To determine the correct two-way frequency table with the marginal frequencies, we will go through the given data step by step.
Given:
- For 9th grade: 2 girls are 15 years old, 0 girls are 16 years old.
- For 10th grade: 10 girls are 16 years old.
We need to fill in the missing value for the 10th grade, 15 years old and calculate the marginal frequencies (totals).
### Step-by-step Calculation:
Step 1: Fill in the missing value for 10th grade, 15 years old.
- Given that the total number of girls is 17 and the sum of known values is [tex]\(2\)[/tex] (from 9th grade, 15 years old) + [tex]\(0\)[/tex] (from 9th grade, 16 years old) + [tex]\(10\)[/tex] (from 10th grade, 16 years old) = [tex]\(12\)[/tex].
- Therefore, the number of 15-year-old girls in 10th grade = [tex]\(17 - 12 = 5\)[/tex].
Now we have:
- 9th grade: 15 years old = 2, 16 years old = 0.
- 10th grade: 15 years old = 5, 16 years old = 10.
Step 2: Calculate the row and column totals (marginal frequencies).
- Row totals (grades):
- [tex]\(9\)[/tex]th grade total = [tex]\(2\)[/tex] (15 years old) + [tex]\(0\)[/tex] (16 years old) = [tex]\(2\)[/tex].
- [tex]\(10\)[/tex]th grade total = [tex]\(5\)[/tex] (15 years old) + [tex]\(10\)[/tex] (16 years old) = [tex]\(15\)[/tex].
- Column totals (ages):
- Total 15 years old = [tex]\(2\)[/tex] (9th grade) + [tex]\(5\)[/tex] (10th grade) = [tex]\(7\)[/tex].
- Total 16 years old = [tex]\(0\)[/tex] (9th grade) + [tex]\(10\)[/tex] (10th grade) = [tex]\(10\)[/tex].
- Grand total = [tex]\(9\)[/tex]th grade total + [tex]\(10\)[/tex]th grade total = [tex]\(2\)[/tex] + [tex]\(15\)[/tex] = [tex]\(17\)[/tex].
Step 3: Construct the two-way frequency table.
Given our calculations, the table should look like this:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & 15 \text{ years old} & 16 \text{ years old} & \text{Total} \\
\hline 9th grade & 2 & 0 & 2 \\
\hline 10th grade & 5 & 10 & 15 \\
\hline \text{Total} & 7 & 10 & 17 \\
\hline
\end{tabular}
\][/tex]
### Conclusion:
The correct two-way frequency table is Option A:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & 15 \text{ years old} & 16 \text{ years old} & \text{Total} \\
\hline 9th grade & 2 & 0 & 2 \\
\hline 10th grade & 5 & 10 & 15 \\
\hline \text{Total} & 7 & 10 & 17 \\
\hline
\end{tabular}
\][/tex]
