Question

the two - way table shows the results of a recent study on the effectiveness of the flu vaccine. let n be the event that a person tested negative for the flu, and let v be the event that the person was vaccinated.

	pos.	neg.	total
not vaccinated	485	600	1,085
total	950	1,371	2,321

answer the questions to determine if events n and v independent. round your answers to the nearest hundredth.
$p(n|v) = \square$
$p(n) = \square$
are events n and v independent events? yes or no? \square

Explanation:

Step1: Calculate \( P(N|V) \)

The formula for conditional probability is \( P(N|V) = \frac{P(N \cap V)}{P(V)} \). From the table, the number of people who are vaccinated and tested negative (\( N \cap V \)) is 771, and the total number of vaccinated people (\( V \)) is 1236. So \( P(N|V) = \frac{771}{1236} \approx 0.62 \) (rounded to the nearest hundredth).

Step2: Calculate \( P(N) \)

The total number of people who tested negative (\( N \)) is 1371, and the total number of people in the study is 2321. So \( P(N) = \frac{1371}{2321} \approx 0.59 \) (rounded to the nearest hundredth).

Step3: Determine independence

Two events \( N \) and \( V \) are independent if \( P(N|V) = P(N) \). Since \( 0.62
eq 0.59 \), the events are not independent.

Answer:

\( P(N|V) \approx \boxed{0.62} \)
\( P(N) \approx \boxed{0.59} \)
Are events \( N \) and \( V \) independent? \(\boxed{\text{No}}\)