Answer :
To determine whether the three age-group populations have different mean body temperatures, we'll perform an ANOVA (Analysis of Variance) test. We'll follow these steps:
1. Formulate Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean body temperatures for all three age groups are equal.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): At least one age group has a different mean body temperature.
2. Calculate Necessary Values:
- [tex]\( N \)[/tex] (Total number of observations) = 18 (6 subjects per group).
- [tex]\( k \)[/tex] (Number of groups) = 3.
3. Compute Group Means:
- Mean for each group is calculated by dividing the sum of temperatures for each group by the number of subjects in that group.
4. Calculate the Overall Mean:
- [tex]\( \text{Overall Mean} = \frac{T_1 + T_2 + T_3}{N} \)[/tex].
5. Compute Sum of Squares Between Groups (SSB):
- [tex]\( \text{SSB} = 6 \left[ (\text{Mean}_1 - \text{Overall Mean})^2 + (\text{Mean}_2 - \text{Overall Mean})^2 + (\text{Mean}_3 - \text{Overall Mean})^2 \right] \)[/tex].
6. Compute Total Sum of Squares (SST):
- [tex]\( \text{SST} = \Sigma x^2 - (N \times (\text{Overall Mean})^2) \)[/tex].
7. Calculate Sum of Squares Within Groups (SSW):
- [tex]\( \text{SSW} = \text{SST} - \text{SSB} \)[/tex].
8. Determine Degrees of Freedom:
- Degrees of freedom between groups: [tex]\( \text{df}_\text{between} = k - 1 = 2 \)[/tex].
- Degrees of freedom within groups: [tex]\( \text{df}_\text{within} = N - k = 15 \)[/tex].
9. Compute Mean Squares:
- Mean Square Between ([tex]\( \text{MSB} \)[/tex]): [tex]\( \text{MSB} = \frac{\text{SSB}}{\text{df}_\text{between}} \)[/tex].
- Mean Square Within ([tex]\( \text{MSW} \)[/tex]): [tex]\( \text{MSW} = \frac{\text{SSW}}{\text{df}_\text{within}} \)[/tex].
10. Calculate the F-Statistic:
- [tex]\( F = \frac{\text{MSB}}{\text{MSW}} \)[/tex].
11. Determine the p-value:
- Using the F-distribution with [tex]\( \text{df}_\text{between} = 2 \)[/tex] and [tex]\( \text{df}_\text{within} = 15 \)[/tex].
Based on the computed values, the results are as follows:
- Sum of Squares Between (SSB): 0.1878
- Sum of Squares Total (SST): 5.2111
- Sum of Squares Within (SSW): 5.0233
- Mean Square Between (MSB): 0.0939
- Mean Square Within (MSW): 0.3349
- F-Statistic: 0.2804
- p-value: 0.7594
12. Conclusion:
At the [tex]\( \alpha = 0.05 \)[/tex] significance level, the p-value of 0.7594 is greater than 0.05. Therefore, we fail to reject the null hypothesis. This indicates there is no statistically significant difference in the mean body temperatures among the three age group populations.
1. Formulate Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The mean body temperatures for all three age groups are equal.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): At least one age group has a different mean body temperature.
2. Calculate Necessary Values:
- [tex]\( N \)[/tex] (Total number of observations) = 18 (6 subjects per group).
- [tex]\( k \)[/tex] (Number of groups) = 3.
3. Compute Group Means:
- Mean for each group is calculated by dividing the sum of temperatures for each group by the number of subjects in that group.
4. Calculate the Overall Mean:
- [tex]\( \text{Overall Mean} = \frac{T_1 + T_2 + T_3}{N} \)[/tex].
5. Compute Sum of Squares Between Groups (SSB):
- [tex]\( \text{SSB} = 6 \left[ (\text{Mean}_1 - \text{Overall Mean})^2 + (\text{Mean}_2 - \text{Overall Mean})^2 + (\text{Mean}_3 - \text{Overall Mean})^2 \right] \)[/tex].
6. Compute Total Sum of Squares (SST):
- [tex]\( \text{SST} = \Sigma x^2 - (N \times (\text{Overall Mean})^2) \)[/tex].
7. Calculate Sum of Squares Within Groups (SSW):
- [tex]\( \text{SSW} = \text{SST} - \text{SSB} \)[/tex].
8. Determine Degrees of Freedom:
- Degrees of freedom between groups: [tex]\( \text{df}_\text{between} = k - 1 = 2 \)[/tex].
- Degrees of freedom within groups: [tex]\( \text{df}_\text{within} = N - k = 15 \)[/tex].
9. Compute Mean Squares:
- Mean Square Between ([tex]\( \text{MSB} \)[/tex]): [tex]\( \text{MSB} = \frac{\text{SSB}}{\text{df}_\text{between}} \)[/tex].
- Mean Square Within ([tex]\( \text{MSW} \)[/tex]): [tex]\( \text{MSW} = \frac{\text{SSW}}{\text{df}_\text{within}} \)[/tex].
10. Calculate the F-Statistic:
- [tex]\( F = \frac{\text{MSB}}{\text{MSW}} \)[/tex].
11. Determine the p-value:
- Using the F-distribution with [tex]\( \text{df}_\text{between} = 2 \)[/tex] and [tex]\( \text{df}_\text{within} = 15 \)[/tex].
Based on the computed values, the results are as follows:
- Sum of Squares Between (SSB): 0.1878
- Sum of Squares Total (SST): 5.2111
- Sum of Squares Within (SSW): 5.0233
- Mean Square Between (MSB): 0.0939
- Mean Square Within (MSW): 0.3349
- F-Statistic: 0.2804
- p-value: 0.7594
12. Conclusion:
At the [tex]\( \alpha = 0.05 \)[/tex] significance level, the p-value of 0.7594 is greater than 0.05. Therefore, we fail to reject the null hypothesis. This indicates there is no statistically significant difference in the mean body temperatures among the three age group populations.