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To determine which two-way frequency table correctly shows the marginal frequencies, we need to complete the given table by correctly filling in the missing values and then calculating the totals for each row and column.
Let's start with the initial table and fill in the missing value:
[tex]\[ \begin{array}{|c|c|c|} \hline & \text{15 years old} & \text{16 years old} \\ \hline \text{9th grade} & 2 & 0 \\ \hline \text{10th grade} & x & 10 \\ \hline \end{array} \][/tex]
Where [tex]\( x \)[/tex] represents the missing value.
By examining the possible values presented in option tables A and B, we derive the following full initial table:
[tex]\[ \begin{array}{|c|c|c|} \hline & \text{15 years old} & \text{16 years old} \\ \hline \text{9th grade} & 2 & 0 \\ \hline \text{10th grade} & 0 & 10 \\ \hline \end{array} \][/tex]
Next, let's calculate the total values for each row and each column. We can summarize our calculations as follows:
Step 1: Calculate Row Totals
- 9th grade total: [tex]\( 2 + 0 = 2 \)[/tex]
- 10th grade total: [tex]\( 0 + 10 = 10 \)[/tex]
Step 2: Calculate Column Totals
- 15 years old total: [tex]\( 2 + 0 = 2 \)[/tex]
- 16 years old total: [tex]\( 0 + 10 = 10 \)[/tex]
Step 3: Calculate the Overall Total
- Overall total: [tex]\( 2 + 10 = 12 \)[/tex]
Given these calculations, our complete frequency table with marginal frequencies rounds up as follows:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 0 & 10 & 10 \\ \hline \text{Total} & 2 & 10 & 12 \\ \hline \end{array} \][/tex]
Now let's match these results with the given options:
Option A:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 9 & 10 & 19 \\ \hline \text{Total} & 11 & 10 & 21 \\ \hline \end{array} \][/tex]
This table does not match our calculated totals.
Option B:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 0 & 10 & 10 \\ \hline \text{Total} & 2 & 10 & 12 \\ \hline \end{array} \][/tex]
This table matches our calculated totals perfectly.
Therefore, the correct two-way frequency table that shows the marginal frequencies is from:
Option B.
Let's start with the initial table and fill in the missing value:
[tex]\[ \begin{array}{|c|c|c|} \hline & \text{15 years old} & \text{16 years old} \\ \hline \text{9th grade} & 2 & 0 \\ \hline \text{10th grade} & x & 10 \\ \hline \end{array} \][/tex]
Where [tex]\( x \)[/tex] represents the missing value.
By examining the possible values presented in option tables A and B, we derive the following full initial table:
[tex]\[ \begin{array}{|c|c|c|} \hline & \text{15 years old} & \text{16 years old} \\ \hline \text{9th grade} & 2 & 0 \\ \hline \text{10th grade} & 0 & 10 \\ \hline \end{array} \][/tex]
Next, let's calculate the total values for each row and each column. We can summarize our calculations as follows:
Step 1: Calculate Row Totals
- 9th grade total: [tex]\( 2 + 0 = 2 \)[/tex]
- 10th grade total: [tex]\( 0 + 10 = 10 \)[/tex]
Step 2: Calculate Column Totals
- 15 years old total: [tex]\( 2 + 0 = 2 \)[/tex]
- 16 years old total: [tex]\( 0 + 10 = 10 \)[/tex]
Step 3: Calculate the Overall Total
- Overall total: [tex]\( 2 + 10 = 12 \)[/tex]
Given these calculations, our complete frequency table with marginal frequencies rounds up as follows:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 0 & 10 & 10 \\ \hline \text{Total} & 2 & 10 & 12 \\ \hline \end{array} \][/tex]
Now let's match these results with the given options:
Option A:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 9 & 10 & 19 \\ \hline \text{Total} & 11 & 10 & 21 \\ \hline \end{array} \][/tex]
This table does not match our calculated totals.
Option B:
[tex]\[ \begin{array}{|c|c|c|c|} \hline & \text{15 years old} & \text{16 years old} & \text{Total} \\ \hline \text{9th grade} & 2 & 0 & 2 \\ \hline \text{10th grade} & 0 & 10 & 10 \\ \hline \text{Total} & 2 & 10 & 12 \\ \hline \end{array} \][/tex]
This table matches our calculated totals perfectly.
Therefore, the correct two-way frequency table that shows the marginal frequencies is from:
Option B.
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