To analyze what term best describes the entry of 0.31 in this table, we need to understand different statistical terms used in two-way tables: joint frequency, joint relative frequency, marginal frequency, and marginal relative frequency.
Here's a step-by-step explanation of each term:
1. Joint Frequency:
- This refers to the count of occurrences where two categorical variables intersect. For example, the number of 9th grade students who prefer dancing.
2. Joint Relative Frequency:
- This is obtained by dividing the joint frequency by the total number of observations. In simpler terms, it represents the proportion of the total number of observations where two categorical variables intersect.
- In our table, the value 0.31 corresponds to the 'Dancing' column for '10th grade students'. This means 0.31 represents the proportion of the total number of 10th grade students who prefer dancing. Since it's a relative frequency specific to an intersection of '10th grade students' and 'Dancing', it is deemed a joint relative frequency.
3. Marginal Frequency:
- Marginal frequency refers to the sum of joint frequencies in a row or column in a two-way table. For example, the total number of students who prefer sports.
4. Marginal Relative Frequency:
- This is obtained by dividing the marginal frequency by the total number of observations. It gives a proportion relative to one categorical variable, either row-wise or column-wise.
- For example, the value in the 'Row totals' or 'Column totals'.
Given the provided table:
[tex]\[
\begin{tabular}{|c|l|l|l|}
\hline \multicolumn{4}{|c|}{ Do you prefer dancing or playing sports? } \\
\hline & Playing sports & Dancing & Row totals \\
\hline $9 ^{\text {th }}$ grade students & 0.25 & 0.27 & 0.52 \\
\hline $1 \mathbf { 0 } ^ { \text { th } } \text { grade students }$ & 0.17 & 0.31 & 0.48 \\
\hline Column totals & 0.42 & 0.58 & 1 \\
\hline
\end{tabular}
\][/tex]
The entry of 0.31 corresponds to the 'Dancing' column for '10th grade students'. It specifically represents the proportion of 10th grade students who prefer dancing over playing sports. Since this entry shows the relative proportion of 10th grade students who prefer dancing, it is best described as a Joint Relative Frequency.
Hence, the term that best describes the entry of 0.31 is:
Joint relative frequency
