To construct a correct two-way frequency table for the given data, we need to determine the relative frequencies of each group. Here is the step-by-step solution:
1. Given Frequencies:
- Grades 9-10: Playing Sports: 11, Dancing: 3, Watching Movies/TV: 6
- Grades 11-12: Playing Sports: 5, Dancing: 16, Watching Movies/TV: 9
2. Calculate Row Totals:
- Total students in Grades 9-10: [tex]\(11 + 3 + 6 = 20\)[/tex]
- Total students in Grades 11-12: [tex]\(5 + 16 + 9 = 30\)[/tex]
3. Calculate Column Totals:
- Total students who prefer Playing Sports: [tex]\(11 + 5 = 16\)[/tex]
- Total students who prefer Dancing: [tex]\(3 + 16 = 19\)[/tex]
- Total students who prefer Watching Movies/TV: [tex]\(6 + 9 = 15\)[/tex]
4. Calculate Overall Total:
- Total students surveyed: [tex]\(20 + 30 = 50\)[/tex]
5. Calculate Relative Frequencies:
- Grades 9-10:
- Playing Sports: [tex]\( \frac{11}{50} = 0.22 \)[/tex]
- Dancing: [tex]\( \frac{3}{50} = 0.06 \)[/tex]
- Watching Movies/TV: [tex]\( \frac{6}{50} = 0.12 \)[/tex]
- Row total: [tex]\( \frac{20}{50} = 0.4 \)[/tex]
- Grades 11-12:
- Playing Sports: [tex]\( \frac{5}{50} = 0.10 \)[/tex]
- Dancing: [tex]\( \frac{16}{50} = 0.32 \)[/tex]
- Watching Movies/TV: [tex]\( \frac{9}{50} = 0.18 \)[/tex]
- Row total: [tex]\( \frac{30}{50} = 0.6 \)[/tex]
6. Column Totals:
- Playing Sports: [tex]\( \frac{16}{50} = 0.32 \)[/tex]
- Dancing: [tex]\( \frac{19}{50} = 0.38 \)[/tex]
- Watching Movies/TV: [tex]\( \frac{15}{50} = 0.30 \)[/tex]
- Row total: 1 (since all relative frequencies sum up to 1)
7. Construct the Two-Way Frequency Table:
```
\begin{tabular}{|l|l|l|l|l|}
\hline \multicolumn{5}{|c|}{ High School Students Leisure Time Activity Preferences } \\
\hline & Playing Sports & Dancing & Watching Movies/TV & Row totals \\
\hline Grades 9-10 & 0.22 & 0.06 & 0.12 & 0.40 \\
\hline Grades 11-12 & 0.10 & 0.32 & 0.18 & 0.60 \\
\hline Column totals & 0.32 & 0.38 & 0.30 & 1 \\
\hline
\end{tabular}
```
From this table, we can see the relative frequency of students’ preferences across different activities and grade levels. The corrected row total for Grades 9-10 is 0.40, not 0.12. This is important for ensuring the consistency and correctness of the table.
