To answer the question regarding which two-way frequency table correctly shows the marginal frequencies, let's first organize the information given in the initial table.
The table shows the distribution of ages and grades among some of the 17 girls on the club soccer team:
| | 15 years old | 16 years old |
|-------|--------------|--------------|
| 9th grade | 2 | 0 |
| 10th grade| | 10 |
We need to correctly display this information in a two-way frequency table, including the marginal frequencies, which are the totals for each row and each column.
Let's break this down:
1. Total 9th grade students:
- [tex]\(2 \)[/tex] (15 years old) [tex]\(+ 0\)[/tex] (16 years old) = [tex]\(2\)[/tex] students.
2. Total 10th grade students:
- We are missing the number of 15-year-old 10th graders. Since the total number of girls is 17 and we have accounted for 2 of them (both 9th grade), there must be [tex]\(17 - 2 = 15\)[/tex] girls in 10th grade.
- Since there are 10 girls in the 16 years old 10th grade category, that leaves [tex]\(15 - 10 = 5\)[/tex] girls who must be in the 15 years old 10th grade category.
3. Total 15 years old students:
- [tex]\(2\)[/tex] (9th grade) [tex]\(+ 5\)[/tex] (10th grade) = [tex]\(7\)[/tex] students.
4. Total 16 years old students:
- [tex]\(0\)[/tex] (9th grade) [tex]\(+ 10\)[/tex] (10th grade) = [tex]\(10\)[/tex] students.
5. Grand Total:
- Each marginal total column and row should add up to [tex]\(17\)[/tex].
Thus, the correct two-way frequency table with marginal frequencies should be as follows:
| | \textbf{15 years old} | \textbf{16 years old} | \textbf{Total} |
|--------------|-----------------------|-----------------------|----------------|
| \textbf{9th grade} | 2 | 0 | 2 |
| \textbf{10th grade} | 5 | 10 | 15 |
| \textbf{Total} | 7 | 10 | 17 |
This matches option A:
A.
\begin{tabular}{|c|c|c|c|}
\hline & \begin{tabular}{c}
15 years \\
old
\end{tabular} & \begin{tabular}{c}
16 years \\
old
\end{tabular} & Total \\
\hline 9th grade & 2 & 0 & 2 \\
\hline 10th grade & 5 & 10 & 15 \\
\hline Total & 7 & 10 & 17 \\
\hline
\end{tabular}
Therefore, the correct two-way frequency table that shows the marginal frequencies is option A.
