To tackle this question, we're going to fill in the missing values and calculate the marginal frequencies. A marginal frequency is the sum total of frequencies in any row or column of the table.
We start with the given data:
- Males, 6' or over: 12
- Males, under 6': 86
- Females, 6' or over: 3
- Total adults: 200
First, we calculate how many females are under 6'. We get this by subtracting the already known counts from the total:
[tex]\[ \text{Females under 6'} = 200 - (12 + 86 + 3) = 200 - 101 = 99 \][/tex]
With this information, our table now becomes:
[tex]\[
\begin{tabular}{|c|c|c|}
\hline
& $6^{\prime}$ or over & Under 6' \\
\hline
Male & 12 & 86 \\
\hline
Female & 3 & 99 \\
\hline
\end{tabular}
\][/tex]
Next, we add the marginal frequencies, which are the row totals and column totals.
1. Row Totals:
- Total males:
[tex]\[ 12 + 86 = 98 \][/tex]
- Total females:
[tex]\[ 3 + 99 = 102 \][/tex]
2. Column Totals:
- Total 6' or over:
[tex]\[ 12 + 3 = 15 \][/tex]
- Total under 6':
[tex]\[ 86 + 99 = 185 \][/tex]
These pieces of information allow us to complete the table, which now looks like:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& $6^{\prime}$ or over & Under 6' & Total \\
\hline
Male & 12 & 86 & 98 \\
\hline
Female & 3 & 99 & 102 \\
\hline
Total & 15 & 185 & 200 \\
\hline
\end{tabular}
\][/tex]
So, the two-way frequency table with the correct marginal frequencies is:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& $6^{\prime}$ or over & Under 6' & Total \\
\hline
Male & 12 & 86 & 98 \\
\hline
Female & 3 & 99 & 102 \\
\hline
Total & 15 & 185 & 200 \\
\hline
\end{tabular}
\][/tex]
