Model Answer
0 min readIntroduction
Effective healthcare delivery hinges on efficient resource allocation and minimizing patient wait times, particularly in emergency situations. Long waiting times in emergency wings can lead to adverse patient outcomes and decreased satisfaction. City administrations increasingly utilize statistical analysis to identify areas for improvement in healthcare services. This question presents a scenario where a city administration aims to determine if there's a statistically significant difference in average waiting times across three hospitals' emergency wings, employing a hypothesis testing framework to guide resource allocation and intervention strategies. Understanding and applying statistical methods like ANOVA is crucial for evidence-based public health management.
Understanding the Problem and Hypotheses
The city administration wants to ascertain if the average waiting times in the emergency wings of the three hospitals are equal or if there's a significant difference. To address this, we employ a one-way ANOVA test. This test is appropriate because we are comparing the means of three independent groups (the three hospitals).
Null Hypothesis (H0): The average waiting times are equal across all three hospitals (μ1 = μ2 = μ3).
Alternative Hypothesis (H1): At least one of the average waiting times is different from the others.
The Logic of ANOVA
ANOVA works by partitioning the total variance in the data into different sources of variation. It compares the variance *between* the groups (hospitals in this case) to the variance *within* the groups. If the between-group variance is significantly larger than the within-group variance, it suggests that the group means are different.
ANOVA Calculation (Conceptual Outline)
While a full calculation isn't expected in an exam setting, the core steps are:
- Calculate the Grand Mean: The average waiting time across all hospitals and all sampled cases.
- Calculate the Sum of Squares Between Groups (SSB): Measures the variability between the hospital means and the grand mean.
- Calculate the Sum of Squares Within Groups (SSW): Measures the variability within each hospital's sample.
- Calculate the Degrees of Freedom: dfbetween = k-1 (where k is the number of groups/hospitals), dfwithin = N-k (where N is the total number of observations).
- Calculate the Mean Squares: MSB = SSB/dfbetween, MSW = SSW/dfwithin
- Calculate the F-statistic: F = MSB/MSW
Decision Rule
The F-statistic is compared to a critical F-value obtained from the F-distribution table, using the chosen significance level (α = 0.05) and the degrees of freedom calculated above.
Decision Rule:
- If F > Fcritical, reject the null hypothesis (H0). This indicates a statistically significant difference in average waiting times.
- If F ≤ Fcritical, fail to reject the null hypothesis (H0). This indicates insufficient evidence to conclude a difference in average waiting times.
Applying the Decision to the Scenario
Without the actual calculated F-statistic and the corresponding F-critical value, we must assume a hypothetical outcome. Let's assume, for the sake of illustration, that after performing the ANOVA test, the calculated F-statistic is 4.5. With α = 0.05 and degrees of freedom (dfbetween = 2, dfwithin = 27), the F-critical value is approximately 3.35 (this value would be obtained from an F-distribution table).
Since 4.5 > 3.35, we would reject the null hypothesis.
Conclusion
Based on this hypothetical outcome, there is evidence at the 0.05 level of significance to conclude that there is a statistically significant difference in the average waiting times in the emergency wings of the three hospitals. The city administration should investigate the reasons for these differences and implement targeted interventions to reduce waiting times in the hospitals with longer averages. This could involve resource reallocation, process improvements, or staff training.
Conclusion
In conclusion, the application of ANOVA provides a robust statistical framework for evaluating differences in average waiting times across the three hospitals. Rejecting the null hypothesis suggests that the city administration's concern about varying waiting times is justified. Further investigation into the specific causes of these differences is crucial for developing effective strategies to improve emergency care delivery and enhance patient outcomes. A data-driven approach, like this, is essential for optimizing public healthcare resources.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.