To determine the modal class interval and estimate the mean age of the employees given the provided frequency table, follow these steps:
### Part (a) Find the Modal Class Interval
1. Identify the Frequencies:
- The frequencies for the age intervals are:
- [tex]\(18 - [tex]\(20 - [tex]\(22 - [tex]\(24 - [tex]\(26
2. Determine the Modal Class Interval:
- The modal class interval is the class interval with the highest frequency.
- From the frequencies, the highest frequency is 8.
3. Corresponding Interval:
- The interval with the highest frequency of 8 is [tex]\(24
Thus, the modal class interval is [tex]\(24
### Part (b) Estimate the Mean Age
To calculate the mean age, we will use the midpoints of each class interval and the frequencies.
1. Calculate midpoints for each interval:
- [tex]\(18 - [tex]\(20 - [tex]\(22 - [tex]\(24 - [tex]\(26
2. Multiply each midpoint by its corresponding frequency:
- [tex]\(19 \times 3 = 57\)[/tex]
- [tex]\(21 \times 2 = 42\)[/tex]
- [tex]\(23 \times 7 = 161\)[/tex]
- [tex]\(25 \times 8 = 200\)[/tex]
- [tex]\(26 \times 0 = 0\)[/tex] (Essentially, this is not factored in since frequency is zero.)
3. Calculate the total sum of these products:
- [tex]\(57 + 42 + 161 + 200 + 0 = 460\)[/tex]
4. Find the total number of employees:
- Total employees [tex]\( = 3 + 2 + 7 + 8 + 0 = 20\)[/tex]
5. Compute the estimated mean age:
- Estimated mean age [tex]\(= \frac{\text{Total sum of products}}{\text{Total number of employees}}\)[/tex]
- [tex]\(= \frac{460}{20}\)[/tex]
- [tex]\(= 23\)[/tex]
While typically the correct steps would yield an actual number, it turns out the specific scenario considers unknowns leading to an unquantifiable mean in the given context — thus showcasing special circumstances for this specific estimation.
Nonetheless, an estimated mean age based on standard operations would be [tex]\(\approx 23\)[/tex].
