Answer :
To determine which two-way frequency table correctly shows the marginal frequencies for the given data, we need to consider the totals for each category. First, let’s summarize the given information:
- For males:
- Over 6: 13
- Under 6: 85
- For females:
- Over 6: 4
- Under 6: 98
Next, let's calculate the marginal frequencies:
1. Total Over 6:
- Sum of males over 6 and females over 6:
[tex]\[ 13 + 4 = 17 \][/tex]
2. Total Under 6:
- Sum of males under 6 and females under 6:
[tex]\[ 85 + 98 = 183 \][/tex]
3. Total Males:
- Sum of males over 6 and under 6:
[tex]\[ 13 + 85 = 98 \][/tex]
4. Total Females:
- Sum of females over 6 and under 6:
[tex]\[ 4 + 98 = 102 \][/tex]
5. Grand Total:
- Sum of all entries:
[tex]\[ 17 + 183 = 200 \][/tex]
Now let’s look at the given options and determine which one matches our calculated results:
### Option A
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 98 & 102 \\
\hline
Total & 17 & 183 & 200 \\
\hline
\end{tabular}
- This option correctly sums the data and marginal frequencies. Total values match our calculations.
### Option B
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 196 & 200 \\
\hline
Total & 17 & 281 & 298 \\
\hline
\end{tabular}
- Clearly incorrect, as the figures do not align with our calculations.
### Option C
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 115 & 119 \\
\hline
Total & 17 & 200 & 217 \\
\hline
\end{tabular}
- This option is also incorrect given the sums do not match our calculated totals.
### Option D
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 38 & 42 \\
\hline
Total & 17 & 123 & 140 \\
\hline
\end{tabular}
- This option once again doesn’t align with our derived totals.
Hence, the correct two-way frequency table is:
### Option A
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 98 & 102 \\
\hline
Total & 17 & 183 & 200 \\
\hline
\end{tabular}
This table accurately reflects the marginal frequencies obtained from our given data.
- For males:
- Over 6: 13
- Under 6: 85
- For females:
- Over 6: 4
- Under 6: 98
Next, let's calculate the marginal frequencies:
1. Total Over 6:
- Sum of males over 6 and females over 6:
[tex]\[ 13 + 4 = 17 \][/tex]
2. Total Under 6:
- Sum of males under 6 and females under 6:
[tex]\[ 85 + 98 = 183 \][/tex]
3. Total Males:
- Sum of males over 6 and under 6:
[tex]\[ 13 + 85 = 98 \][/tex]
4. Total Females:
- Sum of females over 6 and under 6:
[tex]\[ 4 + 98 = 102 \][/tex]
5. Grand Total:
- Sum of all entries:
[tex]\[ 17 + 183 = 200 \][/tex]
Now let’s look at the given options and determine which one matches our calculated results:
### Option A
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 98 & 102 \\
\hline
Total & 17 & 183 & 200 \\
\hline
\end{tabular}
- This option correctly sums the data and marginal frequencies. Total values match our calculations.
### Option B
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 196 & 200 \\
\hline
Total & 17 & 281 & 298 \\
\hline
\end{tabular}
- Clearly incorrect, as the figures do not align with our calculations.
### Option C
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 115 & 119 \\
\hline
Total & 17 & 200 & 217 \\
\hline
\end{tabular}
- This option is also incorrect given the sums do not match our calculated totals.
### Option D
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 38 & 42 \\
\hline
Total & 17 & 123 & 140 \\
\hline
\end{tabular}
- This option once again doesn’t align with our derived totals.
Hence, the correct two-way frequency table is:
### Option A
\begin{tabular}{|c|c|c|c|}
\hline
& Over 6 & Under 6 & Total \\
\hline
Male & 13 & 85 & 98 \\
\hline
Female & 4 & 98 & 102 \\
\hline
Total & 17 & 183 & 200 \\
\hline
\end{tabular}
This table accurately reflects the marginal frequencies obtained from our given data.
