a psychologist at a mental hospital was asked to determine whether there was any clear difference in the length of stay of patients with different categories of diagnosis. looking at the last 4 patients in each of the 3 major categories , the results in terms of weeks and stay where as follows:
Diagnosis Category
Affective Disorders Cognitive Disorders Drug-Related Conditions
7 12 8
6 8 10
5 9 12
6 11 10
(a) Using the .05 level and the five steps of hypothesis testing, is there a significant difference in length of stay among diagnosis categories? (b) Sketch the distributions involved.(c) Figure the effect size for the study. (d) Explain your answer to part (a) to someone who understands everything involved in conducting a t test for independent means but is unfamiliar with the analysis of variance(e) Test the significance of planned contrasts (using the .05 level without a Bonferroni correction) for affective disorders versus drug-related conditions and (f) cognitive disorders versus drug-related conditions. (g) Explain your answers to parts (e) and (f) to a person who understands analysis of variance but is unfamiliar with planned contrasts
(a) To determine if there is a significant difference in length of stay among diagnosis categories, we can use analysis of variance (ANOVA).
Step 1: State the Null Hypothesis (H0) and Alternative Hypothesis (Ha)
The null hypothesis assumes that there is no significant difference in length of stay among the diagnosis categories. The alternate hypothesis assumes that there is a significant difference.
H0: μ1 = μ2 = μ3 (No significant difference)
Ha: At least one mean is different (There is a significant difference)
Step 2: Select the Level of Significance
In this case, the significance level is given as 0.05.
Step 3: Compute the Test Statistic
We need to calculate the F-statistic, which indicates the ratio of between-group variability to within-group variability.
Step 4: Determine the Critical Value
We can use a reference table for F-distribution to find the critical value. The degrees of freedom for the numerator is k-1, where k represents the number of diagnosis categories (3 in this case). The degrees of freedom for the denominator is n-k, where n is the total number of observations (12 in this case).
Step 5: Make a Decision
If the calculated F-value is greater than the critical value, we reject the null hypothesis, indicating a significant difference. Otherwise, we fail to reject the null hypothesis.
(b) To sketch the distributions involved, you can create boxplots for each diagnosis category. The length of stay in weeks will be plotted on the y-axis, and each diagnosis category will be shown on the x-axis. The boxplots allow for easy comparison of the medians, quartiles, and potential outliers.
(c) To figure the effect size for the study, you can calculate eta-squared (η²). Eta-squared represents the proportion of variance explained by the independent variable (diagnosis category). It can be computed by dividing the sum of squares between groups by the total sum of squares. A larger value indicates a larger effect size.
(d) In this study, we conducted an analysis of variance (ANOVA) to determine if there is a significant difference in the length of stay among diagnosis categories. The calculated F-value is then compared to a critical value obtained from the F-distribution table. If the calculated F-value exceeds the critical value, we reject the null hypothesis, indicating that there is a significant difference in length of stay among diagnosis categories.
(e) To test the significance of planned contrasts, we can conduct t-tests between specific diagnosis categories. For affective disorders versus drug-related conditions, we compare the mean length of stay for patients in these two categories. Similarly, we conduct another t-test for cognitive disorders versus drug-related conditions. We can use a significance level of 0.05 to determine if these differences are significant.
(f) To explain the answers to parts (e) and (f) to someone unfamiliar with planned contrasts, we conducted additional t-tests to compare the mean length of stay between specific diagnosis categories. For affective disorders versus drug-related conditions, and cognitive disorders versus drug-related conditions, we compared the means and calculated the t-statistic. By comparing the t-statistic with the critical value obtained from the t-distribution table (using a significance level of 0.05), we determined if the differences in length of stay between these diagnosis categories were statistically significant.
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However, the sample seems to be too small to come to any definite conclusion.