Statistical Methods

Unit 9 - Correlation and regression

9.10 Fitting of length-weight relationship
Examples:Total length (cm) and weight (g) recorded on a sample of 12 fish are given below:
(i) Fit length-weight relationship of the type W = aL^b, where W is weight and L is length of fish
(ii) Test whether b differed significantly differs from 3.
Solution:


(i) Length-weight relationship
= 1.2898 = 1.5310



Then = 1.5310-4.8917= -3.3607

Sy²= 0.12015; Sx² =0.00827
Then Variance of deviations from regression is given by:

S² =

=
= 0.0011
a= Antilog (A+S²/2)
= Antilog [-3.3607+ (0.0011) /2]
= Antilog ( -3.35015) = 0.0004
The length-weight relationship is therefore given by
W = 0.0004 L ^3.7944
(ii)To test whether the sample regression coefficient (b=3.7944) comes from a population with the regression coefficient β=3, the following null hypothesis is set up.
Hypothesis
H_o: β=3; H₁:β≠3
The test statistic used is,

This is distributed as t with n-2 df.
S_b²=[(S_y /S_x)²– b²]

=0.0122

Hence S_b = = 0.1105
Hence
(iii) Statistical decision
The table values of t at 5% and 1% level of significance are 2.228 and 3.169 respectively. As t calculated is 7.2 which are more than the table values of t at both 5% and 1% level of significance, the null hypothesis is rejected.