Nine field density tests give x̄ = 121.4 pcf with s = 3.6 pcf. Construct the 95% confidence interval on the mean (t₀.₀₂₅,₈ = 2.306).

Start with the thinking

• σ is unknown and n is small — use t, not z.
• Degrees of freedom are n − 1 = 8.

Step-by-step solution

1. Standard error — SE = s/√n = 3.6/√9 = 1.20 pcf

2. Margin

   $$E = t \cdot SE = 2.306(1.20) = 2.77 pcf$$

3. Lower limit

   $$121.4 - 2.77 = 118.6 pcf$$

4. Upper limit

   $$121.4 + 2.77 = 124.2 pcf$$

5. Interval — 118.6 pcf ≤ μ ≤ 124.2 pcf

Why the other options are there

• 119.1 to 123.7 (z = 1.96 used)
• 114.1 to 128.7 (√n omitted)

Reference: FE Reference Handbook — Probability and Statistics — Confidence intervals

A probability and statistics problem uses z-score. Given mean (mu) = 2,110 psi; std deviation (sigma) = 560.0 psi; value (x) = 2,170 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

   $$List the givens: mean (mu) = 2,110 psi, std deviation (sigma) = 560.0 psi, value (x) = 2,170 psi$$

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

5. Step 5 — Evaluate:

   $$z = 0.1071$$

6. Step 6 — Check: returning z = 0.1071 to

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

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

Why the other options are there

• 0.2143 — kept a factor of two that cancels in the correct rearrangement.
• 0.0536 — dropped that same factor in the other direction.
• 0.1179 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses Confidence interval margin. Given z critical value (z) = 1.4300; std deviation (sigma) = 0.5500 in; sample size (n) = 93.0000, determine the margin of error (E) in in.

Start with the thinking

• The governing relation printed in this handbook section is Confidence interval margin.
• Everything except E is given, so isolate E 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:

   $$E = z \sigma / \sqrt{n}$$

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

3. Step 3 — List the givens: z critical value (z) = 1.4300, std deviation (sigma) = 0.5500 in, sample size (n) = 93.0000.

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

5. Step 5 — Evaluate:

   $$E = 0.0816\ \text{in}$$

6. Step 6 — Check: returning E = 0.0816 in to

   $$E = z \sigma / \sqrt{n}$$

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

Why the other options are there

• 0.1631 — kept a factor of two that cancels in the correct rearrangement.
• 0.0408 — dropped that same factor in the other direction.
• 0.0897 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses z-score. Given mean (mu) = 2,160 psi; std deviation (sigma) = 250.0 psi; z-score (z) = 1.6400, 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

   $$List the givens: mean (mu) = 2,160 psi, std deviation (sigma) = 250.0 psi, z-score (z) = 1.6400$$

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

5. Step 5 — Evaluate:

   $$x = 2570\ \text{psi}$$

6. Step 6 — Check: returning x = 2,570 psi to

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

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

Why the other options are there

• 5,140 — kept a factor of two that cancels in the correct rearrangement.
• 1,285 — dropped that same factor in the other direction.
• 2,827 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses Confidence interval margin. Given z critical value (z) = 1.8600; sample size (n) = 84.0000; margin of error (E) = 0.0590 in, determine the std deviation (sigma) in in.

Start with the thinking

• The governing relation printed in this handbook section is Confidence interval margin.
• 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:

   $$E = z \sigma / \sqrt{n}$$

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

3. Step 3 — List the givens: z critical value (z) = 1.8600, sample size (n) = 84.0000, margin of error (E) = 0.0590 in.

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

5. Step 5 — Evaluate:

   $$\sigma = 0.2907\ \text{in}$$

6. Step 6 — Check: returning sigma = 0.2907 in to

   $$E = z \sigma / \sqrt{n}$$

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

Why the other options are there

• 0.5814 — kept a factor of two that cancels in the correct rearrangement.
• 0.1454 — dropped that same factor in the other direction.
• 0.3198 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses z-score. Given std deviation (sigma) = 330.0 psi; value (x) = 5,450 psi; z-score (z) = -0.9000, 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

   $$List the givens: std deviation (sigma) = 330.0 psi, value (x) = 5,450 psi, z-score (z) = -0.9000$$

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

5. Step 5 — Evaluate:

   $$\mu = 5747\ \text{psi}$$

6. Step 6 — Check: returning mu = 5,747 psi to

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

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

Why the other options are there

• 11,494 — kept a factor of two that cancels in the correct rearrangement.
• 2,874 — dropped that same factor in the other direction.
• 6,322 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses Confidence interval margin. Given z critical value (z) = 2.0300; std deviation (sigma) = 0.9500 in; margin of error (E) = 0.9130 in, determine the sample size (n).

Start with the thinking

• The governing relation printed in this handbook section is Confidence interval margin.
• Everything except n is given, so isolate n 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:

   $$E = z \sigma / \sqrt{n}$$

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

3. Step 3 — List the givens: z critical value (z) = 2.0300, std deviation (sigma) = 0.9500 in, margin of error (E) = 0.9130 in.

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

5. Step 5 — Evaluate:

   $$n = 4.4617$$

6. Step 6 — Check: returning n = 4.4617 to

   $$E = z \sigma / \sqrt{n}$$

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

Why the other options are there

• 8.9233 — kept a factor of two that cancels in the correct rearrangement.
• 2.2308 — dropped that same factor in the other direction.
• 4.9078 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses z-score. Given mean (mu) = 3,870 psi; value (x) = 6,760 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

   $$List the givens: mean (mu) = 3,870 psi, value (x) = 6,760 psi, z-score (z) = -2.3400$$

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

5. Step 5 — Evaluate:

   $$\sigma = -1235\ \text{psi}$$

6. Step 6 — Check: returning sigma = -1,235 psi to

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

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

Why the other options are there

• -2,470 — kept a factor of two that cancels in the correct rearrangement.
• -617.5 — dropped that same factor in the other direction.
• -1,359 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses Confidence interval margin. Given std deviation (sigma) = 1.2000 in; sample size (n) = 61.0000; margin of error (E) = 0.8830 in, determine the z critical value (z).

Start with the thinking

• The governing relation printed in this handbook section is Confidence interval margin.
• 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:

   $$E = z \sigma / \sqrt{n}$$

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

3. Step 3 — List the givens: std deviation (sigma) = 1.2000 in, sample size (n) = 61.0000, margin of error (E) = 0.8830 in.

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

5. Step 5 — Evaluate:

   $$z = 5.7470$$

6. Step 6 — Check: returning z = 5.7470 to

   $$E = z \sigma / \sqrt{n}$$

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

Why the other options are there

• 11.4941 — kept a factor of two that cancels in the correct rearrangement.
• 2.8735 — dropped that same factor in the other direction.
• 6.3217 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size

A probability and statistics problem uses z-score. Given mean (mu) = 2,500 psi; std deviation (sigma) = 510.0 psi; value (x) = 6,490 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

   $$List the givens: mean (mu) = 2,500 psi, std deviation (sigma) = 510.0 psi, value (x) = 6,490 psi$$

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

5. Step 5 — Evaluate:

   $$z = 7.8235$$

6. Step 6 — Check: returning z = 7.8235 to

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

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

Why the other options are there

• 15.6471 — kept a factor of two that cancels in the correct rearrangement.
• 3.9118 — dropped that same factor in the other direction.
• 8.6059 — rounded an intermediate value before the final step.

Reference: FE Reference Handbook — Probability and Statistics → Confidence Intervals, Sample Distributions and Sample Size