### Paired Sample t-Test for Comparing Two Population MeansThe null hypothesis ($ H_0 $) is $ \mu_1 = \mu_2 $ and the alternative hypothesis ($ H_a $) is specified as $ \mu_1 \neq \mu_2 $. The provided data are from a simple random paired sample from the two populations under consideration.#### Hypotheses:- Null hypothesis ($ H_0 $): $ \mu_1 = \mu_2 $- Alternative hypothesis ($ H_a $): $ \mu_1 \neq \mu_2 $Use the paired t-test to perform the required hypothesis test at a 10% significance level.#### Sample Observations:| Pair | Population 1 | Population 2 ||------|--------------|--------------|| 1 | 6 | 4 || 2 | 7 | 6 || 3 | 12 | 9 || 4 | 14 | 8 || 5 | 22 | 18 || 6 | 11 | 12 || 7 | 4 | 1 |To proceed with the test statistic calculation, use the difference between Population 1 and Population 2.#### Calculation:Find the test statistic $ t $:\[ t = \](Round to three decimal places as needed.)Click the icon below to view the t-table:[View t-table](#)By understanding the differences between the paired observations from the two populations, we can assess whether there is significant evidence to reject the null hypothesis in favor of the alternative hypothesis at the specified significance level. **t-Table**The t-table presented is used in statistics to determine critical values of the t-distribution. The table is arranged with degrees of freedom (df) listed in the first and last columns, and critical values for various levels of significance (alpha) displayed in the middle columns. This table is used commonly for conducting t-tests and other statistical analyses to determine if results are statistically significant.**Degrees of Freedom (df):**- The first and last columns labeled "df" represent the degrees of freedom. - The degrees of freedom range from 1 to 14 in this table.**Significance Levels (α):**- **t₀.₁₀:** Alpha level of 0.10- **t₀.₀₅:** Alpha level of 0.05- **t₀.₀₂₅:** Alpha level of 0.025- **t₀.₀₁:** Alpha level of 0.01- **t₀.₀₀₅:** Alpha level of 0.005**Critical Values:**| df | t₀.₁₀ | t₀.₀₅ | t₀.₀₂₅ | t₀.₀₁ | t₀.₀₀₅ | |-----|--------|--------|--------|--------|----------|| 1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 | | 2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 | | 3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 | | 4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 | | 5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 | | 6 | 1.440 | 1.943 | 2.

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Description: ### Paired Sample t-Test for Comparing Two Population MeansThe null hypothesis ($ H_0 $) is $ \mu_1 = \mu_2 $ and the alternative hypothesis ($ H_a $) is specified as $ \mu_1 \neq \mu_2 $. The provided data are from a simple random paired sa

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The null hypothesis is Ho: H1 = H2 and the alternative hypothesis is as specified. The provided data are from a simple random paired sample from the two populations under consideration. Use the paired t-test to perform the required hypothesis test at the 10% significance level. Observations from Pair Population 1 Population 2 6. 12 E Click the icon to view the t-table. 14 8 5 6 22 18 12 11 4 1 Find the test statistic. Use population 1- population 2 as the difference. t= (Round to three decimal places as needed.)

The null hypothesis is Ho: H1 = H2 and the alternative hypothesis is as specified. The provided data are from a simple random paired sample from the two populations under consideration. Use the paired t-test to perform the required hypothesis test at the 10% significance level. Observations from Pair Population 1 Population 2 6. 12 E Click the icon to view the t-table. 14 8 5 6 22 18 12 11 4 1 Find the test statistic. Use population 1- population 2 as the difference. t= (Round to three decimal places as needed.)

MATLAB: An Introduction with Applications

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### Paired Sample t-Test for Comparing Two Population Means

The null hypothesis ($ H_0 $) is $ \mu_1 = \mu_2 $ and the alternative hypothesis ($ H_a $) is specified as $ \mu_1 \neq \mu_2 $. The provided data are from a simple random paired sample from the two populations under consideration.

#### Hypotheses:
- Null hypothesis ($ H_0 $): $ \mu_1 = \mu_2 $
- Alternative hypothesis ($ H_a $): $ \mu_1 \neq \mu_2 $

Use the paired t-test to perform the required hypothesis test at a 10% significance level.

#### Sample Observations:
| Pair | Population 1 | Population 2 |
|------|--------------|--------------|
| 1 | 6 | 4 |
| 2 | 7 | 6 |
| 3 | 12 | 9 |
| 4 | 14 | 8 |
| 5 | 22 | 18 |
| 6 | 11 | 12 |
| 7 | 4 | 1 |

To proceed with the test statistic calculation, use the difference between Population 1 and Population 2.

#### Calculation:
Find the test statistic $ t $:
\[ t = \]

(Round to three decimal places as needed.)

Click the icon below to view the t-table:

[View t-table](#)

By understanding the differences between the paired observations from the two populations, we can assess whether there is significant evidence to reject the null hypothesis in favor of the alternative hypothesis at the specified significance level.

**t-Table**

The t-table presented is used in statistics to determine critical values of the t-distribution. The table is arranged with degrees of freedom (df) listed in the first and last columns, and critical values for various levels of significance (alpha) displayed in the middle columns. This table is used commonly for conducting t-tests and other statistical analyses to determine if results are statistically significant.

**Degrees of Freedom (df):**
- The first and last columns labeled "df" represent the degrees of freedom.
- The degrees of freedom range from 1 to 14 in this table.

**Significance Levels (α):**
- **t₀.₁₀:** Alpha level of 0.10
- **t₀.₀₅:** Alpha level of 0.05
- **t₀.₀₂₅:** Alpha level of 0.025
- **t₀.₀₁:** Alpha level of 0.01
- **t₀.₀₀₅:** Alpha level of 0.005

**Critical Values:**
| df | t₀.₁₀ | t₀.₀₅ | t₀.₀₂₅ | t₀.₀₁ | t₀.₀₀₅ |
|-----|--------|--------|--------|--------|----------|
| 1 | 3.078 | 6.314 | 12.706 | 31.821 | 63.657 |
| 2 | 1.886 | 2.920 | 4.303 | 6.965 | 9.925 |
| 3 | 1.638 | 2.353 | 3.182 | 4.541 | 5.841 |
| 4 | 1.533 | 2.132 | 2.776 | 3.747 | 4.604 |
| 5 | 1.476 | 2.015 | 2.571 | 3.365 | 4.032 |
| 6 | 1.440 | 1.943 | 2.

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