To solve the problem, let's follow a detailed, step-by-step approach:
1. Understand the Tabular Data:
The table presents two rows of data: the first row contains the ranges of [tex]\( x \)[/tex] and the second row contains corresponding [tex]\( f \)[/tex] values. The table is structured as:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & \text{less than 10} & \text{less than 20} & \text{less than 30} & \text{less than 40} & \text{less than 50} \\
\hline
f & 12 & 19 & 24 & 33 & 40 \\
\hline
\end{array}
\][/tex]
2. Interpret the Table:
- The column headers for [tex]\( x \)[/tex] represent different ranges within which the values fall:
- "less than 10" means values in this column are for [tex]\( x < 10 \)[/tex].
- "less than 20" means values in this column are for [tex]\( 10 \le x < 20 \)[/tex].
- Similarly for other columns: [tex]\( 20 \le x < 30 \)[/tex], [tex]\( 30 \le x < 40 \)[/tex], and [tex]\( 40 \le x < 50 \)[/tex].
- The [tex]\( f \)[/tex] values represent frequencies or counts of occurrences within these ranges:
- For [tex]\( x < 10 \)[/tex], the frequency is 12.
- For [tex]\( 10 \le x < 20 \)[/tex], the cumulative frequency is 19.
- For [tex]\( 20 \le x < 30 \)[/tex], the cumulative frequency is 24.
- For [tex]\( 30 \le x < 40 \)[/tex], the cumulative frequency is 33.
- For [tex]\( 40 \le x < 50 \)[/tex], the cumulative frequency is 40.
3. Analyze the Frequencies:
The key interpretation is understanding that these frequencies might be cumulative. Let’s break down the frequencies as:
- For [tex]\( x < 10 \)[/tex]: the frequency is 12.
- For [tex]\( 10 \le x < 20 \)[/tex]: the frequency from just this interval is [tex]\( 19 - 12 = 7 \)[/tex].
- For [tex]\( 20 \le x < 30 \)[/tex]: the frequency from just this interval is [tex]\( 24 - 19 = 5 \)[/tex].
- For [tex]\( 30 \le x < 40 \)[/tex]: the frequency from just this interval is [tex]\( 33 - 24 = 9 \)[/tex].
- For [tex]\( 40 \le x < 50 \)[/tex]: the frequency from just this interval is [tex]\( 40 - 33 = 7 \)[/tex].
4. Summarize Your Findings:
- The table seems to represent cumulative frequencies.
- Here's the breakdown for intervals from these cumulative frequencies:
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
x & < 10 & 10 \le x < 20 & 20 \le x < 30 & 30 \le x < 40 & 40 \le x < 50 \\
\hline
f & 12 & 7 & 5 & 9 & 7 \\
\hline
\end{array}
\][/tex]
So, the overall interpretation of the table is correct, and the detailed breakdown helps to understand the distribution of frequencies within specified ranges. This approach ensures a clear understanding of how data is distributed across different intervals.
