A probability and statistics problem uses z-score. Given mean (mu) = 3,460 psi; std deviation (sigma) = 150.0 psi; value (x) = 5,880 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

   $$List the givens: mean (mu) = 3,460 psi, std deviation (sigma) = 150.0 psi, value (x) = 5,880 psi$$

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

5. Step 5 — Evaluate:

   $$z = 16.1333$$

6. Step 6 — Check: returning z = 16.1333 to

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

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

Why the other options are there

• 32.2667 — kept a factor of two that cancels in the correct rearrangement.
• 8.0667 — dropped that same factor in the other direction.
• 17.7467 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Probability Functions, Distributions, and Expected Values

A student finds the expected value of a two-outcome gamble. Given outcome 1 (x1) = 2.5000; probability 1 (p1) = 0.3900; outcome 2 (x2) = 15.0000, determine the expected value (EX).

Start with the thinking

• The governing relation printed in this handbook section is Expected value of a discrete random variable.
• Everything except EX is given, so isolate EX 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 expected value of a discrete random variable weights each outcome by its probability.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E[X] = \sum x_i p_i$$

2. Step 2 — Rearrange symbolically for EX:

   $$EX = x_1 p_1 + x_2(1-p_1)$$

3. Step 3

   $$List the givens: outcome 1 (x_{1}) = 2.5000, probability 1 (p_{1}) = 0.3900, outcome 2 (x_{2}) = 15.0000$$

4. Step 4 — Substitute the given values:

   $$EX = x_1 p_1 + x_2(1-p_1)$$

5. Step 5 — Evaluate:

   $$EX = 10.1250$$

6. Step 6 — Check: returning EX = 10.1250 to

   $$E[X] = \sum x_i p_i$$

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

Why the other options are there

• 20.2500 — kept a factor of two that cancels in the correct rearrangement.
• 5.0625 — dropped that same factor in the other direction.
• 11.1375 — rounded an intermediate value before the final step.

Reference: FE Handbook — Expected Values

A probability and statistics problem uses z-score. Given mean (mu) = 2,890 psi; std deviation (sigma) = 260.0 psi; z-score (z) = -0.1400, 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

   $$List the givens: mean (mu) = 2,890 psi, std deviation (sigma) = 260.0 psi, z-score (z) = -0.1400$$

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

5. Step 5 — Evaluate:

   $$x = 2854\ \text{psi}$$

6. Step 6 — Check: returning x = 2,854 psi to

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

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

Why the other options are there

• 5,707 — kept a factor of two that cancels in the correct rearrangement.
• 1,427 — dropped that same factor in the other direction.
• 3,139 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Probability Functions, Distributions, and Expected Values

The expected value of a production yield is estimated from probabilities. Given outcome 1 (x1) = 3.5000; probability 1 (p1) = 0.3500; expected value (EX) = 18.3700, determine the outcome 2 (x2).

Start with the thinking

• The governing relation printed in this handbook section is Expected value of a discrete random variable.
• Everything except x2 is given, so isolate x2 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 expected value of a discrete random variable weights each outcome by its probability.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E[X] = \sum x_i p_i$$

2. Step 2 — Rearrange symbolically for x2:

   $$x_{2} = \dfrac{E[X] - x_1 p_1}{1-p_1}$$

3. Step 3

   $$List the givens: outcome 1 (x_{1}) = 3.5000, probability 1 (p_{1}) = 0.3500, expected value (EX) = 18.3700$$

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{E[X] - x_1 p_1}{1-p_1}$$

5. Step 5 — Evaluate:

   $$x_{2} = 26.3769$$

6. Step 6 — Check: returning x2 = 26.3769 to

   $$E[X] = \sum x_i p_i$$

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

Why the other options are there

• 52.7538 — kept a factor of two that cancels in the correct rearrangement.
• 13.1885 — dropped that same factor in the other direction.
• 29.0146 — rounded an intermediate value before the final step.

Reference: FE Handbook — Expected Values

A probability and statistics problem uses z-score. Given std deviation (sigma) = 350.0 psi; value (x) = 3,890 psi; z-score (z) = 2.9600, 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

   $$List the givens: std deviation (sigma) = 350.0 psi, value (x) = 3,890 psi, z-score (z) = 2.9600$$

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

5. Step 5 — Evaluate:

   $$\mu = 2854\ \text{psi}$$

6. Step 6 — Check: returning mu = 2,854 psi to

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

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

Why the other options are there

• 5,708 — kept a factor of two that cancels in the correct rearrangement.
• 1,427 — dropped that same factor in the other direction.
• 3,139 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Probability Functions, Distributions, and Expected Values

An engineer computes the expected value of a warranty payout distribution. Given outcome 1 (x1) = 3.5000; outcome 2 (x2) = 15.0000; expected value (EX) = 7.3500, determine the probability 1 (p1).

Start with the thinking

• The governing relation printed in this handbook section is Expected value of a discrete random variable.
• Everything except p1 is given, so isolate p1 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 expected value of a discrete random variable weights each outcome by its probability.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E[X] = \sum x_i p_i$$

2. Step 2 — Rearrange symbolically for p1:

   $$p_{1} = \dfrac{x_2 - E[X]}{x_2 - x_1}$$

3. Step 3

   $$List the givens: outcome 1 (x_{1}) = 3.5000, outcome 2 (x_{2}) = 15.0000, expected value (EX) = 7.3500$$

4. Step 4 — Substitute the given values:

   $$p_{1} = \dfrac{x_2 - E[X]}{x_2 - x_1}$$

5. Step 5 — Evaluate:

   $$p_{1} = 0.6652$$

6. Step 6 — Check: returning p1 = 0.6652 to

   $$E[X] = \sum x_i p_i$$

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

Why the other options are there

• 1.3304 — kept a factor of two that cancels in the correct rearrangement.
• 0.3326 — dropped that same factor in the other direction.
• 0.7317 — rounded an intermediate value before the final step.

Reference: FE Handbook — Expected Values

A probability and statistics problem uses z-score. Given mean (mu) = 4,500 psi; value (x) = 1,530 psi; z-score (z) = 1.0400, 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

   $$List the givens: mean (mu) = 4,500 psi, value (x) = 1,530 psi, z-score (z) = 1.0400$$

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

5. Step 5 — Evaluate:

   $$\sigma = -2856\ \text{psi}$$

6. Step 6 — Check: returning sigma = -2,856 psi to

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

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

Why the other options are there

• -5,712 — kept a factor of two that cancels in the correct rearrangement.
• -1,428 — dropped that same factor in the other direction.
• -3,141 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Probability Functions, Distributions, and Expected Values

A student finds the expected value of a two-outcome gamble. Given outcome 1 (x1) = 8.5000; probability 1 (p1) = 0.4400; outcome 2 (x2) = 20.0000, determine the expected value (EX).

Start with the thinking

• The governing relation printed in this handbook section is Expected value of a discrete random variable.
• Everything except EX is given, so isolate EX 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 expected value of a discrete random variable weights each outcome by its probability.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E[X] = \sum x_i p_i$$

2. Step 2 — Rearrange symbolically for EX:

   $$EX = x_1 p_1 + x_2(1-p_1)$$

3. Step 3

   $$List the givens: outcome 1 (x_{1}) = 8.5000, probability 1 (p_{1}) = 0.4400, outcome 2 (x_{2}) = 20.0000$$

4. Step 4 — Substitute the given values:

   $$EX = x_1 p_1 + x_2(1-p_1)$$

5. Step 5 — Evaluate:

   $$EX = 14.9400$$

6. Step 6 — Check: returning EX = 14.9400 to

   $$E[X] = \sum x_i p_i$$

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

Why the other options are there

• 29.8800 — kept a factor of two that cancels in the correct rearrangement.
• 7.4700 — dropped that same factor in the other direction.
• 16.4340 — rounded an intermediate value before the final step.

Reference: FE Handbook — Expected Values

A probability and statistics problem uses z-score. Given mean (mu) = 4,060 psi; std deviation (sigma) = 510.0 psi; value (x) = 6,910 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

   $$List the givens: mean (mu) = 4,060 psi, std deviation (sigma) = 510.0 psi, value (x) = 6,910 psi$$

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

5. Step 5 — Evaluate:

   $$z = 5.5882$$

6. Step 6 — Check: returning z = 5.5882 to

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

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

Why the other options are there

• 11.1765 — kept a factor of two that cancels in the correct rearrangement.
• 2.7941 — dropped that same factor in the other direction.
• 6.1471 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Probability Functions, Distributions, and Expected Values

The expected value of a production yield is estimated from probabilities. Given outcome 1 (x1) = 1.0000; probability 1 (p1) = 0.5700; expected value (EX) = 6.4900, determine the outcome 2 (x2).

Start with the thinking

• The governing relation printed in this handbook section is Expected value of a discrete random variable.
• Everything except x2 is given, so isolate x2 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 expected value of a discrete random variable weights each outcome by its probability.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$E[X] = \sum x_i p_i$$

2. Step 2 — Rearrange symbolically for x2:

   $$x_{2} = \dfrac{E[X] - x_1 p_1}{1-p_1}$$

3. Step 3

   $$List the givens: outcome 1 (x_{1}) = 1.0000, probability 1 (p_{1}) = 0.5700, expected value (EX) = 6.4900$$

4. Step 4 — Substitute the given values:

   $$x_{2} = \dfrac{E[X] - x_1 p_1}{1-p_1}$$

5. Step 5 — Evaluate:

   $$x_{2} = 13.7674$$

6. Step 6 — Check: returning x2 = 13.7674 to

   $$E[X] = \sum x_i p_i$$

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

Why the other options are there

• 27.5349 — kept a factor of two that cancels in the correct rearrangement.
• 6.8837 — dropped that same factor in the other direction.
• 15.1442 — rounded an intermediate value before the final step.

Reference: FE Handbook — Expected Values