An engineer computes a test statistic to compare a small sample mean to a target value. Given sample mean (xbar) = 98.9000; hypothesized mean (mu0) = 91.7000; sample std dev (s) = 3.4000; sample size (n) = 22.0000, determine the test statistic (t).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except t is given, so isolate t 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for t:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 98.9000, hypothesized mean (mu0) = 91.7000, sample std dev (s) = 3.4000, sample size (n) = 22.0000.

4. Step 4 — Substitute the given values:

   $$t = \dfrac{\bar{x}-\mu_0}{3.4000/\sqrt{22.0000}}$$

5. Step 5 — Evaluate:

   $$t = 9.9326$$

6. Step 6 — Check: returning t = 9.9326 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 19.8653 — kept a factor of two that cancels in the correct rearrangement.
• 4.9663 — dropped that same factor in the other direction.
• 10.9259 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

A student calculates the t test statistic for a lab measurement series. Given hypothesized mean (mu0) = 92.5000; sample std dev (s) = 3.3000; sample size (n) = 11.0000; test statistic (t) = 6.3900, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except xbar is given, so isolate xbar 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + t\dfrac{s}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 92.5000, sample std dev (s) = 3.3000, sample size (n) = 11.0000, test statistic (t) = 6.3900.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + 6.3900\dfrac{3.3000}{\sqrt{11.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 98.8580$$

6. Step 6 — Check: returning xbar = 98.8580 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 197.7 — kept a factor of two that cancels in the correct rearrangement.
• 49.4290 — dropped that same factor in the other direction.
• 108.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

The test statistic exceeds the critical value, leading to rejection of the null hypothesis. Given sample mean (xbar) = 104.8; hypothesized mean (mu0) = 98.1000; sample size (n) = 23.0000; test statistic (t) = 5.0500, determine the sample std dev (s).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except s is given, so isolate s 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{n}}{t}$$

3. Step 3 — List the givens: sample mean (xbar) = 104.8, hypothesized mean (mu0) = 98.1000, sample size (n) = 23.0000, test statistic (t) = 5.0500.

4. Step 4 — Substitute the given values:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{23.0000}}{5.0500}$$

5. Step 5 — Evaluate:

   $$s = 6.3628$$

6. Step 6 — Check: returning s = 6.3628 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 12.7256 — kept a factor of two that cancels in the correct rearrangement.
• 3.1814 — dropped that same factor in the other direction.
• 6.9991 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

An engineer computes a test statistic to compare a small sample mean to a target value. Given sample mean (xbar) = 107.0; hypothesized mean (mu0) = 95.2000; sample std dev (s) = 2.1000; sample size (n) = 6.0000, determine the test statistic (t).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except t is given, so isolate t 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for t:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 107.0, hypothesized mean (mu0) = 95.2000, sample std dev (s) = 2.1000, sample size (n) = 6.0000.

4. Step 4 — Substitute the given values:

   $$t = \dfrac{\bar{x}-\mu_0}{2.1000/\sqrt{6.0000}}$$

5. Step 5 — Evaluate:

   $$t = 13.7638$$

6. Step 6 — Check: returning t = 13.7638 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 27.5276 — kept a factor of two that cancels in the correct rearrangement.
• 6.8819 — dropped that same factor in the other direction.
• 15.1402 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

A student calculates the t test statistic for a lab measurement series. Given hypothesized mean (mu0) = 95.8000; sample std dev (s) = 8.5000; sample size (n) = 19.0000; test statistic (t) = 3.6900, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except xbar is given, so isolate xbar 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + t\dfrac{s}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 95.8000, sample std dev (s) = 8.5000, sample size (n) = 19.0000, test statistic (t) = 3.6900.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + 3.6900\dfrac{8.5000}{\sqrt{19.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 103.0$$

6. Step 6 — Check: returning xbar = 103.0 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 206.0 — kept a factor of two that cancels in the correct rearrangement.
• 51.4978 — dropped that same factor in the other direction.
• 113.3 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

The test statistic exceeds the critical value, leading to rejection of the null hypothesis. Given sample mean (xbar) = 98.5000; hypothesized mean (mu0) = 96.2000; sample size (n) = 30.0000; test statistic (t) = 4.2100, determine the sample std dev (s).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except s is given, so isolate s 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{n}}{t}$$

3. Step 3 — List the givens: sample mean (xbar) = 98.5000, hypothesized mean (mu0) = 96.2000, sample size (n) = 30.0000, test statistic (t) = 4.2100.

4. Step 4 — Substitute the given values:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{30.0000}}{4.2100}$$

5. Step 5 — Evaluate:

   $$s = 2.9923$$

6. Step 6 — Check: returning s = 2.9923 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 5.9846 — kept a factor of two that cancels in the correct rearrangement.
• 1.4962 — dropped that same factor in the other direction.
• 3.2915 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

An engineer computes a test statistic to compare a small sample mean to a target value. Given sample mean (xbar) = 104.9; hypothesized mean (mu0) = 95.4000; sample std dev (s) = 7.3000; sample size (n) = 27.0000, determine the test statistic (t).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except t is given, so isolate t 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for t:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 104.9, hypothesized mean (mu0) = 95.4000, sample std dev (s) = 7.3000, sample size (n) = 27.0000.

4. Step 4 — Substitute the given values:

   $$t = \dfrac{\bar{x}-\mu_0}{7.3000/\sqrt{27.0000}}$$

5. Step 5 — Evaluate:

   $$t = 6.7621$$

6. Step 6 — Check: returning t = 6.7621 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 13.5242 — kept a factor of two that cancels in the correct rearrangement.
• 3.3811 — dropped that same factor in the other direction.
• 7.4383 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

A student calculates the t test statistic for a lab measurement series. Given hypothesized mean (mu0) = 99.4000; sample std dev (s) = 3.1000; sample size (n) = 20.0000; test statistic (t) = -3.1900, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except xbar is given, so isolate xbar 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + t\dfrac{s}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 99.4000, sample std dev (s) = 3.1000, sample size (n) = 20.0000, test statistic (t) = -3.1900.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + -3.1900\dfrac{3.1000}{\sqrt{20.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 97.1888$$

6. Step 6 — Check: returning xbar = 97.1888 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• 194.4 — kept a factor of two that cancels in the correct rearrangement.
• 48.5944 — dropped that same factor in the other direction.
• 106.9 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

The test statistic exceeds the critical value, leading to rejection of the null hypothesis. Given sample mean (xbar) = 96.0000; hypothesized mean (mu0) = 94.6000; sample size (n) = 17.0000; test statistic (t) = -3.0800, determine the sample std dev (s).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except s is given, so isolate s 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for s:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{n}}{t}$$

3. Step 3 — List the givens: sample mean (xbar) = 96.0000, hypothesized mean (mu0) = 94.6000, sample size (n) = 17.0000, test statistic (t) = -3.0800.

4. Step 4 — Substitute the given values:

   $$s = \dfrac{(\bar{x}-\mu_0)\sqrt{17.0000}}{-3.0800}$$

5. Step 5 — Evaluate:

   $$s = -1.8741$$

6. Step 6 — Check: returning s = -1.8741 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• -3.7483 — kept a factor of two that cancels in the correct rearrangement.
• -0.9371 — dropped that same factor in the other direction.
• -2.0616 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics

An engineer computes a test statistic to compare a small sample mean to a target value. Given sample mean (xbar) = 96.1000; hypothesized mean (mu0) = 96.4000; sample std dev (s) = 2.3000; sample size (n) = 26.0000, determine the test statistic (t).

Start with the thinking

• The governing relation printed in this handbook section is Test statistic — Student's t.
• Everything except t is given, so isolate t 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.
• This test statistic follows Student's t distribution for small-sample hypothesis testing.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for t:

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 96.1000, hypothesized mean (mu0) = 96.4000, sample std dev (s) = 2.3000, sample size (n) = 26.0000.

4. Step 4 — Substitute the given values:

   $$t = \dfrac{\bar{x}-\mu_0}{2.3000/\sqrt{26.0000}}$$

5. Step 5 — Evaluate:

   $$t = -0.6651$$

6. Step 6 — Check: returning t = -0.6651 to

   $$t = \dfrac{\bar{x}-\mu_0}{s/\sqrt{n}}$$

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

Why the other options are there

• -1.3302 — kept a factor of two that cancels in the correct rearrangement.
• -0.3325 — dropped that same factor in the other direction.
• -0.7316 — rounded an intermediate value before the final step.

Reference: FE Handbook — Test Statistics