Exercises 1) For the next parts, consider the following data set. x y 3 8 4 20 (

Question

Exercises 1) For the next parts, consider the following data set. x y 3 8 4 20 (a) Solve the normal equations to find the least-squares line y ma +b. Calculate the errors ei := yi-(mari + b), 1 i 4. (b) Least squares minimizes the sum of the errors squared. Using this fact, find the minimum value of el + + + (c) Find the height C of the best horizontal line to fit the data set. An exact fit would solve the (unsolvable) system of equations C-0; C 8; C-8; C-20. Find the 4 by 1 matrix A in these equations and solve the normal equation. Draw the data points, the horizontal line at height C, and the four errors e1, e2, es, e4 2) If you look in a statistics book, you will find equations for the least-squares line Assuming the data points are (z1, 1), (r2, 32),... (Ep, Up), find a general form for the least- squares line. In other words, solve for m and b in terms of zi and ya, 1 i S p. Hint: so, for example 1A = Tp 1

Explanation / Answer

a. Traze trend line across the given points, and this line can give you the least squares.

the trend line will use points (3, 13) (2, 9) and use these to find the equation

m=4

9=4(2) +b b=1

y = 4x+1

error for x=1 and y=5

e = 5 - ((4*1)+1) = 0

error for x=4 and y= 17

e = 17 - ((4*4)+1) = 0

b. There is no variation in error sum(e)=0

c.

0 8.00228 8.544 20.3961 8.0622 3.253 2 12.3693 8.59 2 3.481 12.0411 20.485 12.0416 12.39 0

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