Sure, let's walk through the detailed solution to determine the two-way frequency table with the correct marginal frequencies.
First, let's fill in the table and use the given data:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline & 15 years old & 16 years old \\
\hline 9th grade & 2 & 0 \\
\hline 10th grade & 0 & 10 \\
\hline
\end{tabular}
\][/tex]
Next, we'll calculate the marginal frequencies. Marginal frequencies are the totals for each row and column in the table.
1. Total number of 9th graders:
- For 15 years old in 9th grade: [tex]\(2\)[/tex]
- For 16 years old in 9th grade: [tex]\(0\)[/tex]
- Total (for 9th grade): [tex]\(2 + 0 = 2\)[/tex]
2. Total number of 10th graders:
- For 15 years old in 10th grade: [tex]\(0\)[/tex]
- For 16 years old in 10th grade: [tex]\(10\)[/tex]
- Total (for 10th grade): [tex]\(0 + 10 = 10\)[/tex]
3. Total number of 15-year-olds:
- In 9th grade: [tex]\(2\)[/tex]
- In 10th grade: [tex]\(0\)[/tex]
- Total (for 15 years old): [tex]\(2 + 0 = 2\)[/tex]
4. Total number of 16-year-olds:
- In 9th grade: [tex]\(0\)[/tex]
- In 10th grade: [tex]\(10\)[/tex]
- Total (for 16 years old): [tex]\(0 + 10 = 10\)[/tex]
5. Grand total (Total number of girls):
- We add up all the individual entries in the table: [tex]\(2 + 0 + 0 + 10 = 12\)[/tex]
Now, we'll create the two-way frequency table with the marginal frequencies included:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & 15 years old & 16 years old & Total \\
\hline 9th grade & 2 & 0 & 2 \\
\hline 10th grade & 0 & 10 & 10 \\
\hline Total & 2 & 10 & 12 \\
\hline
\end{tabular}
\][/tex]
So, the two-way frequency table that correctly shows the marginal frequencies is as follows:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline & 15 years old & 16 years old & Total \\
\hline 9th grade & 2 & 0 & 2 \\
\hline 10th grade & 0 & 10 & 10 \\
\hline Total & 2 & 10 & 12 \\
\hline
\end{tabular}
\][/tex]
