An engineer computes the residual between a measured and predicted strength. Given observed value (yi) = 91.4000; predicted value (yhat) = 69.6000, determine the residual (e).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except e is given, so isolate e 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for e:

   $$e = y_i - \hat{y}_i$$

3. Step 3

   $$List the givens: observed value (yi) = 91.4000, predicted value (yhat) = 69.6000$$

4. Step 4 — Substitute the given values:

   $$e = y_i - \hat{y}_i$$

5. Step 5 — Evaluate:

   $$e = 21.8000$$

6. Step 6 — Check: returning e = 21.8000 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 43.6000 — kept a factor of two that cancels in the correct rearrangement.
• 10.9000 — dropped that same factor in the other direction.
• 23.9800 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A student checks the residual of a data point on a least squares line. Given predicted value (yhat) = 64.2000; residual (e) = 4.9800, determine the observed value (yi).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yi is given, so isolate yi 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yi:

   $$yi = e_i + \hat{y}_i$$

3. Step 3

   $$List the givens: predicted value (yhat) = 64.2000, residual (e) = 4.9800$$

4. Step 4 — Substitute the given values:

   $$yi = e_i + \hat{y}_i$$

5. Step 5 — Evaluate:

   $$yi = 69.1800$$

6. Step 6 — Check: returning yi = 69.1800 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 138.4 — kept a factor of two that cancels in the correct rearrangement.
• 34.5900 — dropped that same factor in the other direction.
• 76.0980 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A large residual flags an outlier in the calibration data set. Given observed value (yi) = 94.9000; residual (e) = -7.6500, determine the predicted value (yhat).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yhat is given, so isolate yhat 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yhat:

   $$yhat = y_i - e_i$$

3. Step 3

   $$List the givens: observed value (yi) = 94.9000, residual (e) = -7.6500$$

4. Step 4 — Substitute the given values:

   $$yhat = y_i - e_i$$

5. Step 5 — Evaluate:

   $$yhat = 102.6$$

6. Step 6 — Check: returning yhat = 102.6 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 205.1 — kept a factor of two that cancels in the correct rearrangement.
• 51.2750 — dropped that same factor in the other direction.
• 112.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

An engineer computes the residual between a measured and predicted strength. Given observed value (yi) = 47.5000; predicted value (yhat) = 47.7000, determine the residual (e).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except e is given, so isolate e 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for e:

   $$e = y_i - \hat{y}_i$$

3. Step 3

   $$List the givens: observed value (yi) = 47.5000, predicted value (yhat) = 47.7000$$

4. Step 4 — Substitute the given values:

   $$e = y_i - \hat{y}_i$$

5. Step 5 — Evaluate:

   $$e = -0.2000$$

6. Step 6 — Check: returning e = -0.2000 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• -0.4000 — kept a factor of two that cancels in the correct rearrangement.
• -0.1000 — dropped that same factor in the other direction.
• -0.2200 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A student checks the residual of a data point on a least squares line. Given predicted value (yhat) = 49.6000; residual (e) = -9.1400, determine the observed value (yi).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yi is given, so isolate yi 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yi:

   $$yi = e_i + \hat{y}_i$$

3. Step 3

   $$List the givens: predicted value (yhat) = 49.6000, residual (e) = -9.1400$$

4. Step 4 — Substitute the given values:

   $$yi = e_i + \hat{y}_i$$

5. Step 5 — Evaluate:

   $$yi = 40.4600$$

6. Step 6 — Check: returning yi = 40.4600 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 80.9200 — kept a factor of two that cancels in the correct rearrangement.
• 20.2300 — dropped that same factor in the other direction.
• 44.5060 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A large residual flags an outlier in the calibration data set. Given observed value (yi) = 30.0000; residual (e) = -8.0200, determine the predicted value (yhat).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yhat is given, so isolate yhat 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yhat:

   $$yhat = y_i - e_i$$

3. Step 3

   $$List the givens: observed value (yi) = 30.0000, residual (e) = -8.0200$$

4. Step 4 — Substitute the given values:

   $$yhat = y_i - e_i$$

5. Step 5 — Evaluate:

   $$yhat = 38.0200$$

6. Step 6 — Check: returning yhat = 38.0200 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 76.0400 — kept a factor of two that cancels in the correct rearrangement.
• 19.0100 — dropped that same factor in the other direction.
• 41.8220 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

An engineer computes the residual between a measured and predicted strength. Given observed value (yi) = 11.1000; predicted value (yhat) = 39.9000, determine the residual (e).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except e is given, so isolate e 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for e:

   $$e = y_i - \hat{y}_i$$

3. Step 3

   $$List the givens: observed value (yi) = 11.1000, predicted value (yhat) = 39.9000$$

4. Step 4 — Substitute the given values:

   $$e = y_i - \hat{y}_i$$

5. Step 5 — Evaluate:

   $$e = -28.8000$$

6. Step 6 — Check: returning e = -28.8000 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• -57.6000 — kept a factor of two that cancels in the correct rearrangement.
• -14.4000 — dropped that same factor in the other direction.
• -31.6800 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A student checks the residual of a data point on a least squares line. Given predicted value (yhat) = 22.9000; residual (e) = 0.6900, determine the observed value (yi).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yi is given, so isolate yi 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yi:

   $$yi = e_i + \hat{y}_i$$

3. Step 3

   $$List the givens: predicted value (yhat) = 22.9000, residual (e) = 0.6900$$

4. Step 4 — Substitute the given values:

   $$yi = e_i + \hat{y}_i$$

5. Step 5 — Evaluate:

   $$yi = 23.5900$$

6. Step 6 — Check: returning yi = 23.5900 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 47.1800 — kept a factor of two that cancels in the correct rearrangement.
• 11.7950 — dropped that same factor in the other direction.
• 25.9490 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

A large residual flags an outlier in the calibration data set. Given observed value (yi) = 44.9000; residual (e) = -16.5500, determine the predicted value (yhat).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except yhat is given, so isolate yhat 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for yhat:

   $$yhat = y_i - e_i$$

3. Step 3

   $$List the givens: observed value (yi) = 44.9000, residual (e) = -16.5500$$

4. Step 4 — Substitute the given values:

   $$yhat = y_i - e_i$$

5. Step 5 — Evaluate:

   $$yhat = 61.4500$$

6. Step 6 — Check: returning yhat = 61.4500 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 122.9 — kept a factor of two that cancels in the correct rearrangement.
• 30.7250 — dropped that same factor in the other direction.
• 67.5950 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual

An engineer computes the residual between a measured and predicted strength. Given observed value (yi) = 96.0000; predicted value (yhat) = 32.5000, determine the residual (e).

Start with the thinking

• The governing relation printed in this handbook section is Residual of a regression fit.
• Everything except e is given, so isolate e 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 residual is the difference between an observed value and the value predicted by the regression model.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$e_i = y_i - \hat{y}_i$$

2. Step 2 — Rearrange symbolically for e:

   $$e = y_i - \hat{y}_i$$

3. Step 3

   $$List the givens: observed value (yi) = 96.0000, predicted value (yhat) = 32.5000$$

4. Step 4 — Substitute the given values:

   $$e = y_i - \hat{y}_i$$

5. Step 5 — Evaluate:

   $$e = 63.5000$$

6. Step 6 — Check: returning e = 63.5000 to

   $$e_i = y_i - \hat{y}_i$$

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

Why the other options are there

• 127.0 — kept a factor of two that cancels in the correct rearrangement.
• 31.7500 — dropped that same factor in the other direction.
• 69.8500 — rounded an intermediate value before the final step.

Reference: FE Handbook — Residual