A probability and statistics problem uses z-score. Given mean (mu) = 3,420 psi; std deviation (sigma) = 150.0 psi; value (x) = 6,300 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

   $$List the givens: mean (mu) = 3,420 psi, std deviation (sigma) = 150.0 psi, value (x) = 6,300 psi$$

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

5. Step 5 — Evaluate:

   $$z = 19.2000$$

6. Step 6 — Check: returning z = 19.2000 to

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

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

Why the other options are there

• 38.4000 — kept a factor of two that cancels in the correct rearrangement.
• 9.6000 — dropped that same factor in the other direction.
• 21.1200 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Normal Distribution (Gaussian Distribution)

A quality analyst uses the Gaussian distribution to find a z-score for a measurement. Given mean (mu) = 53.0000; std deviation (sigma) = 16.0000; value (x) = 112.0, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is Normal (Gaussian) distribution z-transform.
• 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.
• The normal distribution (Gaussian distribution) standardizes a value x using its mean and standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{x-\mu}{\sigma}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{x-\mu}{\sigma}$$

3. Step 3

   $$List the givens: mean (mu) = 53.0000, std deviation (sigma) = 16.0000, value (x) = 112.0$$

4. Step 4 — Substitute the given values:

   $$z = \dfrac{112.0-53.0000}{16.0000}$$

5. Step 5 — Evaluate:

   $$z = 3.6875$$

6. Step 6 — Check: returning z = 3.6875 to

   $$z = \dfrac{x-\mu}{\sigma}$$

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

Why the other options are there

• 7.3750 — kept a factor of two that cancels in the correct rearrangement.
• 1.8438 — dropped that same factor in the other direction.
• 4.0563 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Distribution (Gaussian Distribution)

A probability and statistics problem uses z-score. Given mean (mu) = 5,570 psi; std deviation (sigma) = 300.0 psi; z-score (z) = -2.1000, 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

   $$List the givens: mean (mu) = 5,570 psi, std deviation (sigma) = 300.0 psi, z-score (z) = -2.1000$$

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

5. Step 5 — Evaluate:

   $$x = 4940\ \text{psi}$$

6. Step 6 — Check: returning x = 4,940 psi to

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

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

Why the other options are there

• 9,880 — kept a factor of two that cancels in the correct rearrangement.
• 2,470 — dropped that same factor in the other direction.
• 5,434 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Normal Distribution (Gaussian Distribution)

The normal distribution model is applied to a batch of dimensional readings. Given mean (mu) = 87.0000; std deviation (sigma) = 5.0000; z-score (z) = -2.9800, determine the value (x).

Start with the thinking

• The governing relation printed in this handbook section is Normal (Gaussian) distribution z-transform.
• 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.
• The normal distribution (Gaussian distribution) standardizes a value x using its mean and standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{x-\mu}{\sigma}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \mu + z\sigma$$

3. Step 3

   $$List the givens: mean (mu) = 87.0000, std deviation (sigma) = 5.0000, z-score (z) = -2.9800$$

4. Step 4 — Substitute the given values:

   $$x = 87.0000 + -2.98005.0000$$

5. Step 5 — Evaluate:

   $$x = 72.1000$$

6. Step 6 — Check: returning x = 72.1000 to

   $$z = \dfrac{x-\mu}{\sigma}$$

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

Why the other options are there

• 144.2 — kept a factor of two that cancels in the correct rearrangement.
• 36.0500 — dropped that same factor in the other direction.
• 79.3100 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Distribution (Gaussian Distribution)

A probability and statistics problem uses z-score. Given std deviation (sigma) = 430.0 psi; value (x) = 4,120 psi; z-score (z) = 1.1100, 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

   $$List the givens: std deviation (sigma) = 430.0 psi, value (x) = 4,120 psi, z-score (z) = 1.1100$$

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

5. Step 5 — Evaluate:

   $$\mu = 3643\ \text{psi}$$

6. Step 6 — Check: returning mu = 3,643 psi to

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

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

Why the other options are there

• 7,285 — kept a factor of two that cancels in the correct rearrangement.
• 1,821 — dropped that same factor in the other direction.
• 4,007 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Normal Distribution (Gaussian Distribution)

An engineer standardizes a strength test result assuming a normal distribution. Given std deviation (sigma) = 12.5000; value (x) = 157.5; z-score (z) = 1.4800, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Normal (Gaussian) distribution z-transform.
• 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.
• The normal distribution (Gaussian distribution) standardizes a value x using its mean and standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{x-\mu}{\sigma}$$

2. Step 2 — Rearrange symbolically for mu:

   $$\mu = x - z\sigma$$

3. Step 3

   $$List the givens: std deviation (sigma) = 12.5000, value (x) = 157.5, z-score (z) = 1.4800$$

4. Step 4 — Substitute the given values:

   $$\mu = 157.5 - 1.480012.5000$$

5. Step 5 — Evaluate:

   $$\mu = 139.0$$

6. Step 6 — Check: returning mu = 139.0 to

   $$z = \dfrac{x-\mu}{\sigma}$$

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

Why the other options are there

• 278.0 — kept a factor of two that cancels in the correct rearrangement.
• 69.5000 — dropped that same factor in the other direction.
• 152.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Distribution (Gaussian Distribution)

A probability and statistics problem uses z-score. Given mean (mu) = 3,940 psi; value (x) = 4,150 psi; z-score (z) = 1.5600, 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

   $$List the givens: mean (mu) = 3,940 psi, value (x) = 4,150 psi, z-score (z) = 1.5600$$

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

5. Step 5 — Evaluate:

   $$\sigma = 134.6\ \text{psi}$$

6. Step 6 — Check: returning sigma = 134.6 psi to

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

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

Why the other options are there

• 269.2 — kept a factor of two that cancels in the correct rearrangement.
• 67.3077 — dropped that same factor in the other direction.
• 148.1 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Normal Distribution (Gaussian Distribution)

A quality analyst uses the Gaussian distribution to find a z-score for a measurement. Given mean (mu) = 191.0; std deviation (sigma) = 11.0000; value (x) = 71.0000, determine the z-score (z).

Start with the thinking

• The governing relation printed in this handbook section is Normal (Gaussian) distribution z-transform.
• 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.
• The normal distribution (Gaussian distribution) standardizes a value x using its mean and standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{x-\mu}{\sigma}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{x-\mu}{\sigma}$$

3. Step 3

   $$List the givens: mean (mu) = 191.0, std deviation (sigma) = 11.0000, value (x) = 71.0000$$

4. Step 4 — Substitute the given values:

   $$z = \dfrac{71.0000-191.0}{11.0000}$$

5. Step 5 — Evaluate:

   $$z = -10.9091$$

6. Step 6 — Check: returning z = -10.9091 to

   $$z = \dfrac{x-\mu}{\sigma}$$

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

Why the other options are there

• -21.8182 — kept a factor of two that cancels in the correct rearrangement.
• -5.4545 — dropped that same factor in the other direction.
• -12.0000 — rounded an intermediate value before the final step.

Reference: FE Handbook — Normal Distribution (Gaussian Distribution)

A probability and statistics problem uses z-score. Given mean (mu) = 5,560 psi; std deviation (sigma) = 180.0 psi; value (x) = 3,000 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

   $$List the givens: mean (mu) = 5,560 psi, std deviation (sigma) = 180.0 psi, value (x) = 3,000 psi$$

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

5. Step 5 — Evaluate:

   $$z = -14.2222$$

6. Step 6 — Check: returning z = -14.2222 to

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

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

Why the other options are there

• -28.4444 — kept a factor of two that cancels in the correct rearrangement.
• -7.1111 — dropped that same factor in the other direction.
• -15.6444 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Normal Distribution (Gaussian Distribution)

The normal distribution model is applied to a batch of dimensional readings. Given mean (mu) = 194.0; std deviation (sigma) = 11.5000; z-score (z) = 1.3300, determine the value (x).

Start with the thinking

• The governing relation printed in this handbook section is Normal (Gaussian) distribution z-transform.
• 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.
• The normal distribution (Gaussian distribution) standardizes a value x using its mean and standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{x-\mu}{\sigma}$$

2. Step 2 — Rearrange symbolically for x:

   $$x = \mu + z\sigma$$

3. Step 3

   $$List the givens: mean (mu) = 194.0, std deviation (sigma) = 11.5000, z-score (z) = 1.3300$$

4. Step 4 — Substitute the given values:

   $$x = 194.0 + 1.330011.5000$$

5. Step 5 — Evaluate:

   $$x = 209.3$$

6. Step 6 — Check: returning x = 209.3 to

   $$z = \dfrac{x-\mu}{\sigma}$$

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

Why the other options are there

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

Reference: FE Handbook — Normal Distribution (Gaussian Distribution)