An engineer finds the 90th fractile of a strength distribution. Given mean (mu) = 75.0000; std deviation (sigma) = 11.5000; standard fractile value (zp) = -0.8500, determine the fractile value (xp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except xp is given, so isolate xp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for xp:

   $$xp = \mu + z_p \sigma$$

3. Step 3 — List the givens: mean (mu) = 75.0000, std deviation (sigma) = 11.5000, standard fractile value (zp) = -0.8500.

4. Step 4 — Substitute the given values:

   $$xp = 75.0000 + z_p 11.5000$$

5. Step 5 — Evaluate:

   $$xp = 65.2250$$

6. Step 6 — Check: returning xp = 65.2250 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 130.5 — kept a factor of two that cancels in the correct rearrangement.
• 32.6125 — dropped that same factor in the other direction.
• 71.7475 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

A student computes a fractile value for a normally distributed load. Given std deviation (sigma) = 18.5000; standard fractile value (zp) = -0.9100; fractile value (xp) = 75.8000, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for mu:

   $$\mu = x_p - z_p \sigma$$

3. Step 3 — List the givens: std deviation (sigma) = 18.5000, standard fractile value (zp) = -0.9100, fractile value (xp) = 75.8000.

4. Step 4 — Substitute the given values:

   $$\mu = x_p - z_p 18.5000$$

5. Step 5 — Evaluate:

   $$\mu = 92.6350$$

6. Step 6 — Check: returning mu = 92.6350 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 185.3 — kept a factor of two that cancels in the correct rearrangement.
• 46.3175 — dropped that same factor in the other direction.
• 101.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

The design fractile of flood flows is estimated from the mean and standard deviation. Given mean (mu) = 68.0000; std deviation (sigma) = 17.0000; fractile value (xp) = 125.0, determine the standard fractile value (zp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except zp is given, so isolate zp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for zp:

   $$zp = \dfrac{x_p - \mu}{\sigma}$$

3. Step 3

   $$List the givens: mean (mu) = 68.0000, std deviation (sigma) = 17.0000, fractile value (xp) = 125.0$$

4. Step 4 — Substitute the given values:

   $$zp = \dfrac{x_p - 68.0000}{17.0000}$$

5. Step 5 — Evaluate:

   $$zp = 3.3529$$

6. Step 6 — Check: returning zp = 3.3529 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 6.7059 — kept a factor of two that cancels in the correct rearrangement.
• 1.6765 — dropped that same factor in the other direction.
• 3.6882 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

An engineer finds the 90th fractile of a strength distribution. Given mean (mu) = 84.0000; std deviation (sigma) = 8.5000; standard fractile value (zp) = -2.0300, determine the fractile value (xp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except xp is given, so isolate xp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for xp:

   $$xp = \mu + z_p \sigma$$

3. Step 3 — List the givens: mean (mu) = 84.0000, std deviation (sigma) = 8.5000, standard fractile value (zp) = -2.0300.

4. Step 4 — Substitute the given values:

   $$xp = 84.0000 + z_p 8.5000$$

5. Step 5 — Evaluate:

   $$xp = 66.7450$$

6. Step 6 — Check: returning xp = 66.7450 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 133.5 — kept a factor of two that cancels in the correct rearrangement.
• 33.3725 — dropped that same factor in the other direction.
• 73.4195 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

A student computes a fractile value for a normally distributed load. Given std deviation (sigma) = 10.5000; standard fractile value (zp) = -0.1500; fractile value (xp) = 187.4, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for mu:

   $$\mu = x_p - z_p \sigma$$

3. Step 3 — List the givens: std deviation (sigma) = 10.5000, standard fractile value (zp) = -0.1500, fractile value (xp) = 187.4.

4. Step 4 — Substitute the given values:

   $$\mu = x_p - z_p 10.5000$$

5. Step 5 — Evaluate:

   $$\mu = 189.0$$

6. Step 6 — Check: returning mu = 189.0 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 378.0 — kept a factor of two that cancels in the correct rearrangement.
• 94.4875 — dropped that same factor in the other direction.
• 207.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

The design fractile of flood flows is estimated from the mean and standard deviation. Given mean (mu) = 76.0000; std deviation (sigma) = 12.0000; fractile value (xp) = 107.1, determine the standard fractile value (zp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except zp is given, so isolate zp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for zp:

   $$zp = \dfrac{x_p - \mu}{\sigma}$$

3. Step 3

   $$List the givens: mean (mu) = 76.0000, std deviation (sigma) = 12.0000, fractile value (xp) = 107.1$$

4. Step 4 — Substitute the given values:

   $$zp = \dfrac{x_p - 76.0000}{12.0000}$$

5. Step 5 — Evaluate:

   $$zp = 2.5917$$

6. Step 6 — Check: returning zp = 2.5917 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 5.1833 — kept a factor of two that cancels in the correct rearrangement.
• 1.2958 — dropped that same factor in the other direction.
• 2.8508 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

An engineer finds the 90th fractile of a strength distribution. Given mean (mu) = 100.0; std deviation (sigma) = 11.5000; standard fractile value (zp) = -0.1500, determine the fractile value (xp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except xp is given, so isolate xp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for xp:

   $$xp = \mu + z_p \sigma$$

3. Step 3 — List the givens: mean (mu) = 100.0, std deviation (sigma) = 11.5000, standard fractile value (zp) = -0.1500.

4. Step 4 — Substitute the given values:

   $$xp = 100.0 + z_p 11.5000$$

5. Step 5 — Evaluate:

   $$xp = 98.2750$$

6. Step 6 — Check: returning xp = 98.2750 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 196.6 — kept a factor of two that cancels in the correct rearrangement.
• 49.1375 — dropped that same factor in the other direction.
• 108.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

A student computes a fractile value for a normally distributed load. Given std deviation (sigma) = 13.5000; standard fractile value (zp) = 0.7200; fractile value (xp) = 159.5, determine the mean (mu).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for mu:

   $$\mu = x_p - z_p \sigma$$

3. Step 3 — List the givens: std deviation (sigma) = 13.5000, standard fractile value (zp) = 0.7200, fractile value (xp) = 159.5.

4. Step 4 — Substitute the given values:

   $$\mu = x_p - z_p 13.5000$$

5. Step 5 — Evaluate:

   $$\mu = 149.8$$

6. Step 6 — Check: returning mu = 149.8 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 299.6 — kept a factor of two that cancels in the correct rearrangement.
• 74.8900 — dropped that same factor in the other direction.
• 164.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

The design fractile of flood flows is estimated from the mean and standard deviation. Given mean (mu) = 147.0; std deviation (sigma) = 14.0000; fractile value (xp) = 40.9000, determine the standard fractile value (zp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except zp is given, so isolate zp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for zp:

   $$zp = \dfrac{x_p - \mu}{\sigma}$$

3. Step 3

   $$List the givens: mean (mu) = 147.0, std deviation (sigma) = 14.0000, fractile value (xp) = 40.9000$$

4. Step 4 — Substitute the given values:

   $$zp = \dfrac{x_p - 147.0}{14.0000}$$

5. Step 5 — Evaluate:

   $$zp = -7.5786$$

6. Step 6 — Check: returning zp = -7.5786 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• -15.1571 — kept a factor of two that cancels in the correct rearrangement.
• -3.7893 — dropped that same factor in the other direction.
• -8.3364 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles

An engineer finds the 90th fractile of a strength distribution. Given mean (mu) = 121.0; std deviation (sigma) = 7.5000; standard fractile value (zp) = 0.6400, determine the fractile value (xp).

Start with the thinking

• The governing relation printed in this handbook section is Fractiles of a normal distribution.
• Everything except xp is given, so isolate xp 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.
• A fractile x_p of a distribution is located a standardized distance z_p from the mean.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$x_p = \mu + z_p \sigma$$

2. Step 2 — Rearrange symbolically for xp:

   $$xp = \mu + z_p \sigma$$

3. Step 3 — List the givens: mean (mu) = 121.0, std deviation (sigma) = 7.5000, standard fractile value (zp) = 0.6400.

4. Step 4 — Substitute the given values:

   $$xp = 121.0 + z_p 7.5000$$

5. Step 5 — Evaluate:

   $$xp = 125.8$$

6. Step 6 — Check: returning xp = 125.8 to

   $$x_p = \mu + z_p \sigma$$

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

Why the other options are there

• 251.6 — kept a factor of two that cancels in the correct rearrangement.
• 62.9000 — dropped that same factor in the other direction.
• 138.4 — rounded an intermediate value before the final step.

Reference: FE Handbook — Fractiles