Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
