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Sagot :
Let's break the problem down step-by-step and fill out the table accordingly.
1. Total number of students: 1000
2. Prevalence rate of anxiety:
- [tex]$20\%$[/tex] of students have test-anxiety.
3. Detection accuracy of the test:
- The test is [tex]$95\%$[/tex] accurate.
Using this information, let's calculate the different categories:
### 1. Calculate the number of students with anxiety
[tex]\[ \text{Number of students with anxiety} = \text{Total students} \times \text{Anxiety prevalence rate} = 1000 \times 0.20 = 200 \][/tex]
### 2. Calculate the number of students without anxiety
[tex]\[ \text{Number of students without anxiety} = \text{Total students} - \text{Number of students with anxiety} = 1000 - 200 = 800 \][/tex]
### 3. True Positives (Students correctly identified as having anxiety)
[tex]\[ \text{True Positives} = \text{Number of students with anxiety} \times \text{Test accuracy} = 200 \times 0.95 = 190 \][/tex]
### 4. False Negatives (Students incorrectly identified as not having anxiety)
[tex]\[ \text{False Negatives} = \text{Number of students with anxiety} \times (1 - \text{Test accuracy}) = 200 \times (1 - 0.95) = 200 \times 0.05 = 10 \][/tex]
### 5. True Negatives (Students correctly identified as not having anxiety)
[tex]\[ \text{True Negatives} = \text{Number of students without anxiety} \times \text{Test accuracy} = 800 \times 0.95 = 760 \][/tex]
### 6. False Positives (Students incorrectly identified as having anxiety)
[tex]\[ \text{False Positives} = \text{Number of students without anxiety} \times (1 - \text{Test accuracy}) = 800 \times (1 - 0.95) = 800 \times 0.05 = 40 \][/tex]
Now we can fill out the table:
[tex]\[ \begin{array}{|c|c|} \hline \text{Category} & \text{Number of Students} \\ \hline \text{Total number of students} & 1000 \\ \hline \text{Number of students with anxiety} & 200 \\ \hline \text{Number of students without anxiety} & 800 \\ \hline \text{True Positives} & 190 \\ \hline \text{False Negatives} & 10 \\ \hline \text{True Negatives} & 760 \\ \hline \text{False Positives} & 40 \\ \hline \end{array} \][/tex]
Therefore, the table is completed as follows:
[tex]\[ \begin{array}{|c|c|} \hline \text{Category} & \text{Number of Students} \\ \hline \text{Total number of students} & 1000 \\ \hline \text{Number of students with anxiety} & 200 \\ \hline \text{Number of students without anxiety} & 800 \\ \hline \text{True Positives (correctly identified as having anxiety)} & 190 \\ \hline \text{False Negatives (incorrectly identified as not having anxiety)} & 10 \\ \hline \text{True Negatives (correctly identified as not having anxiety)} & 760 \\ \hline \text{False Positives (incorrectly identified as having anxiety)} & 40 \\ \hline \end{array} \][/tex]
This completes the detailed, step-by-step solution for filling out the table.
1. Total number of students: 1000
2. Prevalence rate of anxiety:
- [tex]$20\%$[/tex] of students have test-anxiety.
3. Detection accuracy of the test:
- The test is [tex]$95\%$[/tex] accurate.
Using this information, let's calculate the different categories:
### 1. Calculate the number of students with anxiety
[tex]\[ \text{Number of students with anxiety} = \text{Total students} \times \text{Anxiety prevalence rate} = 1000 \times 0.20 = 200 \][/tex]
### 2. Calculate the number of students without anxiety
[tex]\[ \text{Number of students without anxiety} = \text{Total students} - \text{Number of students with anxiety} = 1000 - 200 = 800 \][/tex]
### 3. True Positives (Students correctly identified as having anxiety)
[tex]\[ \text{True Positives} = \text{Number of students with anxiety} \times \text{Test accuracy} = 200 \times 0.95 = 190 \][/tex]
### 4. False Negatives (Students incorrectly identified as not having anxiety)
[tex]\[ \text{False Negatives} = \text{Number of students with anxiety} \times (1 - \text{Test accuracy}) = 200 \times (1 - 0.95) = 200 \times 0.05 = 10 \][/tex]
### 5. True Negatives (Students correctly identified as not having anxiety)
[tex]\[ \text{True Negatives} = \text{Number of students without anxiety} \times \text{Test accuracy} = 800 \times 0.95 = 760 \][/tex]
### 6. False Positives (Students incorrectly identified as having anxiety)
[tex]\[ \text{False Positives} = \text{Number of students without anxiety} \times (1 - \text{Test accuracy}) = 800 \times (1 - 0.95) = 800 \times 0.05 = 40 \][/tex]
Now we can fill out the table:
[tex]\[ \begin{array}{|c|c|} \hline \text{Category} & \text{Number of Students} \\ \hline \text{Total number of students} & 1000 \\ \hline \text{Number of students with anxiety} & 200 \\ \hline \text{Number of students without anxiety} & 800 \\ \hline \text{True Positives} & 190 \\ \hline \text{False Negatives} & 10 \\ \hline \text{True Negatives} & 760 \\ \hline \text{False Positives} & 40 \\ \hline \end{array} \][/tex]
Therefore, the table is completed as follows:
[tex]\[ \begin{array}{|c|c|} \hline \text{Category} & \text{Number of Students} \\ \hline \text{Total number of students} & 1000 \\ \hline \text{Number of students with anxiety} & 200 \\ \hline \text{Number of students without anxiety} & 800 \\ \hline \text{True Positives (correctly identified as having anxiety)} & 190 \\ \hline \text{False Negatives (incorrectly identified as not having anxiety)} & 10 \\ \hline \text{True Negatives (correctly identified as not having anxiety)} & 760 \\ \hline \text{False Positives (incorrectly identified as having anxiety)} & 40 \\ \hline \end{array} \][/tex]
This completes the detailed, step-by-step solution for filling out the table.
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