An engineer performs hypothesis testing on a new concrete mix's mean strength. Given sample mean (xbar) = 107.7; hypothesized mean (mu0) = 95.0000; population std dev (sigma) = 5.0000; sample size (n) = 53.0000, determine the test statistic (z).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 107.7, hypothesized mean (mu0) = 95.0000, population std dev (sigma) = 5.0000, sample size (n) = 53.0000.

4. Step 4 — Substitute the given values:

   $$z = \dfrac{\bar{x}-\mu_0}{5.0000/\sqrt{53.0000}}$$

5. Step 5 — Evaluate:

   $$z = 18.4915$$

6. Step 6 — Check: returning z = 18.4915 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 36.9830 — kept a factor of two that cancels in the correct rearrangement.
• 9.2457 — dropped that same factor in the other direction.
• 20.3406 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

A quality analyst uses hypothesis testing to check if a process mean has shifted. Given hypothesized mean (mu0) = 98.3000; population std dev (sigma) = 3.8000; sample size (n) = 38.0000; test statistic (z) = 9.7500, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 98.3000, population std dev (sigma) = 3.8000, sample size (n) = 38.0000, test statistic (z) = 9.7500.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + 9.7500\dfrac{3.8000}{\sqrt{38.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 104.3$$

6. Step 6 — Check: returning xbar = 104.3 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 208.6 — kept a factor of two that cancels in the correct rearrangement.
• 52.1552 — dropped that same factor in the other direction.
• 114.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

Hypothesis testing on a sample mean determines whether to reject the null hypothesis. Given sample mean (xbar) = 95.7000; population std dev (sigma) = 7.8000; sample size (n) = 58.0000; test statistic (z) = 5.3700, determine the hypothesized mean (mu0).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• Everything except mu0 is given, so isolate mu0 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for mu0:

   $$\mu_{0} = \bar{x} - z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 95.7000, population std dev (sigma) = 7.8000, sample size (n) = 58.0000, test statistic (z) = 5.3700.

4. Step 4 — Substitute the given values:

   $$\mu_{0} = \bar{x} - 5.3700\dfrac{7.8000}{\sqrt{58.0000}}$$

5. Step 5 — Evaluate:

   $$\mu_{0} = 90.2001$$

6. Step 6 — Check: returning mu0 = 90.2001 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 180.4 — kept a factor of two that cancels in the correct rearrangement.
• 45.1000 — dropped that same factor in the other direction.
• 99.2201 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

An engineer performs hypothesis testing on a new concrete mix's mean strength. Given sample mean (xbar) = 99.6000; hypothesized mean (mu0) = 93.1000; population std dev (sigma) = 7.9000; sample size (n) = 17.0000, determine the test statistic (z).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 99.6000, hypothesized mean (mu0) = 93.1000, population std dev (sigma) = 7.9000, sample size (n) = 17.0000.

4. Step 4 — Substitute the given values:

   $$z = \dfrac{\bar{x}-\mu_0}{7.9000/\sqrt{17.0000}}$$

5. Step 5 — Evaluate:

   $$z = 3.3924$$

6. Step 6 — Check: returning z = 3.3924 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 6.7849 — kept a factor of two that cancels in the correct rearrangement.
• 1.6962 — dropped that same factor in the other direction.
• 3.7317 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

A quality analyst uses hypothesis testing to check if a process mean has shifted. Given hypothesized mean (mu0) = 90.0000; population std dev (sigma) = 6.2000; sample size (n) = 34.0000; test statistic (z) = 4.8300, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 90.0000, population std dev (sigma) = 6.2000, sample size (n) = 34.0000, test statistic (z) = 4.8300.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + 4.8300\dfrac{6.2000}{\sqrt{34.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 95.1357$$

6. Step 6 — Check: returning xbar = 95.1357 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 190.3 — kept a factor of two that cancels in the correct rearrangement.
• 47.5678 — dropped that same factor in the other direction.
• 104.6 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

Hypothesis testing on a sample mean determines whether to reject the null hypothesis. Given sample mean (xbar) = 97.7000; population std dev (sigma) = 4.7000; sample size (n) = 28.0000; test statistic (z) = -3.6800, determine the hypothesized mean (mu0).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• Everything except mu0 is given, so isolate mu0 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for mu0:

   $$\mu_{0} = \bar{x} - z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 97.7000, population std dev (sigma) = 4.7000, sample size (n) = 28.0000, test statistic (z) = -3.6800.

4. Step 4 — Substitute the given values:

   $$\mu_{0} = \bar{x} - -3.6800\dfrac{4.7000}{\sqrt{28.0000}}$$

5. Step 5 — Evaluate:

   $$\mu_{0} = 101.0$$

6. Step 6 — Check: returning mu0 = 101.0 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 201.9 — kept a factor of two that cancels in the correct rearrangement.
• 50.4843 — dropped that same factor in the other direction.
• 111.1 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

An engineer performs hypothesis testing on a new concrete mix's mean strength. Given sample mean (xbar) = 99.6000; hypothesized mean (mu0) = 97.6000; population std dev (sigma) = 6.4000; sample size (n) = 63.0000, determine the test statistic (z).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 99.6000, hypothesized mean (mu0) = 97.6000, population std dev (sigma) = 6.4000, sample size (n) = 63.0000.

4. Step 4 — Substitute the given values:

   $$z = \dfrac{\bar{x}-\mu_0}{6.4000/\sqrt{63.0000}}$$

5. Step 5 — Evaluate:

   $$z = 2.4804$$

6. Step 6 — Check: returning z = 2.4804 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 4.9608 — kept a factor of two that cancels in the correct rearrangement.
• 1.2402 — dropped that same factor in the other direction.
• 2.7284 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

A quality analyst uses hypothesis testing to check if a process mean has shifted. Given hypothesized mean (mu0) = 98.2000; population std dev (sigma) = 3.2000; sample size (n) = 34.0000; test statistic (z) = 0.5100, determine the sample mean (xbar).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for xbar:

   $$xbar = \mu_0 + z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: hypothesized mean (mu0) = 98.2000, population std dev (sigma) = 3.2000, sample size (n) = 34.0000, test statistic (z) = 0.5100.

4. Step 4 — Substitute the given values:

   $$xbar = \mu_0 + 0.5100\dfrac{3.2000}{\sqrt{34.0000}}$$

5. Step 5 — Evaluate:

   $$xbar = 98.4799$$

6. Step 6 — Check: returning xbar = 98.4799 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 197.0 — kept a factor of two that cancels in the correct rearrangement.
• 49.2399 — dropped that same factor in the other direction.
• 108.3 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

Hypothesis testing on a sample mean determines whether to reject the null hypothesis. Given sample mean (xbar) = 97.6000; population std dev (sigma) = 3.9000; sample size (n) = 45.0000; test statistic (z) = 1.1100, determine the hypothesized mean (mu0).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• Everything except mu0 is given, so isolate mu0 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for mu0:

   $$\mu_{0} = \bar{x} - z\dfrac{\sigma}{\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 97.6000, population std dev (sigma) = 3.9000, sample size (n) = 45.0000, test statistic (z) = 1.1100.

4. Step 4 — Substitute the given values:

   $$\mu_{0} = \bar{x} - 1.1100\dfrac{3.9000}{\sqrt{45.0000}}$$

5. Step 5 — Evaluate:

   $$\mu_{0} = 96.9547$$

6. Step 6 — Check: returning mu0 = 96.9547 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 193.9 — kept a factor of two that cancels in the correct rearrangement.
• 48.4773 — dropped that same factor in the other direction.
• 106.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing

An engineer performs hypothesis testing on a new concrete mix's mean strength. Given sample mean (xbar) = 104.7; hypothesized mean (mu0) = 93.6000; population std dev (sigma) = 8.4000; sample size (n) = 29.0000, determine the test statistic (z).

Start with the thinking

• The governing relation printed in this handbook section is Hypothesis testing — sample z statistic.
• 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.
• Hypothesis testing compares a sample mean to a hypothesized mean using the standardized z test statistic.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

2. Step 2 — Rearrange symbolically for z:

   $$z = \dfrac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$$

3. Step 3 — List the givens: sample mean (xbar) = 104.7, hypothesized mean (mu0) = 93.6000, population std dev (sigma) = 8.4000, sample size (n) = 29.0000.

4. Step 4 — Substitute the given values:

   $$z = \dfrac{\bar{x}-\mu_0}{8.4000/\sqrt{29.0000}}$$

5. Step 5 — Evaluate:

   $$z = 7.1161$$

6. Step 6 — Check: returning z = 7.1161 to

   $$z = \dfrac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

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

Why the other options are there

• 14.2322 — kept a factor of two that cancels in the correct rearrangement.
• 3.5581 — dropped that same factor in the other direction.
• 7.8277 — rounded an intermediate value before the final step.

Reference: FE Handbook — Hypothesis Testing