Creating a Conditional Relative Frequency Table

[tex]\[
\begin{tabular}{|c|c|c|c|}
\cline{2-4}
& \(\leq 3\) hrs/wk & \(\ \textgreater \ 3\) hrs/wk & Total \\
\hline
\ \textless \ 1 Acre & 180 & 120 & 300 \\
\hline
1 Acre & 40 & 160 & 200 \\
\hline
Total & 220 & 280 & 500 \\
\hline
\end{tabular}
\][/tex]

The frequency table represents data gathered about how much time some farmers spend tending to their land each week. Complete the conditional relative frequency table by identifying the values for each letter:

[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
& \(\leq 3\) hrs/wk & \(\ \textgreater \ 3\) hrs/wk & Total \\
\hline
\ \textless \ 1 Acre & \(a\) & \(b\) & 1.0 \\
\hline
1 Acre & \(c\) & \(d\) & 1.0 \\
\hline
Total & 0.44 & 0.56 & 1.0 \\
\hline
\end{tabular}
\][/tex]

[tex]\(a = \square \)[/tex]
[tex]\(b = \square \)[/tex]
[tex]\(c = \square \)[/tex]
[tex]\(d = \square \)[/tex]



Answer :

Sure, let's go through the steps to complete the conditional relative frequency table based on the provided data.

Given the frequency table of hours worked per week on less than 1 acre and on 1 acre:

[tex]\[ \begin{tabular}{|c|c|c|c|} \cline{2-4} & \text{hrs/wk} & \text{hrs/wk} & \text{Total} \\ \hline <1 \text{ Acre} & 180 & 120 & 300 \\ \hline 1 \text{ Acre} & 40 & 160 & 200 \\ \hline \text{Total} & 220 & 280 & 500 \\ \hline \end{tabular} \][/tex]

To create the conditional relative frequency table by row, we need to compute the relative frequencies for each row.

### Step 1: Calculate the conditional relative frequencies for "<1 Acre"

To find [tex]\( a \)[/tex] (hrs/wk for < 1 Acre):
[tex]\[ a = \frac{180}{300} = 0.6 \][/tex]

To find [tex]\( b \)[/tex] (hrs/wk for < 1 Acre):
[tex]\[ b = \frac{120}{300} = 0.4 \][/tex]

### Step 2: Calculate the conditional relative frequencies for "1 Acre"

To find [tex]\( c \)[/tex] (hrs/wk for 1 Acre):
[tex]\[ c = \frac{40}{200} = 0.2 \][/tex]

To find [tex]\( d \)[/tex] (hrs/wk for 1 Acre):
[tex]\[ d = \frac{160}{200} = 0.8 \][/tex]

### Step 3: Fill in the conditional relative frequency table

Now we can fill in the values into the conditional relative frequency table.

[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & \text{0-3 hrs/wk} & \text{>3 hrs/wk} & \text{Total} \\ \hline <1 \text{ Acre} & 0.6 & 0.4 & 1.0 \\ \hline 1 \text{ Acre} & 0.2 & 0.8 & 1.0 \\ \hline \text{Total} & 0.44 & 0.56 & 1.0 \\ \hline \end{tabular} \][/tex]

Here, we've calculated the conditional relative frequencies for farmers working less than one acre of land and for farmers working one acre of land. The totals across each row sum to 1.0, which confirms that the values are correctly calculated as conditional relative frequencies.