Question

We have reason to believe that seed dispersal should follow an inverse square law, with number of seeds proportional to (1/r2) where r is the distance from the source. The data are

Distance ( r ), m 5 10 15 25

Number of seeds 200 65 24 11

Doing calculations by hand or calculator, does the distribution of seeds follow the expected distribution ?

Answer 1

y <- c(200,65,24,11 )
> x <- c(5,10,15,25 )
> invx2 <- 1/x^2
> model <- lm ( y ~ invx2 )
> summary(model)

Call:
lm(formula = y ~ invx2)

Residuals:
     1      2      3      4 
-1.771  9.566 -4.335 -3.460 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)    6.655      5.368    1.24   0.3408   
invx2       4877.909    258.674   18.86   0.0028 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.918 on 2 degrees of freedom
Multiple R-squared:  
0.9944, Adjusted R-squared:  0.9916 
F-statistic: 355.6 on 1 and 2 DF,  p-value: 0.0028

y^ = 6.655 + 4877.909/r^2

p-value = 0.0028 < alpha

hence we reject the null hypothesis

we conclude that the model is significant

the distribution of seeds follow the expected distribution