Science:Math Exam Resources/Courses/MATH221/April 2009/Question 03
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Question 03  

The population (in hundreds) of a colony of rabbits in year is given in the table:
Find the equation of the least squares line that best ﬁts the data and use it to estimate the population at time . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

How does the line of best fit for relate to the normal equations ? 
Hint 2 

How can you turn the given data into a linear system, ? 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Recall that the least squares solution to a linear problem is the solution to the normal form equation problem: Therefore, the first thing we need to do is setup the original linear system problem. From the table we are given and data and so we would like each to satisfy the linear equations, If we let then solving the 4 linear systems is equivalent to . Computing and we have
We can solve this problem by Gaussian elimination, Subtract 3 multiples of the first row from the second row to get Therefore we have that and so therefore . The line of best fit is given by the equation and when , . Therefore, the population at will approximately by 9.8. Note that this is a reasonable answer since the population at was 9. 