Questions: Recent research indicates that the effectiveness of antidepressant medication is directly related to the severity of the depression (Khan, Brodhead, Kolts Brown, 2005). Based on pretreatment depression scores, patients were divided into four groups based on their level of depression. After receiving the antidepressant medication, depression scores were measured again and the amount of improvement was recorded for each patient. The following data are similar to the results of the study. Low Moderate High Moderate Moderately Severe Severe 3.3 1 3.8 3.2 1.3 2.5 1.7 1.8 1 2.2 5.4 5.2 0.5 2.8 3.8 3.5 3.6 2.8 2.4 4.9 0 2.5 3.5 3.4 From this table, conduct an one-way ANOVA. Calculate the F-ratio and p-value. Be sure to round your answers to three decimal places. Assume all population and ANOVA requirements are met. F-ratio: p-value:

Recent research indicates that the effectiveness of antidepressant medication is directly related to the severity of the depression (Khan, Brodhead, Kolts Brown, 2005). Based on pretreatment depression scores, patients were divided into four groups based on their level of depression. After receiving the antidepressant medication, depression scores were measured again and the amount of improvement was recorded for each patient. The following data are similar to the results of the study. Low Moderate High Moderate Moderately Severe Severe 3.3 1 3.8 3.2 1.3 2.5 1.7 1.8 1 2.2 5.4 5.2 0.5 2.8 3.8 3.5 3.6 2.8 2.4 4.9 0 2.5 3.5 3.4 From this table, conduct an one-way ANOVA. Calculate the F-ratio and p-value. Be sure to round your answers to three decimal places. Assume all population and ANOVA requirements are met. F-ratio: p-value:
Transcript text: Recent research indicates that the effectiveness of antidepressant medication is directly related to the severity of the depression (Khan, Brodhead, Kolts & Brown, 2005). Based on pretreatment depression scores, patients were divided into four groups based on their level of depression. After receiving the antidepressant medication, depression scores were measured again and the amount of improvement was recorded for each patient. The following data are similar to the results of the study. Low Moderate High Moderate Moderately Severe Severe 3.3 1 3.8 3.2 1.3 2.5 1.7 1.8 1 2.2 5.4 5.2 0.5 2.8 3.8 3.5 3.6 2.8 2.4 4.9 0 2.5 3.5 3.4 From this table, conduct an one-way ANOVA. Calculate the $F$ -ratio and $p$-value. Be sure to round your answers to three decimal places. Assume all population and ANOVA requirements are met. F-ratio: $\square$ $p$-value: $\square$
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Solution

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Solution Steps

Step 1: Calculate Sum of Squares

The total sum of squares between groups is calculated as follows:

\[ SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 16.765 \]

The total sum of squares within groups is calculated as:

\[ SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 = 29.275 \]

Step 2: Calculate Mean Squares

The mean square between groups is given by:

\[ MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{16.765}{3} = 5.588 \]

The mean square within groups is calculated as:

\[ MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{29.275}{20} = 1.464 \]

Step 3: Calculate F-ratio

The F-ratio is calculated using the mean squares:

\[ F = \frac{MS_{between}}{MS_{within}} = \frac{5.588}{1.464} = 3.818 \]

Step 4: Calculate p-value

The p-value is determined using the F-distribution:

\[ P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(3.818; 3, 20) = 0.026 \]

Final Answer

The results of the one-way ANOVA are as follows:

F-ratio: \(\boxed{3.818}\)

p-value: \(\boxed{0.026}\)

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