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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 )= \square[/tex]
[tex]P( N )= \square[/tex]

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



Answer :

GinnyAnswer
To determine if events [tex]\(N\)[/tex] (testing negative for the flu) and [tex]\(V\)[/tex] (being vaccinated) are independent, we need to compute the following probabilities and compare them:

1. [tex]\(P(N \mid V)\)[/tex]: The probability of testing negative given that a person was vaccinated.
2. [tex]\(P(N)\)[/tex]: The probability of testing negative in the overall population.

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

This is the probability of a person testing negative given that they were vaccinated. Using the table provided:

- Number of vaccinated individuals who tested negative ([tex]\( N \)[/tex] given [tex]\( V \)[/tex]): 771
- Total number of vaccinated individuals ([tex]\( V \)[/tex]): 1236

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

When rounded to the nearest hundredth:

[tex]\[ P(N \mid V) \approx 0.62 \][/tex]

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

This is the probability of a person testing negative in the overall population. Using the table provided:

- Total number of individuals who tested negative ([tex]\( N \)[/tex]): 1371
- Total population: 2321

[tex]\[ P(N) = \frac{\text{Total number of individuals who tested negative}}{\text{Total population}} = \frac{1371}{2321} \][/tex]

When rounded to the nearest hundredth:

[tex]\[ P(N) \approx 0.59 \][/tex]

Step 3: Compare [tex]\( P(N \mid V) \)[/tex] and [tex]\( P(N) \)[/tex] to check for independence

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

Given that:

[tex]\[ P(N \mid V) \approx 0.62 \][/tex]
[tex]\[ P(N) \approx 0.59 \][/tex]

Since [tex]\( 0.62 \neq 0.59 \)[/tex], the events [tex]\(N\)[/tex] and [tex]\(V\)[/tex] are not independent.

Answers:

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

Are events [tex]\(N\)[/tex] and [tex]\(V\)[/tex] independent events?

No.

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