Answer :
To determine the relative frequencies from the given frequency table, let's follow a step-by-step process. The table summarizes the counts of occurrences of two categories, [tex]\( U \)[/tex] and [tex]\( V \)[/tex], within two groups, [tex]\( S \)[/tex] and [tex]\( T \)[/tex], and their totals.
### Frequency Table
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & U & V & Total \\ \hline S & 5 & 8 & 13 \\ \hline T & 4 & 2 & 6 \\ \hline Total & 9 & 10 & 19 \\ \hline \end{tabular} \][/tex]
We need to convert these frequencies into relative frequencies (as percentages). A relative frequency is computed by dividing each cell's frequency by the overall total and then multiplying by 100 to convert it to a percentage.
1. Calculate relative frequencies:
Calculate the total number of observations:
[tex]\[ \text{Total observations} = 19 \][/tex]
The relative frequency [tex]\( R \)[/tex] for each cell can be computed using:
[tex]\[ R = \left( \frac{\text{Frequency of the cell}}{19} \right) \times 100 \][/tex]
2. Compute relative frequencies for each cell:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & U & V & Total \\ \hline S & \left( \frac{5}{19} \right) \times 100 & \left( \frac{8}{19} \right) \times 100 & \left( \frac{13}{19} \right) \times 100 \\ \hline T & \left( \frac{4}{19} \right) \times 100 & \left( \frac{2}{19} \right) \times 100 & \left( \frac{6}{19} \right) \times 100 \\ \hline Total & \left( \frac{9}{19} \right) \times 100 & \left( \frac{10}{19} \right) \times 100 \\ \hline \end{tabular} \][/tex]
3. Calculate and simplify each cell into percentages:
- For [tex]\( S \)[/tex]:
[tex]\[ \begin{align*} R_{S,U} & = \left( \frac{5}{19} \right) \times 100 \approx 26.32\% \\ R_{S,V} & = \left( \frac{8}{19} \right) \times 100 \approx 42.11\% \\ R_{S, \text{Total}} & = \left( \frac{13}{19} \right) \times 100 \approx 68.42\% \end{align*} \][/tex]
- For [tex]\( T \)[/tex]:
[tex]\[ \begin{align*} R_{T,U} & = \left( \frac{4}{19} \right) \times 100 \approx 21.05\% \\ R_{T,V} & = \left( \frac{2}{19} \right) \times 100 \approx 10.53\% \\ R_{T, \text{Total}} & = \left( \frac{6}{19} \right) \times 100 \approx 31.58\% \end{align*} \][/tex]
- For [tex]\( \text{Total} \)[/tex]:
[tex]\[ \begin{align*} R_{\text{Total},U} & = \left( \frac{9}{19} \right) \times 100 \approx 47.37\% \\ R_{\text{Total},V} & = \left( \frac{10}{19} \right) \times 100 \approx 52.63\% \end{align*} \][/tex]
4. Identify [tex]\( k \)[/tex] from the relative frequency table:
Specifically, [tex]\( k \)[/tex] represents the relative frequency percentage of the value associated with [tex]\( V \)[/tex] in group [tex]\( T \)[/tex]:
[tex]\[ R_{T,V} = \left( \frac{2}{19} \right) \times 100 \approx 10.53\% \][/tex]
When rounded to the nearest percent:
[tex]\[ R_{T,V} \approx 11\% \][/tex]
5. Conclusion:
After rounding, the value of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{11\%} \][/tex]
Thus, the correct value for [tex]\( k \)[/tex] in the relative frequency table is [tex]\( 11\% \)[/tex].
### Frequency Table
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & U & V & Total \\ \hline S & 5 & 8 & 13 \\ \hline T & 4 & 2 & 6 \\ \hline Total & 9 & 10 & 19 \\ \hline \end{tabular} \][/tex]
We need to convert these frequencies into relative frequencies (as percentages). A relative frequency is computed by dividing each cell's frequency by the overall total and then multiplying by 100 to convert it to a percentage.
1. Calculate relative frequencies:
Calculate the total number of observations:
[tex]\[ \text{Total observations} = 19 \][/tex]
The relative frequency [tex]\( R \)[/tex] for each cell can be computed using:
[tex]\[ R = \left( \frac{\text{Frequency of the cell}}{19} \right) \times 100 \][/tex]
2. Compute relative frequencies for each cell:
[tex]\[ \begin{tabular}{|c|c|c|c|} \hline & U & V & Total \\ \hline S & \left( \frac{5}{19} \right) \times 100 & \left( \frac{8}{19} \right) \times 100 & \left( \frac{13}{19} \right) \times 100 \\ \hline T & \left( \frac{4}{19} \right) \times 100 & \left( \frac{2}{19} \right) \times 100 & \left( \frac{6}{19} \right) \times 100 \\ \hline Total & \left( \frac{9}{19} \right) \times 100 & \left( \frac{10}{19} \right) \times 100 \\ \hline \end{tabular} \][/tex]
3. Calculate and simplify each cell into percentages:
- For [tex]\( S \)[/tex]:
[tex]\[ \begin{align*} R_{S,U} & = \left( \frac{5}{19} \right) \times 100 \approx 26.32\% \\ R_{S,V} & = \left( \frac{8}{19} \right) \times 100 \approx 42.11\% \\ R_{S, \text{Total}} & = \left( \frac{13}{19} \right) \times 100 \approx 68.42\% \end{align*} \][/tex]
- For [tex]\( T \)[/tex]:
[tex]\[ \begin{align*} R_{T,U} & = \left( \frac{4}{19} \right) \times 100 \approx 21.05\% \\ R_{T,V} & = \left( \frac{2}{19} \right) \times 100 \approx 10.53\% \\ R_{T, \text{Total}} & = \left( \frac{6}{19} \right) \times 100 \approx 31.58\% \end{align*} \][/tex]
- For [tex]\( \text{Total} \)[/tex]:
[tex]\[ \begin{align*} R_{\text{Total},U} & = \left( \frac{9}{19} \right) \times 100 \approx 47.37\% \\ R_{\text{Total},V} & = \left( \frac{10}{19} \right) \times 100 \approx 52.63\% \end{align*} \][/tex]
4. Identify [tex]\( k \)[/tex] from the relative frequency table:
Specifically, [tex]\( k \)[/tex] represents the relative frequency percentage of the value associated with [tex]\( V \)[/tex] in group [tex]\( T \)[/tex]:
[tex]\[ R_{T,V} = \left( \frac{2}{19} \right) \times 100 \approx 10.53\% \][/tex]
When rounded to the nearest percent:
[tex]\[ R_{T,V} \approx 11\% \][/tex]
5. Conclusion:
After rounding, the value of [tex]\( k \)[/tex] is:
[tex]\[ \boxed{11\%} \][/tex]
Thus, the correct value for [tex]\( k \)[/tex] in the relative frequency table is [tex]\( 11\% \)[/tex].
