Question

Consider the following: (a) suppose you are given the following x, y data pairs. x 2 5 6 y 6 4 8 find the least-squares equation for these data. (use 3 decimal places.) (b) now suppose you are given these x, y data pairs. x 6 4 8 y 2 5 6 find the least-squares equation for these data. (use 3 decimal places.) (c) in the data for parts (a) and (b), did we simply exchange the x and y values of each data pair? (d) solve your answer from part (a) for x. (use 3 decimal places.)

Answer 1

For data pairs (a):
`\(y \approx 0.710x + 3.487\)`. For data pairs (b):
`\(y \approx 0.584x + 0.315\)`. No, data in (a) and (b) differ. Solve for x in (a):
`\(x \approx 3.701\).`

To find the least-squares equation for the given data pairs, we'll use the formulas for the slope (m) and y-intercept (b) in the equation y = mx + b.

Part (a):

Given data pairs:
`\(x = [2, 5, 6]\)` and
`\(y = [6, 4, 8]\)`

Step 1: Calculate the mean of x
`(\( \bar{x} \)) and y (\( \bar{y} \))`

`\[\bar{x} = (2 + 5 + 6)/(3) = 4.333, \quad \bar{y} = (6 + 4 + 8)/(3) = 6.000\]`

Step 2: Calculate the slope (m):

`\[m = \frac{\sum_(i=1)^(n) (x_i - \bar{x})(y_i - \bar{y})}{\sum_(i=1)^(n) (x_i - \bar{x})^2}\]`

`\[m = ((2-4.333)(6-6) + (5-4.333)(4-6) + (6-4.333)(8-6))/((2-4.333)^2 + (5-4.333)^2 + (6-4.333)^2)\]`

`\[m \approx (1.667 - 0.666 + 3.334)/(6.111) \approx (4.335)/(6.111) \approx 0.710\]`

Step 3: Calculate the y-intercept (b):

`\[b = \bar{y} - m \cdot \bar{x}\]`

`\[b = 6.000 - 0.710 \cdot 4.333 \approx 3.487\]`

Step 4: Write the least-squares equation:

`\[y \approx 0.710x + 3.487\]`

Part (b):

Given data pairs:
`\(x = [6, 4, 8]\) and \(y = [2, 5, 6]\)`

Repeat the same steps as in Part (a).

Step 1:

`\[\bar{x} = (6 + 4 + 8)/(3) = 6.000, \quad \bar{y} = (2 + 5 + 6)/(3) = 4.333\]`

Step 2:

`\[m \approx ((6-6)(2-4.333) + (4-6)(5-4.333) + (8-6)(6-4.333))/((6-6)^2 + (4-6)^2 + (8-6)^2)\]`

`\[m \approx (-2.333 + 1.333 + 3.334)/(4) \approx (2.334)/(4) \approx 0.584\]`

Step 3:

`\[b = \bar{y} - m \cdot \bar{x} \approx 4.333 - 0.584 \cdot 6 \approx 0.315\]`

Step 4:

`\[y \approx 0.584x + 0.315\]`

Part (c):

No, we did not simply exchange the x and y values of each data pair. The data in part (a) and part (b) have different values.

Part (d):

To solve the equation from Part (a) for x:

`\[x = (y - b)/(m)\]`

Substitute the values
`\(y = 6\) (from part (a)), \(b = 3.487\), and \(m = 0.710\):`

`\[x \approx (6 - 3.487)/(0.710) \approx 3.701\]`

So, the solution for x is approximately
`\(3.701\)` (rounded to 3 decimal places).