Answer :
Let's analyze the given two-way table and the data provided.
The table summarizes the findings of the free virus detection program used on 500 computers:
[tex]\[ \begin{tabular}{|c|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Virus Reported & Virus not Reported \\ \hline Infected & 28 & 12 \\ \hline Not Infected & 94 & 366 \\ \hline \end{tabular} \][/tex]
From the table, we see:
- `Infected and virus reported` = 28 computers
- `Infected and virus not reported` = 12 computers
- `Not infected and virus reported` = 94 computers
- `Not infected and virus not reported` = 366 computers
To determine the probability of a false positive, we focus on the computers that are not infected:
- The total number of not infected computers = 94 (reported) + 366 (not reported) = 460.
The probability of a false positive is the proportion of not infected computers that were incorrectly reported as infected:
[tex]\[ \frac{\text{Number of not infected computers reported as infected}}{\text{Total number of not infected computers}} = \frac{94}{460} \][/tex]
To convert this probability to a percentage:
[tex]\[ \frac{94}{460} \times 100 = 20.434782608695652 \][/tex]
Therefore, the probability of a false positive is approximately [tex]\(20.43\%\)[/tex].
Given this result, let's evaluate the provided options:
A. The magazine's review suggests Nate should trust the program's report because the probability that the scan result is a false positive is only [tex]$7.11 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$7.11\%$[/tex].
B. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$22.95 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$22.95\%$[/tex].
C. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$92.89 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$92.89\%$[/tex].
D. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$77.05 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$77.05\%. None of the answers in the options are matching the calculated result. Thus, we should conclude that none of the provided answers correctly reflect the actual false positive probability of approximately $[/tex]20.43\%$. However, if we assume the choice provided is to encourage switching programs, option B is the closest to the true value though not exact. Nate's decision should be influenced by the high false positive rate indicated in our calculations.
The table summarizes the findings of the free virus detection program used on 500 computers:
[tex]\[ \begin{tabular}{|c|c|c|} \cline { 2 - 3 } \multicolumn{1}{c|}{} & Virus Reported & Virus not Reported \\ \hline Infected & 28 & 12 \\ \hline Not Infected & 94 & 366 \\ \hline \end{tabular} \][/tex]
From the table, we see:
- `Infected and virus reported` = 28 computers
- `Infected and virus not reported` = 12 computers
- `Not infected and virus reported` = 94 computers
- `Not infected and virus not reported` = 366 computers
To determine the probability of a false positive, we focus on the computers that are not infected:
- The total number of not infected computers = 94 (reported) + 366 (not reported) = 460.
The probability of a false positive is the proportion of not infected computers that were incorrectly reported as infected:
[tex]\[ \frac{\text{Number of not infected computers reported as infected}}{\text{Total number of not infected computers}} = \frac{94}{460} \][/tex]
To convert this probability to a percentage:
[tex]\[ \frac{94}{460} \times 100 = 20.434782608695652 \][/tex]
Therefore, the probability of a false positive is approximately [tex]\(20.43\%\)[/tex].
Given this result, let's evaluate the provided options:
A. The magazine's review suggests Nate should trust the program's report because the probability that the scan result is a false positive is only [tex]$7.11 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$7.11\%$[/tex].
B. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$22.95 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$22.95\%$[/tex].
C. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$92.89 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$92.89\%$[/tex].
D. The magazine's review suggests Nate should use a different detection program because the probability that the scan result is a false positive is [tex]$77.05 \%$[/tex].
- This statement is incorrect because the probability is not [tex]$77.05\%. None of the answers in the options are matching the calculated result. Thus, we should conclude that none of the provided answers correctly reflect the actual false positive probability of approximately $[/tex]20.43\%$. However, if we assume the choice provided is to encourage switching programs, option B is the closest to the true value though not exact. Nate's decision should be influenced by the high false positive rate indicated in our calculations.
