Sure, let's go through each part of the question step-by-step.
### Part (a): Frequency Distribution Table
To construct the frequency distribution table, we need to count the number of estimations that fall into each of the specified intervals. The intervals given are 55-59, 60-64, 65-69, 70-74, 75-79, and 80-84.
#### Step-by-step Classification
- Interval 55-59:
- The estimations that fall into this interval are: 56, 58, 58.
- Frequency = 3
- Interval 60-64:
- The estimations that fall into this interval are: 60, 60, 61, 62, 63, 64, 64.
- Frequency = 7
- Interval 65-69:
- The estimations that fall into this interval are: 65, 65, 65, 66, 66, 66, 66, 67, 67, 68, 68, 68, 69, 69, 69.
- Frequency = 15
- Interval 70-74:
- The estimations that fall into this interval are: 70, 70, 70, 70, 72, 72, 72, 72, 73, 73, 74, 74, 74.
- Frequency = 14
- Interval 75-79:
- The estimations that fall into this interval are: 75, 75, 75, 76, 78, 79.
- Frequency = 6
- Interval 80-84:
- The estimations that fall into this interval are: 80, 80, 81, 83.
- Frequency = 4
Using these frequencies, we can construct the frequency distribution table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Interval} & \text{Frequency} \\
\hline
55 - 59 & 3 \\
60 - 64 & 7 \\
65 - 69 & 15 \\
70 - 74 & 14 \\
75 - 79 & 6 \\
80 - 84 & 4 \\
\hline
\end{array}
\][/tex]
### Part (b): Modal Class
The modal class is the interval that has the highest frequency. From our frequency distribution table, we can see:
- Interval 55-59: Frequency = 3
- Interval 60-64: Frequency = 7
- Interval 65-69: Frequency = 15
- Interval 70-74: Frequency = 14
- Interval 75-79: Frequency = 6
- Interval 80-84: Frequency = 4
The interval with the highest frequency is 65-69 with a frequency of 15.
Therefore, the modal class is [tex]\((65, 69)\)[/tex] with a frequency of [tex]\(15\)[/tex].
