A distance with an accepted value of 98.00 ft is observed as 100.3 ft. The instrument contributes a standard uncertainty of ±0.12 ft and the target centring ±0.10 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 100.3 - 98.00 = 2.300 ft$$

3. Relative error — 2.3469%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.12^{2} + 0.10^{2}) = 0.1562 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.3124 ft$$

Why the other options are there

• ±0.220 ft (uncertainties added directly)
• -2.300 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 106.0 ft is observed as 106.9 ft. The instrument contributes a standard uncertainty of ±0.09 ft and the target centring ±0.13 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 106.9 - 106.0 = 0.900 ft$$

3. Relative error — 0.8491%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.09^{2} + 0.13^{2}) = 0.1581 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.3162 ft$$

Why the other options are there

• ±0.220 ft (uncertainties added directly)
• -0.900 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 198.5 ft is observed as 198.5 ft. The instrument contributes a standard uncertainty of ±0.14 ft and the target centring ±0.05 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 198.5 - 198.5 = 0.000 ft$$

3. Relative error — 0.0000%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.14^{2} + 0.05^{2}) = 0.1487 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.2973 ft$$

Why the other options are there

• ±0.190 ft (uncertainties added directly)
• 0.000 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 139.0 ft is observed as 140.0 ft. The instrument contributes a standard uncertainty of ±0.09 ft and the target centring ±0.09 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 140.0 - 139.0 = 1.000 ft$$

3. Relative error — 0.7194%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.09^{2} + 0.09^{2}) = 0.1273 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.2546 ft$$

Why the other options are there

• ±0.180 ft (uncertainties added directly)
• -1.000 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 172.5 ft is observed as 173.7 ft. The instrument contributes a standard uncertainty of ±0.04 ft and the target centring ±0.04 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 173.7 - 172.5 = 1.200 ft$$

3. Relative error — 0.6957%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.04^{2} + 0.04^{2}) = 0.0566 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.1131 ft$$

Why the other options are there

• ±0.080 ft (uncertainties added directly)
• -1.200 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 72.00 ft is observed as 71.80 ft. The instrument contributes a standard uncertainty of ±0.18 ft and the target centring ±0.16 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 71.80 - 72.00 = -0.200 ft$$

3. Relative error — 0.2778%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.18^{2} + 0.16^{2}) = 0.2408 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.4817 ft$$

Why the other options are there

• ±0.340 ft (uncertainties added directly)
• 0.200 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 150.5 ft is observed as 152.8 ft. The instrument contributes a standard uncertainty of ±0.13 ft and the target centring ±0.02 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 152.8 - 150.5 = 2.300 ft$$

3. Relative error — 1.5282%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.13^{2} + 0.02^{2}) = 0.1315 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.2631 ft$$

Why the other options are there

• ±0.150 ft (uncertainties added directly)
• -2.300 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 127.0 ft is observed as 126.1 ft. The instrument contributes a standard uncertainty of ±0.11 ft and the target centring ±0.09 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 126.1 - 127.0 = -0.900 ft$$

3. Relative error — 0.7087%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.11^{2} + 0.09^{2}) = 0.1421 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.2843 ft$$

Why the other options are there

• ±0.200 ft (uncertainties added directly)
• 0.900 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 131.0 ft is observed as 132.5 ft. The instrument contributes a standard uncertainty of ±0.15 ft and the target centring ±0.18 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 132.5 - 131.0 = 1.500 ft$$

3. Relative error — 1.1450%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.15^{2} + 0.18^{2}) = 0.2343 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.4686 ft$$

Why the other options are there

• ±0.330 ft (uncertainties added directly)
• -1.500 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error

A distance with an accepted value of 198.5 ft is observed as 200.7 ft. The instrument contributes a standard uncertainty of ±0.06 ft and the target centring ±0.19 ft. Report the measurement error, the residual, and the combined standard uncertainty.

Start with the thinking

• Error is observation minus true value; the residual is the same magnitude with the sign convention of the adjustment.
• Independent uncertainties combine in quadrature, never by simple addition.

Step-by-step solution

1. Formula

   $$e = observed - true$$

2. Substituting

   $$e = 200.7 - 198.5 = 2.200 ft$$

3. Relative error — 1.1083%

4. Formula

   $$u_c = \sqrt(\sigma_{1}^{2} + \sigma_{2}^{2})$$

5. Substituting

   $$u_c = \sqrt(0.06^{2} + 0.19^{2}) = 0.1992 ft$$

6. 95% expanded uncertainty

   $$U = 2u_c = 0.3985 ft$$

Why the other options are there

• ±0.250 ft (uncertainties added directly)
• -2.200 ft (sign reversed)

Reference: FE Reference Handbook — Probability and Statistics → Measurement Error