To determine if events [tex]\( N \)[/tex] (testing negative for the flu) and [tex]\( V \)[/tex] (being vaccinated) are independent, we need to compare [tex]\( P(N \mid V) \)[/tex] and [tex]\( P(N) \)[/tex]. If these probabilities are equal, then the events are considered independent. Otherwise, they are dependent.
### Step-by-step Solution:
1. Calculate [tex]\( P(N \mid V) \)[/tex]:
- This is the conditional probability of testing negative for the flu given that a person was vaccinated.
- From the table, the number of people who were vaccinated and tested negative is 771.
- The total number of vaccinated people is 1,236.
[tex]\[
P(N \mid V) = \frac{\text{Number of vaccinated people who tested negative}}{\text{Total number of vaccinated people}} = \frac{771}{1236}
\][/tex]
[tex]\[
P(N \mid V) \approx 0.62
\][/tex]
2. Calculate [tex]\( P(N) \)[/tex]:
- This is the probability of testing negative for the flu out of the entire population.
- From the table, the total number of people who tested negative is 1,371.
- The grand total number of people in the study is 2,321.
[tex]\[
P(N) = \frac{\text{Total number of people who tested negative}}{\text{Total number of people in the study}} = \frac{1371}{2321}
\][/tex]
[tex]\[
P(N) \approx 0.59
\][/tex]
3. Determine if events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent:
- Compare [tex]\( P(N \mid V) \)[/tex] and [tex]\( P(N) \)[/tex].
- If [tex]\( P(N \mid V) = P(N) \)[/tex], then the events are independent.
- Here, [tex]\( P(N \mid V) \approx 0.62 \)[/tex] and [tex]\( P(N) \approx 0.59 \)[/tex], which means they are not equal.
Therefore:
[tex]\[
\text{Are \( N \) and \( V \) independent?} \quad \text{No}
\][/tex]
### Summary
[tex]\[
P(N \mid V) \approx 0.62
\][/tex]
[tex]\[
P(N) \approx 0.59
\][/tex]
[tex]\[
\text{Are events \( N \) and \( V \) independent?} \quad \text{No}
\][/tex]
So, the values filled in the boxes would be:
- [tex]\( P(N \mid V) = 0.62 \)[/tex]
- [tex]\( P(N) = 0.59 \)[/tex]
- Are events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] independent? No
