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Sagot :
To solve this problem, we need to construct a two-way frequency table with the marginal frequencies for the ages and grades given.
The provided table data is:
- 15 years old and 9th grade: 2
- 15 years old and 10th grade: 6 (not directly given, but inferred from the criteria)
- 16 years old and 9th grade: 0
- 16 years old and 10th grade: 10
Step-by-step, here is how we calculate the marginal frequencies:
1. Calculate the total number of 15 years old girls:
- 15 years old and 9th grade: 2
- 15 years old and 10th grade: 6
- Total 15 years old = 2 + 6 = 8
2. Calculate the total number of 16 years old girls:
- 16 years old and 9th grade: 0
- 16 years old and 10th grade: 10
- Total 16 years old = 0 + 10 = 10
3. Calculate the total number of 9th grade girls:
- 15 years old and 9th grade: 2
- 16 years old and 9th grade: 0
- Total 9th grade = 2
4. Calculate the total number of 10th grade girls:
- 15 years old and 10th grade: 6
- 16 years old and 10th grade: 10
- Total 10th grade = 6 + 10 = 16
5. Calculate the grand total:
- Sum of all girls = 2 (9th grade, 15 years old) + 0 (9th grade, 16 years old) + 6 (10th grade, 15 years old) + 10 (10th grade, 16 years old)
- Total = 2 + 0 + 6 + 10 = 18
With these calculations, our table should look like this:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 8 & 10 & 18 \\ \hline \end{tabular} \][/tex]
Let's match our results to the answer choices provided:
Choice A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 16 & 10 & 26 \\ \hline Total & 18 & 10 & 28 \\ \hline \end{tabular} \][/tex]
Total values do not match our calculations.
Choice B:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 8 & 10 & 18 \\ \hline \end{tabular} \][/tex]
All values match our calculations:
2 (for 9th grade total)
16 (for 10th grade total)
8 (for 15 years old total)
10 (for 16 years old total)
18 (grand total)
Choice C:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 8 & 10 & 18 \\ \hline Total & 10 & 10 & 20 \\ \hline \end{tabular} \][/tex]
Total values do not match our calculations.
Choice D:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 10 & 10 & 18 \\ \hline \end{tabular} \][/tex]
Values mostly match but the 'Total of 15 years old' is incorrect.
Therefore, the correct two-way frequency table correctly showing the marginal frequencies is given in choice B.
The provided table data is:
- 15 years old and 9th grade: 2
- 15 years old and 10th grade: 6 (not directly given, but inferred from the criteria)
- 16 years old and 9th grade: 0
- 16 years old and 10th grade: 10
Step-by-step, here is how we calculate the marginal frequencies:
1. Calculate the total number of 15 years old girls:
- 15 years old and 9th grade: 2
- 15 years old and 10th grade: 6
- Total 15 years old = 2 + 6 = 8
2. Calculate the total number of 16 years old girls:
- 16 years old and 9th grade: 0
- 16 years old and 10th grade: 10
- Total 16 years old = 0 + 10 = 10
3. Calculate the total number of 9th grade girls:
- 15 years old and 9th grade: 2
- 16 years old and 9th grade: 0
- Total 9th grade = 2
4. Calculate the total number of 10th grade girls:
- 15 years old and 10th grade: 6
- 16 years old and 10th grade: 10
- Total 10th grade = 6 + 10 = 16
5. Calculate the grand total:
- Sum of all girls = 2 (9th grade, 15 years old) + 0 (9th grade, 16 years old) + 6 (10th grade, 15 years old) + 10 (10th grade, 16 years old)
- Total = 2 + 0 + 6 + 10 = 18
With these calculations, our table should look like this:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 8 & 10 & 18 \\ \hline \end{tabular} \][/tex]
Let's match our results to the answer choices provided:
Choice A:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 16 & 10 & 26 \\ \hline Total & 18 & 10 & 28 \\ \hline \end{tabular} \][/tex]
Total values do not match our calculations.
Choice B:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 8 & 10 & 18 \\ \hline \end{tabular} \][/tex]
All values match our calculations:
2 (for 9th grade total)
16 (for 10th grade total)
8 (for 15 years old total)
10 (for 16 years old total)
18 (grand total)
Choice C:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 8 & 10 & 18 \\ \hline Total & 10 & 10 & 20 \\ \hline \end{tabular} \][/tex]
Total values do not match our calculations.
Choice D:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & 15 years old & 16 years old & Total \\ \hline 9th grade & 2 & 0 & 2 \\ \hline 10th grade & 6 & 10 & 16 \\ \hline Total & 10 & 10 & 18 \\ \hline \end{tabular} \][/tex]
Values mostly match but the 'Total of 15 years old' is incorrect.
Therefore, the correct two-way frequency table correctly showing the marginal frequencies is given in choice B.
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