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

MATH221 April 2009

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Question 03
The population $P(t)$ (in hundreds) of a colony of rabbits in year $t$ is given in the table: $t$ 0 2 4 6 $P$ 5 6 8 9 Find the equation $P(t)=a+bt$ of the least squares line that best fits the data and use it to estimate the population at time $t=7$.

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 $A\mathbf {x} =\mathbf {b}$ relate to the normal equations $A^{T}A\mathbf {x} =A^{T}\mathbf {b}$?

Hint 2
How can you turn the given data into a linear system, $A\mathbf {x} =\mathbf {b}$?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Recall that the least squares solution to a linear problem $A\mathbf {x} =\mathbf {b}$ is the solution to the normal form equation problem: $A^{T}A\mathbf {x} =A^{T}\mathbf {b} .$ Therefore, the first thing we need to do is setup the original linear system problem. From the table we are given $t$ and $P(t)$ data and so we would like each to satisfy the linear equations, ${\begin{aligned}a+0b&=5\\a+2b&=6\\a+4b&=8\\a+6b&=9.\end{aligned}}$ If we let $A={\begin{bmatrix}1&0\\1&2\\1&4\\1&6\end{bmatrix}},\quad \mathbf {b} ={\begin{bmatrix}5\\6\\8\\9\\\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}a\\b\end{bmatrix}}$ then solving the 4 linear systems is equivalent to $A\mathbf {x} =\mathbf {b}$. Computing $A^{T}A$ and $A^{T}\mathbf {b}$ we have $A^{T}A={\begin{bmatrix}4&12\\12&56\end{bmatrix}},\quad A^{T}\mathbf {b} ={\begin{bmatrix}28\\98\end{bmatrix}}.$ We can solve this problem by Gaussian elimination, $\left[{\begin{array}{cc|c}4&12&28\\12&56&98\end{array}}\right].$ Subtract 3 multiples of the first row from the second row to get $\left[{\begin{array}{cc|c}4&12&28\\0&20&14\end{array}}\right].$ Therefore we have that $b=7/10$ and $4a+12b=4a+42/5=28$ so therefore $a=49/10$. The line of best fit is given by the equation $P(t)=49/10+7/10t$ and when $t=7$, $P(t)=49/10+49/10=49/5=9.8$. Therefore, the population at $t=7$ will approximately by 9.8. Note that this is a reasonable answer since the population at $t=6$ was 9.

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