A relative frequency table is made from data in a frequency table.

\begin{table}

\captionsetup{labelformat=empty}

\caption{Frequency Table}

U V Total

S 5 8 13

T 4 2 6

Total 9 10 19

\end{table}

\begin{table}

\captionsetup{labelformat=empty}

\caption{Relative Frequency Table}

U V Total

S 26\% 42\% 68\%

T 21\% $k$ 32\%

Total 47\% 53\% 100\%

\end{table}

What is the value of $k$ in the relative frequency table? Round the answer to the nearest percent.

A11\%
B20\%
C33\%

To find the value of $k$, we look at the frequency table to identify the frequency for the intersection of row T and column V. In the provided frequency table, the count for cell (T, V) is $2$. The grand total of all observations in the table is $19$. The relative frequency represents the proportion of this specific cell count compared to the grand total.

Calculate the relative frequency $k$

The relative frequency is calculated by dividing the specific frequency of the cell by the grand total and then multiplying by $100\%$ to express it as a percentage. The formula is:
\[k = \left( \frac{\text{Frequency of cell (T, V)}}{\text{Grand Total}} \right) \times 100\%\]
Substituting the values:
\[k = \left( \frac{2}{19} \right) \times 100\% \approx 0.10526 \times 100\% = 10.526\%\]

Round to the nearest percent and verify the answer

Now we round the result to the nearest percent. Since $10.526\%$ has a decimal part of $0.526$, which is equal to or greater than $0.5$, we round up to the nearest whole percentage point.
\[10.526\% \approx 11\%\]
Comparing this result with the given choices:

A$20\%$

B$11\%$

C$33\%$

We can clearly see that the correct choice is b. Therefore, $k = 11\%$.

Examples

Relative frequency tables are essential in statistics for interpreting large datasets. For example, if you have data on student preferences for school lunch options (like pizza vs. salad) across different grades, you can convert those raw counts into percentages to easily compare which group prefers which option proportionally, regardless of the different group sizes. This helps teachers and school administrators make informed decisions about nutrition programs.