Regression, differences and their formulas

• November 20th 2009, 09:58 AM
MathBane
Regression, differences and their formulas
The question:

For this exercise, round all regression parameters to three decimal places.

In the fishery sciences it is important to determine the length of a fish as a function of its age. One common approach, the von Bertalanffy model, uses a decreasing exponential function of age to describe the growth in length yet to be attained; in other words, the difference between the maximum length and the current length is supposed to decay exponentially with age. The following table shows the length L, in inches, at age t, in years, of the North Sea sole.

t = age
L = Length
1 |3.7
2 |7.5
3 |10
4 |11.5
5 |12.7
6 |13.5
7 |14
8 |14.4
Suppose the maximum length attained by the sole is 15.0 inches.

(a) Make a table showing, for each age, the difference D between the maximum length and the actual length L of the sole.

t = age D = Difference
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |0.6
(I was only able to get row 8 by using the information underneath the previous table.)

(b) Find the exponential function that approximates D. (Round all regression parameters to three decimal places.)

D =

(c) Find a formula expressing the length L of a sole as a function of its age t. (Round all parameters to three decimal places.)

L =
(I thought this would be $4.877(1.176)^t$, but that was incorrect.)
• November 20th 2009, 11:14 AM
masters
Quote:

Originally Posted by MathBane
The question:

For this exercise, round all regression parameters to three decimal places.

In the fishery sciences it is important to determine the length of a fish as a function of its age. One common approach, the von Bertalanffy model, uses a decreasing exponential function of age to describe the growth in length yet to be attained; in other words, the difference between the maximum length and the current length is supposed to decay exponentially with age. The following table shows the length L, in inches, at age t, in years, of the North Sea sole.

t = age L = Length
1 |3.7
2 |7.5
3 |10
4 |11.5
5 |12.7
6 |13.5
7 |14
8 |14.4
Suppose the maximum length attained by the sole is 15.0 inches.

(a) Make a table showing, for each age, the difference D between the maximum length and the actual length L of the sole.

t = age D = Difference
1 |11.3
2 | 7.5
3 | 5
4 | 3.5
5 | 2.3
6 | 1.5
7 | 1
8 |0.6
(I was only able to get row 8 by using the information underneath the previous table.) Why?

(b) Find the exponential function that approximates D. (Round all regression parameters to three decimal places.)

D = 17.466(.662)^t

(c) Find a formula expressing the length L of a sole as a function of its age t. (Round all parameters to three decimal places.)

L = a + b ln (t)
L = 3.939 + 5.261 ln (t)
(I thought this would be $4.877(1.176)^t$, but that was incorrect.)

Hi Mathbane,

The Difference table is exponential regression, but the Length table is logarithmic.