Use linear regression to analyze the relationship between time since introduction and the natural log of snake population size.

Question

Years after Introduction(X)       Number(Y)        (Xi - X)    (Yi - Y)    (Xi - X)(Yi - Y)

1                                                  4                  -4.5       -167.3            752.9

2                                                  12                -3.5       -159.3            557.6

3                                                  23                -2.5       -148.3            370.8

4                                                  46                -1.5       -125.3            188

5                                                  94                -0.5       -77.3              38.7

6                                                 110                 0.5       -61.3             -30.7

7                                                 200                1.5         28.7              43.1

8                                                 345                2.5         173.7            434.3

9                                                 369                3.5         197.7            692

10                                               510                4.5         338.7            1524.2 

5.5(mean)                                   171.3(mean)

3.03(St. Dev)                               178.1(St.Dev)

 

 

1) Use linear regression to analyze the relationship between time since introduction and the
natural log of snake population size.
2) Calculate estimates of population growth rate (r) and initial population size (N0)
3) These data are not a perfect proxy for the size of the snake population at large. As public
awareness of the snake population has grown, so has the effort to find and kill these snakes.
Over time, more people have become involved, and more area has been searched. One
estimate figures that, on average, the effort has grown by a factor of 1.3 each year. If this is true
then an appropriate null hypothesis for r is ln(1.3) = 0.262. Test the hypothesis that your
estimate of r is significantly different than the hypothesized value of 0.262.

 

Please help explain how to do this. I'm not sure where to start. If you could include the answer that would be great, that way I can check to see if I am doing this problem right. 

 

 

Thank you!

Answer 1

I entered the data for years and number into two columns in Excel.

I used the Excel LN function (natural log) to create a third column with ln(number).

I had Excel do a scatter plot of the ln(number) column vs the years column.

I clicked on this plot to get a trend line and equation  .

 

The equation was  y = 0.5195 x + 1.4793    (R^2 =  .9563 - a good fit)

  Note that exp(1.4793) = 4.3899

 

The exponential model is obtained by exponentiation of both sides of the equation. 

This results in:

 

number = 4.3899 exp(.5195 x years).

 

So the rate is  .5195 per year and the initial population is 4.3899   (about 4 snakes)