A probability and statistics problem uses z-score. Given mean (mu) = 3,610 psi; std deviation (sigma) = 580.0 psi; value (x) = 2,130 psi, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except z is given, so isolate z 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that z stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 3,610 psi, std deviation (sigma) = 580.0 psi, value (x) = 2,130 psi$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$z = -2.5517$$

6. Step 6 — Check: returning z = -2.5517 to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• -5.1034 — kept a factor of two that cancels in the correct rearrangement.
• -1.2759 — dropped that same factor in the other direction.
• -2.8069 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 5,100 psi; std deviation (sigma) = 350.0 psi; z-score (z) = 0.3200, determine the value (x) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except x is given, so isolate x 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that x stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 5,100 psi, std deviation (sigma) = 350.0 psi, z-score (z) = 0.3200$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$x = 5212\ \text{psi}$$

6. Step 6 — Check: returning x = 5,212 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 10,424 — kept a factor of two that cancels in the correct rearrangement.
• 2,606 — dropped that same factor in the other direction.
• 5,733 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given std deviation (sigma) = 480.0 psi; value (x) = 2,690 psi; z-score (z) = 3.0000, determine the mean (mu) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except mu is given, so isolate mu 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that mu stands alone on the left-hand side.

3. Step 3

   $$List the givens: std deviation (sigma) = 480.0 psi, value (x) = 2,690 psi, z-score (z) = 3.0000$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\mu = 1250\ \text{psi}$$

6. Step 6 — Check: returning mu = 1,250 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 2,500 — kept a factor of two that cancels in the correct rearrangement.
• 625.0 — dropped that same factor in the other direction.
• 1,375 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 4,550 psi; value (x) = 6,880 psi; z-score (z) = 1.0300, determine the std deviation (sigma) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except sigma is given, so isolate sigma 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that sigma stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 4,550 psi, value (x) = 6,880 psi, z-score (z) = 1.0300$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\sigma = 2262\ \text{psi}$$

6. Step 6 — Check: returning sigma = 2,262 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 4,524 — kept a factor of two that cancels in the correct rearrangement.
• 1,131 — dropped that same factor in the other direction.
• 2,488 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 4,720 psi; std deviation (sigma) = 560.0 psi; value (x) = 2,010 psi, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except z is given, so isolate z 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that z stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 4,720 psi, std deviation (sigma) = 560.0 psi, value (x) = 2,010 psi$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$z = -4.8393$$

6. Step 6 — Check: returning z = -4.8393 to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• -9.6786 — kept a factor of two that cancels in the correct rearrangement.
• -2.4196 — dropped that same factor in the other direction.
• -5.3232 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 2,220 psi; std deviation (sigma) = 350.0 psi; z-score (z) = -0.3800, determine the value (x) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except x is given, so isolate x 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that x stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 2,220 psi, std deviation (sigma) = 350.0 psi, z-score (z) = -0.3800$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$x = 2087\ \text{psi}$$

6. Step 6 — Check: returning x = 2,087 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 4,174 — kept a factor of two that cancels in the correct rearrangement.
• 1,044 — dropped that same factor in the other direction.
• 2,296 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given std deviation (sigma) = 270.0 psi; value (x) = 3,200 psi; z-score (z) = 1.3100, determine the mean (mu) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except mu is given, so isolate mu 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that mu stands alone on the left-hand side.

3. Step 3

   $$List the givens: std deviation (sigma) = 270.0 psi, value (x) = 3,200 psi, z-score (z) = 1.3100$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\mu = 2846\ \text{psi}$$

6. Step 6 — Check: returning mu = 2,846 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 5,693 — kept a factor of two that cancels in the correct rearrangement.
• 1,423 — dropped that same factor in the other direction.
• 3,131 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 3,830 psi; value (x) = 3,290 psi; z-score (z) = 2.7800, determine the std deviation (sigma) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except sigma is given, so isolate sigma 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that sigma stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 3,830 psi, value (x) = 3,290 psi, z-score (z) = 2.7800$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$\sigma = -194.2\ \text{psi}$$

6. Step 6 — Check: returning sigma = -194.2 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• -388.5 — kept a factor of two that cancels in the correct rearrangement.
• -97.1223 — dropped that same factor in the other direction.
• -213.7 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 3,250 psi; std deviation (sigma) = 320.0 psi; value (x) = 2,140 psi, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except z is given, so isolate z 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that z stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 3,250 psi, std deviation (sigma) = 320.0 psi, value (x) = 2,140 psi$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$z = -3.4688$$

6. Step 6 — Check: returning z = -3.4688 to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• -6.9375 — kept a factor of two that cancels in the correct rearrangement.
• -1.7344 — dropped that same factor in the other direction.
• -3.8156 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations

A probability and statistics problem uses z-score. Given mean (mu) = 4,160 psi; std deviation (sigma) = 140.0 psi; z-score (z) = 0.7200, determine the value (x) in psi.

Start with the thinking

• The governing relation printed in this handbook section is z-score.
• Everything except x is given, so isolate x 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.
• Probability and Statistics items reward recognising the unknown before touching a calculator.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = (x - \mu) / \sigma$$

2. Step 2 — Rearrange the relation so that x stands alone on the left-hand side.

3. Step 3

   $$List the givens: mean (mu) = 4,160 psi, std deviation (sigma) = 140.0 psi, z-score (z) = 0.7200$$

4. Step 4 — Substitute the given values into the rearranged relation.

5. Step 5 — Evaluate:

   $$x = 4261\ \text{psi}$$

6. Step 6 — Check: returning x = 4,261 psi to

   $$z = (x - \mu) / \sigma$$

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

Why the other options are there

• 8,522 — kept a factor of two that cancels in the correct rearrangement.
• 2,130 — dropped that same factor in the other direction.
• 4,687 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Approximations