Question

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

x 1 3 6

y 5 5 8 Find the least-squares equation for these data. (Use 3 decimal places.) y hat = + x (b)

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

x 5 5 8

y 1 3 6

Find the least-squares equation for these data. (Use 3 decimal places.) y hat = + x (c)

In the data for parts (a) and (b), did we simply exchange the x and y values of each data pair? Yes or No?

(d) Solve your answer from part (a) for x. (Use 3 decimal places.) x = + y

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

(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 will produce the same least-squares equation every time.

-Switching x and y values will not necessarily produce the same least-squares equation every time.

-Switching x and y values will never produce the same least-squares equation every time.

Answer 1

(a)

From the given data, the following Table is calculated:

X	Y	XY	X2
1	5	5	1
3	5	15	9
6	8	48	36
Total = 10	18	68	46

Least Squares Equation is given by:

(b)

From the given data, the following Table is calculated:

X	Y	XY	X2
5	1	5	25
5	3	15	25
8	6	48	64
Total = 18	10	68	114

Least Squares Equation is given by:

(c)

Answer is:

Yes

(d)

(i)

From part (a), we have:

Solving for x, we get:

i.e.,

(ii)

No, We do not get the least-squares equation of part (b) with the symbols x and y exchanged

(e)

Correct option:

Switching x and y values will never produce the same least-squares equation every time.