**Core Concept**
The question requires the student to understand the difference between categorical and numerical data in statistical analysis, specifically in the context of comparing nosocomial infection prevention strategies between hospitals and nursing homes. The correct answer involves using a statistical test suitable for analyzing categorical data, which is essential for evaluating the effectiveness of infection control measures.

**Why the Correct Answer is Right**
The Chi-squared test is the most appropriate statistical method for comparing the distribution of categorical data (in this case, effective vs. ineffective nosocomial infection prevention strategies) between two or more groups (hospitals vs. nursing homes). This non-parametric test assesses the likelihood that the observed differences in categorical data occurred by chance, allowing the Department of Health Services to determine whether the difference between hospitals and nursing homes is statistically significant.

**Why Each Wrong Option is Incorrect**
**Option B:** One-way analysis of variance (ANOVA) is a parametric test used for comparing the means of three or more groups, which is not suitable for categorical data. ANOVA assumes normally distributed data and equal variances between groups, which are not applicable in this scenario.

**Option C:** The paired t-test is used for comparing two related groups, which is not relevant here since we are comparing two distinct groups (hospitals and nursing homes).

**Option D:** Pearson correlation measures the linear relationship between two continuous variables, which is not applicable in this scenario since we are dealing with categorical data.

**Clinical Pearl / High-Yield Fact**
When comparing categorical data between groups, it is essential to use a non-parametric test like the Chi-squared test to avoid making assumptions about the distribution of data. This is crucial in public health and epidemiology, where categorical data are often used to evaluate the effectiveness of interventions.

**✓ Correct Answer: A. Chi-squared test**