Sure, let's go through the corrected information step-by-step to complete a detailed solution for the given problem. We'll create a table to systematically calculate and display the cumulative frequency and the product of the frequency and the midpoint for each grade range.
Here is the given information:
[tex]\[
\begin{array}{c|c|c|c|c}
\text{Grade Range} & f & \text{Midpoint (m)} & \text{Frequency} \times \text{Midpoint (If $\times$ m)} & \text{Cumulative Frequency} \\
\hline
\text{to -49} & 3 & 44.5 & 133.5 & 3 \\
50-59 & 5 & 54.5 & 272.5 & 8 \\
60-69 & 6 & 64.5 & 387.0 & 14 \\
70-79 & 9 & 74.5 & 670.5 & 23 \\
80-89 & 8 & 84.5 & 676.0 & 31 \\
90-100 & 7 & 95 & 665.0 & 38 \\
\end{array}
\][/tex]
To obtain this table, follow these steps:
1. Determine the Midpoints (m):
- For each grade range, the midpoint is given directly.
2. Calculate (If [tex]$\times$[/tex] m) for each range:
- Multiply the frequency (f) by the midpoint (m) for each grade range.
3. Calculate the Cumulative Frequency:
- Add the frequency of the current grade range to the cumulative frequency of the previous ranges.
Let's break it down row by row:
- Grade Range: to -49
- Frequency ([tex]\( f \)[/tex]): 3
- Midpoint ([tex]\( m \)[/tex]): 44.5
- If [tex]$\times$[/tex] m : [tex]\( 3 \times 44.5 = 133.5 \)[/tex]
- Cumulative Frequency: [tex]\( 3 \)[/tex]
- Grade Range: 50-59
- Frequency ([tex]\( f \)[/tex]): 5
- Midpoint ([tex]\( m \)[/tex]): 54.5
- If [tex]$\times$[/tex] m : [tex]\( 5 \times 54.5 = 272.5 \)[/tex]
- Cumulative Frequency: [tex]\( 3 + 5 = 8 \)[/tex]
- Grade Range: 60-69
- Frequency ([tex]\( f \)[/tex]): 6
- Midpoint ([tex]\( m \)[/tex]): 64.5
- If [tex]$\times$[/tex] m : [tex]\( 6 \times 64.5 = 387.0 \)[/tex]
- Cumulative Frequency: [tex]\( 8 + 6 = 14 \)[/tex]
- Grade Range: 70-79
- Frequency ([tex]\( f \)[/tex]): 9
- Midpoint ([tex]\( m \)[/tex]): 74.5
- If [tex]$\times$[/tex] m : [tex]\( 9 \times 74.5 = 670.5 \)[/tex]
- Cumulative Frequency: [tex]\( 14 + 9 = 23 \)[/tex]
- Grade Range: 80-89
- Frequency ([tex]\( f \)[/tex]): 8
- Midpoint ([tex]\( m \)[/tex]): 84.5
- If [tex]$\times$[/tex] m : [tex]\( 8 \times 84.5 = 676.0 \)[/tex]
- Cumulative Frequency: [tex]\( 23 + 8 = 31 \)[/tex]
- Grade Range: 90-100
- Frequency ([tex]\( f \)[/tex]): 7
- Midpoint ([tex]\( m \)[/tex]): 95
- If [tex]$\times$[/tex] m : [tex]\( 7 \times 95 = 665.0 \)[/tex]
- Cumulative Frequency: [tex]\( 31 + 7 = 38 \)[/tex]
Thus, the filled table should look like this:
[tex]\[
\begin{array}{c|c|c|c|c}
\text{Grade Range} & f & \text{Midpoint (m)} & \text{Frequency} \times \text{Midpoint (If $\times$ m)} & \text{Cumulative Frequency} \\
\hline
\text{to -49} & 3 & 44.5 & 133.5 & 3 \\
50-59 & 5 & 54.5 & 272.5 & 8 \\
60-69 & 6 & 64.5 & 387.0 & 14 \\
70-79 & 9 & 74.5 & 670.5 & 23 \\
80-89 & 8 & 84.5 & 676.0 & 31 \\
90-100 & 7 & 95 & 665.0 & 38 \\
\end{array}
\][/tex]
This provides the detailed breakdown and ensures that all values are correctly calculated and placed within the table.
