Question

In: Statistics and Probability

(a) Suppose you are given the following (x, y) data pairs. x 1 3 4 y...

(a) Suppose you are given the following (x, y) data pairs.

x 1 3 4
y 4 3 7

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =  +  x

(b) Now suppose you are given these (x, y) data pairs.

x 4 3 7
y 1 3 4

Find the least-squares equation for these data (rounded to three digits after the decimal).
ŷ =  +  x

(c) In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair?

Yes

No    


(d) Solve your answer from part (a) for x (rounded to three digits after the decimal).
x =  +  y

Do you get the least-squares equation of part (b) with the symbols x and y exchanged?

Yes

No    


(e) In general, suppose we have the least-squares equation y = a + bx for a set of data pairs (x, y). If we solve this equation for x, will we necessarily get the least-squares equation for the set of data pairs (y, x), (with x and y exchanged)? Explain using parts (a) through (d).

Switching x and y values sometimes produces the same least-squares equation and sometimes it is different.

In general, switching x and y values produces a different least-squares equation.   

In general, switching x and y values produces the same least-squares equation.

Solutions

Expert Solution

(a)

Following table shows the calculations:

X Y X^2 Y^2 XY
1 4 1 16 4
3 3 9 9 9
4 7 16 49 28
Total 8 14 26 74 41

(b)

Following table shows the calculations:

X Y X^2 Y^2 XY
4 1 16 1 4
3 3 9 9 9
7 4 49 16 28
Total 14 8 74 26 41

(c)

Yes

(d)

y'= 0.786*x+ 2.571

x = (y' - 2.571) / 0.786 = 0.1272*y' - 3.271

No

(e)

In general, switching x and y values produces the same least-squares equation.


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