Certainly! Let's go through the steps to create a stem-and-leaf plot and fill out the frequency table with cumulative frequencies, relative frequencies, and cumulative relative frequencies.
### a) Stem-and-Leaf Plot
To create a stem-and-leaf plot, we will split each grade into a "stem" (the leading digit(s)) and a "leaf" (the trailing digit).
Given grades:
[tex]\[ 96, 66, 65, 82, 85, 82, 87, 76, 80, 85, 83, 69, 79, 70, 83, 63, 81, 94, 71, 83, 99, 75, 73, 83, 86 \][/tex]
Stem-and-Leaf Plot:
- 60-69: Stem = 6, Leaves = {3, 5, 6, 9}
- 70-79: Stem = 7, Leaves = {0, 1, 3, 5, 6, 9}
- 80-89: Stem = 8, Leaves = {0, 1, 2, 2, 3, 3, 3, 3, 5, 5, 6, 7}
- 90-99: Stem = 9, Leaves = {4, 6, 9}
Organized:
[tex]\[
\begin{array}{c|l}
\text{Stem} & \text{Leaves} \\
\hline
6 & 3, 5, 6, 9 \\
7 & 0, 1, 3, 5, 6, 9 \\
8 & 0, 1, 2, 2, 3, 3, 3, 3, 5, 5, 6, 7 \\
9 & 4, 6, 9 \\
\end{array}
\][/tex]
### b) Frequency Table
Next, we will fill out the table for frequencies, cumulative frequencies, relative frequencies, and cumulative relative frequencies.
\begin{tabular}{|c|c|c|c|c|}
\hline
Class & Frequency & \begin{tabular}{c}
Cumulative \\
Frequency
\end{tabular} & \begin{tabular}{c}
Relative \\
Frequency
\end{tabular} & \begin{tabular}{c}
Cumulative Relative \\
Frequency
\end{tabular} \\
\hline
[tex]$60-69$[/tex] & 4 & 4 & 0.16 & 0.16 \\
\hline
[tex]$70-79$[/tex] & 6 & 10 & 0.24 & 0.40 \\
\hline
[tex]$80-89$[/tex] & 12 & 22 & 0.48 & 0.88 \\
\hline
[tex]$90-99$[/tex] & 3 & 25 & 0.12 & 1.00 \\
\hline
Total & 25 & 25 & 1.00 & 1.00 \\
\hline
\end{tabular}
Here's how we calculated the values:
- Frequency: The number of grades in each class range.
- [tex]$60-69$[/tex]: 4 grades
- [tex]$70-79$[/tex]: 6 grades
- [tex]$80-89$[/tex]: 12 grades
- [tex]$90-99$[/tex]: 3 grades
- Cumulative Frequency: The running total of frequencies up to that class range.
- [tex]$60-69$[/tex]: 4
- [tex]$70-79$[/tex]: 4 + 6 = 10
- [tex]$80-89$[/tex]: 10 + 12 = 22
- [tex]$90-99$[/tex]: 22 + 3 = 25
- Relative Frequency: The frequency divided by the total number of grades (25).
- [tex]$60-69$[/tex]: \( \frac{4}{25} = 0.16 \)
- [tex]$70-79$[/tex]: \( \frac{6}{25} = 0.24 \)
- [tex]$80-89$[/tex]: \( \frac{12}{25} = 0.48 \)
- [tex]$90-99$[/tex]: \( \frac{3}{25} = 0.12 \)
- Cumulative Relative Frequency: The running total of relative frequencies up to that class range.
- [tex]$60-69$[/tex]: 0.16
- [tex]$70-79$[/tex]: 0.16 + 0.24 = 0.40
- [tex]$80-89$[/tex]: 0.40 + 0.48 = 0.88
- [tex]$90-99$[/tex]: 0.88 + 0.12 = 1.00
This is the detailed solution to create a stem-and-leaf plot and fill out the frequency table with the provided exam grades.
