logistic equation biologically where possible.
EXERCISE 2- Answer this question by using hand and R/Excel - show working.
For a certain corn crop, the yield Y(x) increases with an increasing level of fertiliser x applied, where x is measured in kg ha-1 and the model is
Y(x) = 5000 - (3000/1+0.01x)
(i) Use R or Excel to plot the function for 0 ≤ x ≤ 300. What is the modelled corn yield when no fertiliser is applied? What does the shape of the curve imply about the affect of fertiliser on yield?
(ii) a. What does Y'(100) represent on the curve?
b. Given Y'(x) = 30/(1+0.01x)2, find the exact value of Y'(100).
c. Use numerical differentiation to find an approximation to Y'(100) by using the points x = 100 and x = 100.001.
Note: Keep as many decimal places as possible in your calculations, to minimise rounding error in your final answer.
d. Compare the exact value to the approximate value by finding the relative error, where Relative error = |Exact value - Approximation/Exact value|. Comment on the effectiveness of the approximation.
(iii) a. Find the area under the curve Y(x) between 100 and 150, by using the following formula: F(150) - F(100), where F(x) = 5000x - 300000 ln(1+0.01x)
b. Use numerical differentiation to show that the area is approximately 182916.7. Steps: (1) Draw a picture showing the 2 trapezoids. (2) Calculate the area of the 2 trapezoids.
c. Find the relative error and comment.