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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.

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
\begin{tabular}{c}
Low \\
Moderate
\end{tabular} &
\begin{tabular}{c}
High \\
Moderate
\end{tabular} &
\begin{tabular}{c}
Moderately \\
Severe
\end{tabular} &
Severe \\
\hline
1.5 & 1.2 & 2.7 & 4.4 \\
\hline
2.4 & 1.7 & 2.2 & 4.4 \\
\hline
1.7 & 4.2 & 4 & 3.5 \\
\hline
1.4 & 3.6 & 2.8 & 3.6 \\
\hline
2 & 1.8 & 3.9 & 4.9 \\
\hline
3.9 & 3.8 & 3.4 & 2 \\
\hline
\end{tabular}
\][/tex]

From this table, conduct a one-way ANOVA. Calculate the [tex]$F$[/tex]-ratio and [tex]$p$[/tex]-value. Be sure to round your answers to three decimal places. Assume all population and ANOVA requirements are met.

[tex]$F$[/tex]-ratio: [tex]$\square$[/tex]

[tex]$p$[/tex]-value: [tex]$\square$[/tex]



Answer :

GinnyAnswer
To address the research question using a one-way ANOVA, we need to determine if there are significant differences in improvements in depression scores among the four groups: Low, High, Moderately Severe, and Severe. Let's conduct our one-way ANOVA step by step.

### Step-by-Step Solution:

1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): There are no significant differences in the means of improvement across the four groups.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): At least one group mean improvement is significantly different from the others.

2. Collect the Data:
- Low: [tex]\( [1.5, 2.4, 1.7, 1.4, 2.0, 3.9] \)[/tex]
- High: [tex]\( [1.2, 1.7, 4.2, 3.6, 1.8, 3.8] \)[/tex]
- Moderately Severe: [tex]\( [2.7, 2.2, 4.0, 2.8, 3.9, 3.4] \)[/tex]
- Severe: [tex]\( [4.4, 4.4, 3.5, 3.6, 4.9, 2.0] \)[/tex]

3. Calculate the ANOVA:
- Using an appropriate statistical method (typically within a software or a calculator designed for statistical computation such as SPSS, R, or Python's `scipy.stats` library), we compute the F-ratio and the p-value.

4. F-ratio and p-value Results:
- F-ratio: Rounded to three decimal places, the F-ratio is 2.851.
- p-value: Rounded to three decimal places, the p-value is 0.063.

5. Interpret the Results:
- F-ratio: The F-statistic is a measure of the variance between the group means relative to the variance within the groups. Typically, higher F-values suggest that the group means are not all the same.
- p-value: The p-value indicates the probability of observing the test statistic or more extreme when the null hypothesis is true. A common significance level [tex]\(\alpha\)[/tex] is 0.05. Since the p-value (0.063) is slightly higher than 0.05, we do not have enough evidence to reject the null hypothesis at the [tex]\( \alpha = 0.05 \)[/tex] level.

### Conclusion:
There is not enough statistical evidence at the 0.05 significance level to say that there is a significant difference in the mean improvements in depression scores among the four levels of depression severity.

### Final Answers:
- F-ratio: [tex]\(2.851\)[/tex]
- p-value: [tex]\(0.063\)[/tex]

These results suggest that, while there appear to be some differences in the mean improvements, these differences are not statistically significant based on the ANOVA test conducted.

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