Science:Math Exam Resources/Courses/MATH102/December 2015/Question 05

MATH102 December 2015

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Question 05
Which of the following lines $y=ax+b$ provides the best fit to the data in the least squares sense?

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

You can probably identify the line that best fits the data just by looking, but it is better to justify your intuition with a calculation. The least-squares line is the one that minimizes the sum of the squares of the differences in y-values between the points and the line. For example, consider picture (a). This picture contains 5 data points: (1,4), (2,3), (3,2), (4,1), and (5,3). The line passes through the points (1,3), (2,3), (3,3), (4,3), and (5,3). The differences in the y-values for these points are -1, 0, 1, 2, and 0. We square each of these numbers to arrive at 1, 0, 1, 4, 0, and add to get 6. Try performing the same calculation for the remaining 3 lines and figure out which one will minimize this quantity.

Checking a solution serves two purposes: helping you if, after having used the hint, 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. You can probably identify the line that best fits the data just by looking, but it is better to justify your intuition with a calculation. The least-squares line is the one that minimizes the sum of the squares of the differences in y-values between the points and the line. We start with picture (a). This picture contains 5 data points: (1,4), (2,3), (3,2), (4,1), and (5,3). The line passes through the points (1,3), (2,3), (3,3), (4,3), and (5,3). The differences in the y-values for these points are -1, 0, 1, 2, and 0. We square each of these numbers to arrive at 1, 0, 1, 4, 0, and add to get 6. The line in (b) passes through (1,3.5), (2,3), (3,2.5), (4,2), and (5,1.5). The differences in y-values are therefore -0.5, 0, 0.5, 1, and -1.5. Squaring gives 0.25, 0, 0.25, 1, and 2.25. We add these numbers and arrive at 3.75. The line in (c) passes through (1,4), (2,3), (3,2), (4,1), and (5,0). The differences in y-values are therefore 0, 0, 0, 0, -3. Squaring gives 0, 0, 0, 0, 9. Adding gives 9. The line in (d) passes through (1,5), (2,4), (3,3), (4,2), and (5,1). The differences in y-values are therefore 1, 1, 1, 1, and -2. Squaring these gives 1, 1, 1, 1, 4. We add these together to get 8. The smallest of these numbers, 3.75, is achieved by (b). answer: $\color {blue}(b)$