Question

1. An important advance in ecological research from tho late 1960's was the heory of Islnd Biogeography, which demonstrated a relationship between the size of islands and the number ot species of different groups that could be found in the islands. One of the irst examples founut was the number of amphibian and reptile species on islands in the West Indies, The data and a scatterplot of number of species versus island area (in square miles) is given helow T. An important advance in ecological research from the late 19' Area Number Redonda Saba Montserrit Puerto Rico Jamaica Hispaniola Cuba 13 104 8816 11669 77793 103723 10 84 76 (a) Clearly, this relationship is not linear. Find a logarithmic transformation that produces a clearly linear relationship. Support your choice with appropriate graphical or numerical analysis of the data. 2.89(L.) Leth ow er mode се Peue) (b) Use your model to predict the number of amphibian and reptiles species on the island of Martinique, which has an area of 436 square miles ous moole) lo 430 900344531

I need help doing this I'm really bad at getting the a and be. can you answer and explain

Answer 1

a)

R software is used

x is Area and y is Number of Species

> x=c(3,13,104,8816,11669,77793,103723)
> y=c(3,5,9,40,39,84,76)
> plot(x,y,type='l')

Make log transformation on both x and y i.e. Area and Number of Species respectively.

> log_y = log(y)
> log_x=log(x)

> plot(log_x,log_y,type='l')

>

This shows there is linear relationship between log(x) and log(y).

b)

Fit linear regression model on log(x) and log(y).

log(x) is independent variable i.e. regressor.

log(y) is dependent variable i.e. response.

> model=lm(log_y~log_x)
> model

Call:
lm(formula = log_y ~ log_x)

Coefficients:
(Intercept) log_x
0.7615 0.3165

Linearizable function for this model is $y = \beta_0 * x^{\beta_1}$

From this we conclude that our model is  log(y) = log($\beta_0$) + $\beta_1$*log(x)   

> predict=predict(model,data.frame(log_x=log(436)))
> predict
2.685127 This is value for log(y) i.e. log(Martinique)

For finding value of Martinique exponential(log(Martinique))
> exp(predict)
14.66007

Therefore For Martinique number of Species = 14.66007