A probability and statistics problem uses z-score. Given mean (mu) = 4,370 psi; std deviation (sigma) = 460.0 psi; value (x) = 6,180 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

   $$List the givens: mean (mu) = 4,370 psi, std deviation (sigma) = 460.0 psi, value (x) = 6,180 psi$$

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

5. Step 5 — Evaluate:

   $$z = 3.9348$$

6. Step 6 — Check: returning z = 3.9348 to

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

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

Why the other options are there

• 7.8696 — kept a factor of two that cancels in the correct rearrangement.
• 1.9674 — dropped that same factor in the other direction.
• 4.3283 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Standard Deviation Charts

An analyst computes the upper control limit for a standard deviation chart. Given control chart factor B4 (B4) = 1.7900; average sample std dev (sbar) = 4.8000, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Standard deviation control chart limits.
• Everything except UCL is given, so isolate UCL 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.
• Standard deviation charts monitor process variability using control limits based on the average sample standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_s = B_4 \bar{s}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = B_4 \bar{s}$$

3. Step 3 — List the givens: control chart factor B4 (B4) = 1.7900, average sample std dev (sbar) = 4.8000.

4. Step 4 — Substitute the given values:

   $$UCL = B_4 \bar{s}$$

5. Step 5 — Evaluate:

   $$UCL = 8.5920$$

6. Step 6 — Check: returning UCL = 8.5920 to

   $$UCL_s = B_4 \bar{s}$$

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

Why the other options are there

• 17.1840 — kept a factor of two that cancels in the correct rearrangement.
• 4.2960 — dropped that same factor in the other direction.
• 9.4512 — rounded an intermediate value before the final step.

Reference: FE Handbook — Standard Deviation Charts

A probability and statistics problem uses z-score. Given mean (mu) = 2,860 psi; std deviation (sigma) = 150.0 psi; z-score (z) = 2.8900, 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

   $$List the givens: mean (mu) = 2,860 psi, std deviation (sigma) = 150.0 psi, z-score (z) = 2.8900$$

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

5. Step 5 — Evaluate:

   $$x = 3294\ \text{psi}$$

6. Step 6 — Check: returning x = 3,294 psi to

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

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

Why the other options are there

• 6,587 — kept a factor of two that cancels in the correct rearrangement.
• 1,647 — dropped that same factor in the other direction.
• 3,623 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Standard Deviation Charts

The standard deviation chart shows increased variability beyond the control limit. Given average sample std dev (sbar) = 4.6500; upper control limit (UCL) = 21.0300, determine the control chart factor B4 (B4).

Start with the thinking

• The governing relation printed in this handbook section is Standard deviation control chart limits.
• Everything except B4 is given, so isolate B4 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.
• Standard deviation charts monitor process variability using control limits based on the average sample standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_s = B_4 \bar{s}$$

2. Step 2 — Rearrange symbolically for B4:

   $$B_{4} = \dfrac{UCL}{\bar{s}}$$

3. Step 3 — List the givens: average sample std dev (sbar) = 4.6500, upper control limit (UCL) = 21.0300.

4. Step 4 — Substitute the given values:

   $$B_{4} = \dfrac{21.0300}{\bar{s}}$$

5. Step 5 — Evaluate:

   $$B_{4} = 4.5226$$

6. Step 6 — Check: returning B4 = 4.5226 to

   $$UCL_s = B_4 \bar{s}$$

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

Why the other options are there

• 9.0452 — kept a factor of two that cancels in the correct rearrangement.
• 2.2613 — dropped that same factor in the other direction.
• 4.9748 — rounded an intermediate value before the final step.

Reference: FE Handbook — Standard Deviation Charts

A probability and statistics problem uses z-score. Given std deviation (sigma) = 280.0 psi; value (x) = 4,280 psi; z-score (z) = -2.0100, 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

   $$List the givens: std deviation (sigma) = 280.0 psi, value (x) = 4,280 psi, z-score (z) = -2.0100$$

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

5. Step 5 — Evaluate:

   $$\mu = 4843\ \text{psi}$$

6. Step 6 — Check: returning mu = 4,843 psi to

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

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

Why the other options are there

• 9,686 — kept a factor of two that cancels in the correct rearrangement.
• 2,421 — dropped that same factor in the other direction.
• 5,327 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Standard Deviation Charts

A quality engineer builds a standard deviation chart for a filling process. Given control chart factor B4 (B4) = 1.6600; upper control limit (UCL) = 17.1900, determine the average sample std dev (sbar).

Start with the thinking

• The governing relation printed in this handbook section is Standard deviation control chart limits.
• Everything except sbar is given, so isolate sbar 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.
• Standard deviation charts monitor process variability using control limits based on the average sample standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_s = B_4 \bar{s}$$

2. Step 2 — Rearrange symbolically for sbar:

   $$sbar = \dfrac{UCL}{B_4}$$

3. Step 3 — List the givens: control chart factor B4 (B4) = 1.6600, upper control limit (UCL) = 17.1900.

4. Step 4 — Substitute the given values:

   $$sbar = \dfrac{17.1900}{B_4}$$

5. Step 5 — Evaluate:

   $$sbar = 10.3554$$

6. Step 6 — Check: returning sbar = 10.3554 to

   $$UCL_s = B_4 \bar{s}$$

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

Why the other options are there

• 20.7108 — kept a factor of two that cancels in the correct rearrangement.
• 5.1777 — dropped that same factor in the other direction.
• 11.3910 — rounded an intermediate value before the final step.

Reference: FE Handbook — Standard Deviation Charts

A probability and statistics problem uses z-score. Given mean (mu) = 2,740 psi; value (x) = 1,570 psi; z-score (z) = -2.3400, 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

   $$List the givens: mean (mu) = 2,740 psi, value (x) = 1,570 psi, z-score (z) = -2.3400$$

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

5. Step 5 — Evaluate:

   $$\sigma = 500.0\ \text{psi}$$

6. Step 6 — Check: returning sigma = 500.0 psi to

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

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

Why the other options are there

• 1,000 — kept a factor of two that cancels in the correct rearrangement.
• 250.0 — dropped that same factor in the other direction.
• 550.0 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Standard Deviation Charts

An analyst computes the upper control limit for a standard deviation chart. Given control chart factor B4 (B4) = 2.1100; average sample std dev (sbar) = 8.9500, determine the upper control limit (UCL).

Start with the thinking

• The governing relation printed in this handbook section is Standard deviation control chart limits.
• Everything except UCL is given, so isolate UCL 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.
• Standard deviation charts monitor process variability using control limits based on the average sample standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_s = B_4 \bar{s}$$

2. Step 2 — Rearrange symbolically for UCL:

   $$UCL = B_4 \bar{s}$$

3. Step 3 — List the givens: control chart factor B4 (B4) = 2.1100, average sample std dev (sbar) = 8.9500.

4. Step 4 — Substitute the given values:

   $$UCL = B_4 \bar{s}$$

5. Step 5 — Evaluate:

   $$UCL = 18.8845$$

6. Step 6 — Check: returning UCL = 18.8845 to

   $$UCL_s = B_4 \bar{s}$$

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

Why the other options are there

• 37.7690 — kept a factor of two that cancels in the correct rearrangement.
• 9.4422 — dropped that same factor in the other direction.
• 20.7729 — rounded an intermediate value before the final step.

Reference: FE Handbook — Standard Deviation Charts

A probability and statistics problem uses z-score. Given mean (mu) = 5,390 psi; std deviation (sigma) = 300.0 psi; value (x) = 1,910 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

   $$List the givens: mean (mu) = 5,390 psi, std deviation (sigma) = 300.0 psi, value (x) = 1,910 psi$$

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

5. Step 5 — Evaluate:

   $$z = -11.6000$$

6. Step 6 — Check: returning z = -11.6000 to

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

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

Why the other options are there

• -23.2000 — kept a factor of two that cancels in the correct rearrangement.
• -5.8000 — dropped that same factor in the other direction.
• -12.7600 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Standard Deviation Charts

The standard deviation chart shows increased variability beyond the control limit. Given average sample std dev (sbar) = 5.4000; upper control limit (UCL) = 15.5300, determine the control chart factor B4 (B4).

Start with the thinking

• The governing relation printed in this handbook section is Standard deviation control chart limits.
• Everything except B4 is given, so isolate B4 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.
• Standard deviation charts monitor process variability using control limits based on the average sample standard deviation.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$UCL_s = B_4 \bar{s}$$

2. Step 2 — Rearrange symbolically for B4:

   $$B_{4} = \dfrac{UCL}{\bar{s}}$$

3. Step 3 — List the givens: average sample std dev (sbar) = 5.4000, upper control limit (UCL) = 15.5300.

4. Step 4 — Substitute the given values:

   $$B_{4} = \dfrac{15.5300}{\bar{s}}$$

5. Step 5 — Evaluate:

   $$B_{4} = 2.8759$$

6. Step 6 — Check: returning B4 = 2.8759 to

   $$UCL_s = B_4 \bar{s}$$

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

Why the other options are there

• 5.7519 — kept a factor of two that cancels in the correct rearrangement.
• 1.4380 — dropped that same factor in the other direction.
• 3.1635 — rounded an intermediate value before the final step.

Reference: FE Handbook — Standard Deviation Charts