To complete the two-way frequency table and find the number of students enrolled in either algebra or physics, we can break down the information provided.
From the sample of 50 students:
- Students taking Algebra (A) = 39
- Students taking Physics (P) = 16
- Students taking both Algebra and Physics (A ∩ P) = 7
Step 1: Calculate the number of students taking only Algebra and only Physics
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Students taking only Algebra (A only) = Students taking Algebra - Students taking both \[ A \text{ only} = 39 - 7 = 32 \]
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Students taking only Physics (P only) = Students taking Physics - Students taking both \[ P \text{ only} = 16 - 7 = 9 \]
Step 2: Fill in the table
Now we can complete the table with the calculated values:
\[ \begin{array}{|c|c|c|} \hline & \textbf{Algebra} & \textbf{Not in Algebra} & \textbf{Total} \ \hline \textbf{Physics} & 7 & 9 & 16 \ \hline \textbf{Not in Physics} & 32 & 2 & 34 \ \hline \textbf{Total} & 39 & 11 & 50 \ \hline \end{array} \]
Breakdown:
- Total students = 50
- Students "not in Algebra" (total - algebra students) = 50 - 39 = 11
- Students "not in Physics" (total - physics students) = 50 - 16 = 34
The numbers we used for students:
- Algebra and Physics (7)
- Only Physics (9)
- Only Algebra (32)
- Not in Algebra and Not in Physics = 50 - (students in Algebra + students in Physics - students in both) = 50 - (39 + 16 - 7) = 50 - 48 = 2.
Step 3: Calculate the number of students enrolled in either Algebra or Physics
To find the number of students enrolled in either algebra or physics, we can use the principle of inclusion-exclusion:
\[ \text{Total in either Algebra or Physics} = \text{Students in Algebra} + \text{Students in Physics} - \text{Students in both} \]
So we have: \[ = 39 + 16 - 7 = 48 \]
Thus, the number of students enrolled in either Algebra or Physics is 48.
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