- The conditional probability is \(P(N|V) = \frac{771}{1236} \approx \boxed{0.62}\).
- The marginal probability is \(P(N) = \frac{1371}{2321} \approx \boxed{0.59}\).
- Since \(P(N|V)
eq P(N)\), the events are not independent: \(\boxed{\text{No}}\).
Explanation
- Data and Independence Analysis
To determine whether the events \(N\) (testing negative) and \(V\) (being vaccinated) are independent, we must compare the conditional probability \(P(N|V)\) with the marginal probability \(P(N)\).
From the provided study data, we have the following counts:
- Total number of people = \(2,321\)
- Total number of vaccinated people (\(V\)) = \(1,236\)
- Total number of people who tested negative (\(N\)) = \(1,371\)
- Number of people who were both vaccinated and tested negative (\(V \cap N\)) = \(771\)
Events are independent if and only if \(P(N|V) = P(N)\). Otherwise, they are dependent.
- Calculate the conditional probability \(P(N|V)\)
The conditional probability \(P(N|V)\) represents the probability that a person tests negative given that they are vaccinated. This is calculated as the ratio of people who are both vaccinated and tested negative to the total number of vaccinated people:
\[P(N|V) = \frac{\text{Number of } (V \cap N)}{\text{Number of } V} = \frac{771}{1,236} \approx 0.62378...\]
Rounding to the nearest hundredth, we get \(P(N|V) \approx 0.62\).
- Calculate the marginal probability \(P(N)\)
The marginal probability \(P(N)\) represents the overall probability that a person tested negative, regardless of their vaccination status. This is the ratio of the total number of people who tested negative to the total number of participants in the study:
\[P(N) = \frac{\text{Total number of } N}{\text{Total number of people}} = \frac{1,371}{2,321} \approx 0.59069...\]
Rounding to the nearest hundredth, we get \(P(N) \approx 0.59\).
- Conclusion on independence
We have \(P(N|V) \approx 0.62\) and \(P(N) \approx 0.59\). Since these two probabilities are not equal (\(0.62
eq 0.59\)), the event of testing negative is dependent on whether a person was vaccinated. Therefore, the events are not independent.
Examples
In public health studies, checking for independence is crucial. For example, if testing negative were independent of vaccination, it would suggest the vaccine had no statistical relationship with the test outcome. Since they are dependent here, public health officials can further analyze the data to determine if the vaccine provides a protective effect against a positive flu test. This methodology serves as a foundation for evaluating treatment efficacy in clinical trials.