To complete the conditional relative frequency table by row, let's first understand the given frequencies and the total for each category. The original frequency table you've provided is:
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & \begin{tabular}{c}
[tex]$\leq 3$[/tex] \\
hrs. [tex]$/$[/tex] wk.
\end{tabular} & \begin{tabular}{c}
[tex]${data-answer}gt;3$[/tex] \\
hrs. [tex]$/$[/tex] wk.
\end{tabular} & Total \\
\hline[tex]${data-answer}lt;1$[/tex] Acre & 180 & 120 & 300 \\
\hline[tex]$\geq 1$[/tex] Acre & 40 & 160 & 200 \\
\hline Total & 220 & 280 & 500 \\
\hline
\end{tabular}
We are asked to find the conditional relative frequencies for each category, represented by letters 'a', 'b', 'c', and 'd', and complete the table. The goal is to determine the conditional relative frequencies for each row, which is the frequency divided by the row total.
1. Compute 'a' and 'b':
- \( a \) represents the relative frequency of farmers with less than 1 acre of land that spend 3 or fewer hours per week:
[tex]\[
a = \frac{180}{300} = 0.60
\][/tex]
- \( b \) represents the relative frequency of farmers with less than 1 acre of land that spend more than 3 hours per week:
[tex]\[
b = \frac{120}{300} = 0.40
\][/tex]
2. Compute 'c' and 'd':
- \( c \) represents the relative frequency of farmers with 1 or more acres of land who spend 3 or fewer hours per week:
[tex]\[
c = \frac{40}{200} = 0.20
\][/tex]
- \( d \) represents the relative frequency of farmers with 1 or more acres of land who spend more than 3 hours per week:
[tex]\[
d = \frac{160}{200} = 0.80
\][/tex]
After placing these values into the row-conditional relative frequency table, we get:
[tex]\[
\begin{array}{l}
a = 0.60 \\
b = 0.40 \\
c = 0.20 \\
d = 0.80
\end{array}
\][/tex]
\begin{tabular}{|c|c|c|c|}
\cline { 2 - 4 } \multicolumn{1}{c|}{} & \begin{tabular}{c}
[tex]$\leq 3$[/tex] \\
hrs./wk.
\end{tabular} & \begin{tabular}{c}
[tex]${data-answer}gt;3$[/tex] \\
hrs./wk.
\end{tabular} & Total \\
\hline[tex]${data-answer}lt;1$[/tex] Acre & 0.60 & 0.40 & 1.0 \\
\hline[tex]$\geq 1$[/tex] Acre & 0.20 & 0.80 & 1.0 \\
\hline Total & 0.44 & 0.56 & 1.0 \\
\hline
\end{tabular}
Thus, the table is now correctly filled with the required conditional relative frequencies for each row.
