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The two-way table shows the results of a recent study on the effectiveness of the flu vaccine. Let [tex]$N$[/tex] be the event that a person tested negative for the flu, and let [tex]$V$[/tex] be the event that the person was vaccinated.

\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & Pos. & Neg. & Total \\
\hline Vaccinated & 465 & 771 & 1,236 \\
\hline \begin{tabular}{c}
Not \\
Vaccinated
\end{tabular} & 485 & 600 & 1,085 \\
\hline Total & 950 & 1,371 & 2,321 \\
\hline
\end{tabular}

Answer the questions to determine if events [tex]$N$[/tex] and [tex]$V$[/tex] are independent. Round your answers to the nearest hundredth.

[tex]$P( N \mid V )=$[/tex] [tex]$\square$[/tex]
[tex]$P( N )=$[/tex] [tex]$\square$[/tex]

Are events [tex]$N$[/tex] and [tex]$V$[/tex] independent events? Yes or no?



Answer :

GinnyAnswer
Let's work through the problem step by step:

1. Calculate [tex]\( P(N \mid V) \)[/tex]:

- [tex]\( P(N \mid V) \)[/tex] is the probability of a person testing negative for the flu given that they were vaccinated.
- From the table, the number of vaccinated people (event [tex]\( V \)[/tex]) is [tex]\( 1236 \)[/tex].
- The number of vaccinated people who tested negative for the flu is [tex]\( 771 \)[/tex].

Therefore:
[tex]\[ P(N \mid V) = \frac{\text{Number of vaccinated people who tested negative}}{\text{Number of vaccinated people}} = \frac{771}{1236} \][/tex]

Using given results:
[tex]\[ P(N \mid V) = 0.62 \][/tex]

2. Calculate [tex]\( P(N) \)[/tex]:

- [tex]\( P(N) \)[/tex] is the overall probability of a person testing negative for the flu.
- From the table, the total population is [tex]\( 2321 \)[/tex].
- The total number of people who tested negative for the flu is [tex]\( 1371 \)[/tex].

Therefore:
[tex]\[ P(N) = \frac{\text{Number of people who tested negative}}{\text{Total population}} = \frac{1371}{2321} \][/tex]

Using given results:
[tex]\[ P(N) = 0.59 \][/tex]

3. Determine if events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent:

- Events [tex]\( N \)[/tex] and [tex]\( V \)[/tex] are independent if and only if [tex]\( P(N \mid V) = P(N) \)[/tex].

Comparing:
[tex]\[ P(N \mid V) = 0.62 \][/tex]
[tex]\[ P(N) = 0.59 \][/tex]

Since [tex]\( 0.62 \neq 0.59 \)[/tex]:
[tex]\[ \text{Events } N \text{ and } V \text{ are not independent.} \][/tex]

4. Provide the final answers:

[tex]\[ P(N \mid V) = 0.62 \][/tex]
[tex]\[ P(N) = 0.59 \][/tex]
[tex]\[ \text{Are events } N \text{ and } V \text{ independent? No.} \][/tex]

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