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A sample of 245 high school students from grades [tex]$9-10$[/tex] and [tex]$11-12$[/tex] were asked to choose the kind of band to have play at a school dance: rap, rock, or country. [tex]$51\%$[/tex] of the sample is grades [tex]$9-10$[/tex]. From the sample, it is determined that [tex]$16.3\%$[/tex] were students in grades [tex]$9-10$[/tex] who prefer a rap band, and [tex]$22.4\%$[/tex] were students in grades [tex]$9-10$[/tex] who prefer a country band. [tex]$14.3\%$[/tex] were students in grades [tex]$11-12$[/tex] who prefer a country band, while [tex]$10.2\%$[/tex] were students in grades [tex]$11-12$[/tex] who prefer a rock band. The most popular choice was rap, with [tex]$40.8\%$[/tex] of students making it their preferred option.

Which of the following is a correct two-way relative frequency table for the data?

\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{5}{|c|}{ School Dance Band Preference } \\
\hline
& Rap & Rock & Country & Row totals \\
\hline
Grades 9-10 & 16.3 & 12.2 & 22.4 & 51 \\
\hline
Grades 11-12 & 24.5 & 10.2 & 14.3 & 49 \\
\hline
Column totals & 40.8 & 22.4 & 36.7 & 100 \\
\hline
\end{tabular}

\begin{tabular}{|l|c|c|c|l|}
\hline
\multicolumn{5}{|c|}{ School Dance Band Preference } \\
\hline
& Rap & Rock & Country & Row totals \\
\hline
Grades 9-10 & [tex]$16.3\%$[/tex] & [tex]$12.3\%$[/tex] & [tex]$22.4\%$[/tex] & [tex]$51\%$[/tex] \\
\hline
Grades 11-12 & [tex]$24.5\%$[/tex] & [tex]$10.2\%$[/tex] & [tex]$14.3\%$[/tex] & [tex]$49\%$[/tex] \\
\hline
Column totals & [tex]$40.8\%$[/tex] & [tex]$22.5\%$[/tex] & [tex]$36.7\%$[/tex] & [tex]$100\%$[/tex] \\
\hline
\end{tabular}

\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{5}{|c|}{ School Dance Band Preference } \\
\hline
& Rap & Rock & Country & Row totals \\
\hline
Grades 9-10 & 40 & 30 & 55 & 125 \\
\hline
Grades 11-12 & 60 & 25 & 35 & 120 \\
\hline
Column totals & 100 & 55 & 90 & 245 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
Let's break down the given data and construct the two-way relative frequency table step by step.

1. Given Data:
- Total students = 245
- Percentage of students in grades 9-10 = 51%
- Percentage of students in grades 9-10 who prefer rap = 16.3%
- Percentage of students in grades 9-10 who prefer country = 22.4%
- Percentage of students in grades 11-12 who prefer country = 14.3%
- Percentage of students in grades 11-12 who prefer rock = 10.2%
- Overall percentage of students who prefer rap = 40.8%

2. Percentage of students in grades 11-12:
[tex]\[ \text{Percentage of students in grades 11-12} = 100\% - 51\% = 49\% \][/tex]

3. Calculate the percentage of students in grades 11-12 who prefer rap:
[tex]\[ \text{Overall percentage who prefer rap} = 40.8\% \][/tex]
[tex]\[ \text{Percentage of grades 9-10 who prefer rap} = 16.3\% \][/tex]
[tex]\[ \text{Percentage of grades 11-12 who prefer rap} = 40.8\% - 16.3\% = 24.5\% \][/tex]

4. Calculate the percentage of students in grades 9-10 who prefer rock:
[tex]\[ \text{Total percentage of students} = 100\% \][/tex]
[tex]\[ \text{Sum of all preferences for grades 9-10} = 16.3\% + x + 22.4\% \][/tex]
Since grades 9-10 totals to 51%:
[tex]\[ 16.3\% + x + 22.4\% = 51\% \][/tex]
[tex]\[ x = 51\% - 16.3\% - 22.4\% = 12.3\% \][/tex]

5. Calculate the overall percentage of students who prefer rock:
[tex]\[ \text{Sum of preferences for rock} = 12.3\% (grades 9-10) + 10.2\% (grades 11-12) = 22.5\% \][/tex]

6. Calculate the overall percentage of students who prefer country:
[tex]\[ \text{Sum of preferences for country} = 22.4\% (grades 9-10) + 14.3\% (grades 11-12) = 36.7\% \][/tex]

7. Construct the two-way relative frequency table:

\begin{tabular}{|l|c|c|c|l|}
\hline \multicolumn{5}{|c|}{ School Dance Band Preference } \\
\hline & Rap & Rock & Country & Row totals \\
\hline Grades 9-10 & [tex]$16.3 \%$[/tex] & [tex]$12.3 \%$[/tex] & [tex]$22.4 \%$[/tex] & [tex]$51 \%$[/tex] \\
\hline Grades 11-12 & [tex]$24.5 \%$[/tex] & [tex]$10.2 \%$[/tex] & [tex]$14.3 \%$[/tex] & [tex]$49 \%$[/tex] \\
\hline Column totals & [tex]$40.8 \%$[/tex] & [tex]$22.5 \%$[/tex] & [tex]$36.7 \%$[/tex] & [tex]$100 \%$[/tex] \\
\hline
\end{tabular}

Thus, the correct table is the second one:
\begin{tabular}{|l|c|c|c|l|}
\hline \multicolumn{5}{|c|}{ School Dance Band Preference } \\
\hline & Rap & Rock & Country & Row totals \\
\hline Grades 9-10 & [tex]$16.3 \%$[/tex] & [tex]$12.3 \%$[/tex] & [tex]$22.4 \%$[/tex] & [tex]$51 \%$[/tex] \\
\hline Grades 11-12 & [tex]$24.5 \%$[/tex] & [tex]$10.2 \%$[/tex] & [tex]$14.3 \%$[/tex] & [tex]$49 \%$[/tex] \\
\hline Column totals & [tex]$40.8 \%$[/tex] & [tex]$22.5 \%$[/tex] & [tex]$36.7 \%$[/tex] & [tex]$100 \%$[/tex] \\
\hline
\end{tabular}

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