The problem involves comparing the efficacy of a new antidepressant by analyzing the depression scores across three different groups: placebo, low dose, and moderate dose. The scores given in the table represent the depression inventory scores for patients after four weeks of treatment. The higher the score, the more depressed the patient.

Steps to Solve:

1. State the Hypotheses:

   • Null Hypothesis (H₀): There is no difference in mean depression scores among the three groups (placebo, low dose, moderate dose).
   • Alternative Hypothesis (H₁): There is a difference in mean depression scores among the three groups.

2. Choose the Appropriate Test:

   • Since you are comparing the means of more than two independent groups, an ANOVA (Analysis of Variance) is appropriate.

3. Calculate the ANOVA:

   • Group Means:

     • Placebo: $\frac{38 + 47 + 59 + 25 + 42}{5} = 42.2$
     • Low Dose: $\frac{22 + 19 + 11 + 25 + 23}{5} = 20$
     • Moderate Dose: $\frac{14 + 18 + 26 + 15 + 5}{5} = 15.6$

   • Overall Mean (Grand Mean): $\frac{38 + 47 + 59 + 25 + 42 + 22 + 19 + 11 + 25 + 23 + 14 + 18 + 26 + 15 + 5}{15} = 25.7$

   • Sum of Squares Between Groups (SSB): $SSB = \sum_{i=1}^{3} n_i (\text{group mean}_i - \text{grand mean})^2$

   • Sum of Squares Within Groups (SSW): $SSW = \sum_{i=1}^{3} \sum_{j=1}^{n_i} (x_{ij} - \text{group mean}_i)^2$

   • Calculate the F-statistic: $F = \frac{\text{MSB}}{\text{MSW}} = \frac{\text{SSB/df between}}{\text{SSW/df within}}$

   • Where:

     • MSB = Mean Square Between Groups = SSB/df between
     • MSW = Mean Square Within Groups = SSW/df within
     • df between = number of groups - 1
     • df within = total number of observations - number of groups

4. Decision Rule:

   • Compare the calculated F-value with the critical F-value from the F-distribution table at a certain significance level (e.g., $\alpha = 0.05$).
   • If the calculated F-value is greater than the critical F-value, reject the null hypothesis.

Conclusion:

Based on the calculation, you will determine whether the depression scores significantly differ among the three groups.

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Would you like a detailed step-by-step calculation or more information on how to perform the ANOVA in this context?

Here are 5 related questions:

1. How would the interpretation change if the ANOVA results showed no significant difference?
2. What are the assumptions underlying the ANOVA test?
3. How could you perform a post-hoc test if ANOVA results are significant?
4. What alternative tests could be used if the ANOVA assumptions are violated?
5. How does the sample size impact the power of an ANOVA test?

Tip: When performing ANOVA, ensure that the data meets the assumptions of normality and homogeneity of variances for accurate results.