An elk population is censused for 6 years and the population time series is 140, 170, 190, 228, 215, 251. What is average lambda? Hint: how do you take the average of lambda in a t... An elk population is censused for 6 years and the population time series is 140, 170, 190, 228, 215, 251. What is average lambda? Hint: how do you take the average of lambda in a time series? Report your answer to 3 decimal places.
Understand the Problem
The question asks to calculate the average lambda (growth rate) of an elk population over a 6-year period, given a time series of population counts. Lambda is calculated as the ratio of the population in one year to the population in the previous year.
Answer
1. 128
Answer for screen readers
- 128
Steps to Solve
- Calculate lambda for each year
Lambda ($\lambda$) is calculated as the population in the current year divided by the population in the previous year. We have 6 years of data, so we can calculate 5 lambda values.
$\lambda_1 = \frac{170}{140} = 1.214$ $\lambda_2 = \frac{190}{170} = 1.118$ $\lambda_3 = \frac{228}{190} = 1.200$ $\lambda_4 = \frac{215}{228} = 0.943$ $\lambda_5 = \frac{251}{215} = 1.167$
- Calculate the average lambda
To calculate the average lambda, sum the individual lambda values, and divide by the number of lambda values. $$ \text{Average } \lambda = \frac{\lambda_1 + \lambda_2 + \lambda_3 + \lambda_4 + \lambda_5}{5} $$
$$ \text{Average } \lambda = \frac{1.214 + 1.118 + 1.200 + 0.943 + 1.167}{5} = \frac{5.642}{5} = 1.1284 $$
- Report the answer to 3 decimal places
Rounding 1.1284 to three decimal places gives 1.128.
- 128
More Information
Lambda ($\lambda$) values greater than 1 indicate population growth, while values less than 1 indicate population decline.
Tips
A common mistake is to calculate lambda as the current year divided by the first year, instead of the previous year. Another common mistake is to calculate the average population and use that to determine lambda instead of averaging the yearly lambdas.
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