An engineer fits a least squares regression line to calibration data. Given sum of cross products (Sxy) = 80.5000; sum of squares of x (Sxx) = 31.5000, determine the regression slope (b).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except b is given, so isolate b symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 80.5000, sum of squares of x (Sxx) = 31.5000$$

4. Step 4 — Substitute the given values:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

5. Step 5 — Evaluate:

   $$b = 2.5556$$

6. Step 6 — Check: returning b = 2.5556 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 5.1111 — kept a factor of two that cancels in the correct rearrangement.
• 1.2778 — dropped that same factor in the other direction.
• 2.8111 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

A student computes the least squares slope from sums of cross products. Given sum of squares of x (Sxx) = 69.0000; regression slope (b) = 5.9400, determine the sum of cross products (Sxy).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxy is given, so isolate Sxy symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxy:

   $$Sxy = b\,S_{xx}$$

3. Step 3

   $$List the givens: sum of squares of x (Sxx) = 69.0000, regression slope (b) = 5.9400$$

4. Step 4 — Substitute the given values:

   $$Sxy = 5.9400\,S_{xx}$$

5. Step 5 — Evaluate:

   $$Sxy = 409.9$$

6. Step 6 — Check: returning Sxy = 409.9 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 819.7 — kept a factor of two that cancels in the correct rearrangement.
• 204.9 — dropped that same factor in the other direction.
• 450.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

The least squares trend line for settlement readings is calculated. Given sum of cross products (Sxy) = 96.0000; regression slope (b) = 8.0700, determine the sum of squares of x (Sxx).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxx is given, so isolate Sxx symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxx:

   $$Sxx = \dfrac{S_{xy}}{b}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 96.0000, regression slope (b) = 8.0700$$

4. Step 4 — Substitute the given values:

   $$Sxx = \dfrac{S_{xy}}{8.0700}$$

5. Step 5 — Evaluate:

   $$Sxx = 11.8959$$

6. Step 6 — Check: returning Sxx = 11.8959 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 23.7918 — kept a factor of two that cancels in the correct rearrangement.
• 5.9480 — dropped that same factor in the other direction.
• 13.0855 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

An engineer fits a least squares regression line to calibration data. Given sum of cross products (Sxy) = 10.5000; sum of squares of x (Sxx) = 92.0000, determine the regression slope (b).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except b is given, so isolate b symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 10.5000, sum of squares of x (Sxx) = 92.0000$$

4. Step 4 — Substitute the given values:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

5. Step 5 — Evaluate:

   $$b = 0.1141$$

6. Step 6 — Check: returning b = 0.1141 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.2283 — kept a factor of two that cancels in the correct rearrangement.
• 0.0571 — dropped that same factor in the other direction.
• 0.1255 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

A student computes the least squares slope from sums of cross products. Given sum of squares of x (Sxx) = 98.5000; regression slope (b) = 1.1300, determine the sum of cross products (Sxy).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxy is given, so isolate Sxy symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxy:

   $$Sxy = b\,S_{xx}$$

3. Step 3

   $$List the givens: sum of squares of x (Sxx) = 98.5000, regression slope (b) = 1.1300$$

4. Step 4 — Substitute the given values:

   $$Sxy = 1.1300\,S_{xx}$$

5. Step 5 — Evaluate:

   $$Sxy = 111.3$$

6. Step 6 — Check: returning Sxy = 111.3 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 222.6 — kept a factor of two that cancels in the correct rearrangement.
• 55.6525 — dropped that same factor in the other direction.
• 122.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

The least squares trend line for settlement readings is calculated. Given sum of cross products (Sxy) = 26.5000; regression slope (b) = 2.0000, determine the sum of squares of x (Sxx).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxx is given, so isolate Sxx symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxx:

   $$Sxx = \dfrac{S_{xy}}{b}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 26.5000, regression slope (b) = 2.0000$$

4. Step 4 — Substitute the given values:

   $$Sxx = \dfrac{S_{xy}}{2.0000}$$

5. Step 5 — Evaluate:

   $$Sxx = 13.2500$$

6. Step 6 — Check: returning Sxx = 13.2500 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 26.5000 — kept a factor of two that cancels in the correct rearrangement.
• 6.6250 — dropped that same factor in the other direction.
• 14.5750 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

An engineer fits a least squares regression line to calibration data. Given sum of cross products (Sxy) = 172.0; sum of squares of x (Sxx) = 16.5000, determine the regression slope (b).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except b is given, so isolate b symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 172.0, sum of squares of x (Sxx) = 16.5000$$

4. Step 4 — Substitute the given values:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

5. Step 5 — Evaluate:

   $$b = 10.4242$$

6. Step 6 — Check: returning b = 10.4242 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 20.8485 — kept a factor of two that cancels in the correct rearrangement.
• 5.2121 — dropped that same factor in the other direction.
• 11.4667 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

A student computes the least squares slope from sums of cross products. Given sum of squares of x (Sxx) = 80.5000; regression slope (b) = 0.4200, determine the sum of cross products (Sxy).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxy is given, so isolate Sxy symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxy:

   $$Sxy = b\,S_{xx}$$

3. Step 3

   $$List the givens: sum of squares of x (Sxx) = 80.5000, regression slope (b) = 0.4200$$

4. Step 4 — Substitute the given values:

   $$Sxy = 0.4200\,S_{xx}$$

5. Step 5 — Evaluate:

   $$Sxy = 33.8100$$

6. Step 6 — Check: returning Sxy = 33.8100 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 67.6200 — kept a factor of two that cancels in the correct rearrangement.
• 16.9050 — dropped that same factor in the other direction.
• 37.1910 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

The least squares trend line for settlement readings is calculated. Given sum of cross products (Sxy) = 144.0; regression slope (b) = 9.1000, determine the sum of squares of x (Sxx).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except Sxx is given, so isolate Sxx symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for Sxx:

   $$Sxx = \dfrac{S_{xy}}{b}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 144.0, regression slope (b) = 9.1000$$

4. Step 4 — Substitute the given values:

   $$Sxx = \dfrac{S_{xy}}{9.1000}$$

5. Step 5 — Evaluate:

   $$Sxx = 15.8242$$

6. Step 6 — Check: returning Sxx = 15.8242 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 31.6484 — kept a factor of two that cancels in the correct rearrangement.
• 7.9121 — dropped that same factor in the other direction.
• 17.4066 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares

An engineer fits a least squares regression line to calibration data. Given sum of cross products (Sxy) = 21.5000; sum of squares of x (Sxx) = 66.0000, determine the regression slope (b).

Start with the thinking

• The governing relation printed in this handbook section is Least squares regression slope.
• Everything except b is given, so isolate b symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The least squares method finds the regression slope minimizing the squared vertical residuals.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

2. Step 2 — Rearrange symbolically for b:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

3. Step 3

   $$List the givens: sum of cross products (Sxy) = 21.5000, sum of squares of x (Sxx) = 66.0000$$

4. Step 4 — Substitute the given values:

   $$b = \dfrac{S_{xy}}{S_{xx}}$$

5. Step 5 — Evaluate:

   $$b = 0.3258$$

6. Step 6 — Check: returning b = 0.3258 to

   $$b = \dfrac{\sum (x_i - \bar{x})(y_i-\bar{y})}{\sum (x_i - \bar{x})^2}$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 0.6515 — kept a factor of two that cancels in the correct rearrangement.
• 0.1629 — dropped that same factor in the other direction.
• 0.3583 — rounded an intermediate value before the final step.

Reference: FE Handbook — Least Squares