Answer :
To address the research problem, we'll follow the steps outlined and go through the process of hypothesis testing using one-way ANOVA, determine the necessity for post hoc tests, compute eta squared, and finally, summarize our findings in a source table.
### a. One-Way ANOVA Hypothesis Test
Hypotheses:
- Null Hypothesis (H0): The means of the stress levels are equal across all three groups (Non-Commuters, Short Distance, Long Distance).
- Alternative Hypothesis (H1): At least one group mean is different.
F-Value and P-Value:
After conducting the ANOVA test, we obtained:
- F-Value: 6.7848101265822764
- P-Value: 0.01068419503705755
Decision Rule:
- Level of significance ([tex]\(\alpha\)[/tex]) = 0.05.
- Compare the p-value to [tex]\(\alpha\)[/tex]:
- If [tex]\(p \leq 0.05\)[/tex], reject the null hypothesis.
- If [tex]\(p > 0.05\)[/tex], fail to reject the null hypothesis.
Conclusion:
Since the p-value (0.01068419503705755) is less than the significance level (0.05), we reject the null hypothesis. This indicates that there is a statistically significant difference in the mean stress levels among the three groups.
### b. Need for Post Hoc Tests
Since we rejected the null hypothesis in the ANOVA test, this suggests that at least one group's mean is different from the others. Therefore, we need to perform post hoc tests to determine specifically which groups are different from each other. Post hoc tests are necessary when the ANOVA test indicates a significant effect, as it helps to pinpoint the specific pairs of groups that have significant differences.
### c. Compute Eta Squared (η²)
Eta Squared Calculation:
- Grand Mean = Average of all observations combined.
- Total Sum of Squares (SS_Total) = 67.33333333333334
- Between-Groups Sum of Squares (SS_Between) = 35.73333333333333
Eta squared (η²) measures the proportion of the total variance that is attributed to the factor.
[tex]\[ \eta^2 = \frac{SS_{Between}}{SS_{Total}} = \frac{35.73333333333333}{67.33333333333334} = 0.5306930693069305 \][/tex]
### d. Summary of Findings in a Source Table
| Source | Sum of Squares | Degrees of Freedom | Mean Square | F-value | P-value |
|-------------------|----------------|--------------------|---------------------|--------------------|---------------------|
| Between Groups | 35.73333333333333 | 2 | 17.866666666666664 | 6.7848101265822764 | 0.01068419503705755 |
| Within Groups | 31.600000000000016 | 12 | 2.6333333333333346 | - | - |
| Total | 67.33333333333334 | 14 | - | - | - |
### Summary
- The ANOVA test reveals a statistically significant difference in stress levels among non-commuters, short-distance commuters, and long-distance commuters in New York City ([tex]\(p = 0.01068419503705755\)[/tex]).
- Post hoc tests are required to identify which specific group's mean stress levels are different.
- The eta squared value (η²) indicates that approximately 53.07% of the variance in stress levels can be explained by the commuting distance factor.
- The source table summarizes the key statistics for the ANOVA test, including sum of squares, degrees of freedom, mean squares, F-value, and p-value.
### a. One-Way ANOVA Hypothesis Test
Hypotheses:
- Null Hypothesis (H0): The means of the stress levels are equal across all three groups (Non-Commuters, Short Distance, Long Distance).
- Alternative Hypothesis (H1): At least one group mean is different.
F-Value and P-Value:
After conducting the ANOVA test, we obtained:
- F-Value: 6.7848101265822764
- P-Value: 0.01068419503705755
Decision Rule:
- Level of significance ([tex]\(\alpha\)[/tex]) = 0.05.
- Compare the p-value to [tex]\(\alpha\)[/tex]:
- If [tex]\(p \leq 0.05\)[/tex], reject the null hypothesis.
- If [tex]\(p > 0.05\)[/tex], fail to reject the null hypothesis.
Conclusion:
Since the p-value (0.01068419503705755) is less than the significance level (0.05), we reject the null hypothesis. This indicates that there is a statistically significant difference in the mean stress levels among the three groups.
### b. Need for Post Hoc Tests
Since we rejected the null hypothesis in the ANOVA test, this suggests that at least one group's mean is different from the others. Therefore, we need to perform post hoc tests to determine specifically which groups are different from each other. Post hoc tests are necessary when the ANOVA test indicates a significant effect, as it helps to pinpoint the specific pairs of groups that have significant differences.
### c. Compute Eta Squared (η²)
Eta Squared Calculation:
- Grand Mean = Average of all observations combined.
- Total Sum of Squares (SS_Total) = 67.33333333333334
- Between-Groups Sum of Squares (SS_Between) = 35.73333333333333
Eta squared (η²) measures the proportion of the total variance that is attributed to the factor.
[tex]\[ \eta^2 = \frac{SS_{Between}}{SS_{Total}} = \frac{35.73333333333333}{67.33333333333334} = 0.5306930693069305 \][/tex]
### d. Summary of Findings in a Source Table
| Source | Sum of Squares | Degrees of Freedom | Mean Square | F-value | P-value |
|-------------------|----------------|--------------------|---------------------|--------------------|---------------------|
| Between Groups | 35.73333333333333 | 2 | 17.866666666666664 | 6.7848101265822764 | 0.01068419503705755 |
| Within Groups | 31.600000000000016 | 12 | 2.6333333333333346 | - | - |
| Total | 67.33333333333334 | 14 | - | - | - |
### Summary
- The ANOVA test reveals a statistically significant difference in stress levels among non-commuters, short-distance commuters, and long-distance commuters in New York City ([tex]\(p = 0.01068419503705755\)[/tex]).
- Post hoc tests are required to identify which specific group's mean stress levels are different.
- The eta squared value (η²) indicates that approximately 53.07% of the variance in stress levels can be explained by the commuting distance factor.
- The source table summarizes the key statistics for the ANOVA test, including sum of squares, degrees of freedom, mean squares, F-value, and p-value.
