Question

The data file TreeAgeDiamSugarMaple.txt is given below:

Diamet Age
2.86 1.61
4.29 3.23
8.57 6.45
14.29 8.06
15.71 14.52
17.14 22.58
20.00 24.19
54.29 37.10
60.00 40.32
88.57 32.26
100.00 46.77
102.86 51.61
105.71 58.07
108.57 37.10
114.29 37.90
177.14 78.23
178.57 100.81
252.86 76.61
254.29 90.32
311.43 134.68
397.14 96.77
491.43 151.61
560.00 163.71
531.43 140.32
532.86 112.10
545.71 109.68
541.43 123.39

a.) Determine a good polynomial regression model for this data using the AIC and/or BIC criteria. (Fit all polynomial regression models up to a maximum degree of 8 and then choose the best one.)

b.) For the second degree polynomial model, test the null hypothesis that the coefficient of x^2 is 0. Use a level of significance: α = 0.05. Report the p-value as well (note that this doesn't involve α).

c.) Produce a plot of the residuals versus the predicted values for the model chosen in part (a) above. Is there anything remarkable about the plot?

d.) Using the model in part (a), predict the age of a tree with a diameter of 110 by using an interval in which you have 95% confidence (i.e. obtain a 95% prediction interval for Y at x = 110.)

Answer 1

Answer(a):

degree of polynomial	b0	b1	b2	b3	b4	b5	b6	b7	b8	AIC
linear	-46.5429	3.8045								315.88
quadratic	-34.8373	3.280646	0.003414							317.65
cubic	14.20339	-0.93858	0.072253	-0.00029						315.76
4th degree	7.82E+00	7.29E-02	4.10E-02	2.49E-05	-9.86E-07					317.69
5th degree	-3.53E+01	9.99E+00	-4.08E-01	7.44E-03	-5.19E-05	1.24E-07				317.75
6th degree	2.39E+01	-7.60E+00	7.06E-01	-2.05E-02	2.73E-04	-1.64E-06	3.60E-09			316.74
7th degree	3.45E+01	-1.16E+01	1.05E+00	-3.25E-02	4.80E-04	-3.49E-06	1.19E-08	-1.45E-11		318.66
8th degree	-2.51E+01	1.51E+01	-1.89E+00	1.03E-01	-2.67E-03	3.71E-05	-2.80E-07	1.09E-09	-1.69E-12	318.76

we fitted polynomial models up to max of 8th degree and 3rd degree polynomial (cubic) was found to best on the basis of lowest AIC value.