Question: Nelson collects the running times of 50 athletes and records the data
in the table below.
Time ($x$ seconds) Frequency
$35 < x \leq 40$ 4
$40 < x \leq 45$ 7
$45 < x \leq 50$ 30
$50 < x \leq 55$ 6
$55 < x \leq 60$ 3
Complete the cumulative frequency table.
Time ($x$ seconds) Cumulative Frequency
$35 < x \leq 40$
$35 < x \leq 45$
$35 < x \leq 50$
$35 < x \leq 55$
$35 < x \leq 60$
1. **State the problem:** We have a frequency table of running times for 50 athletes with class intervals and frequencies given. We need to complete the cumulative frequency table.
2. **Recall the formula:** The cumulative frequency for a class interval is the sum of the frequencies of all previous intervals plus the frequency of the current interval.
3. **Calculate cumulative frequencies step-by-step:**
- For $35 < x \leq 40$: cumulative frequency = $4$
- For $35 < x \leq 45$: cumulative frequency = $4 + 7 = 11$
- For $35 < x \leq 50$: cumulative frequency = $11 + 30 = 41$
- For $35 < x \leq 55$: cumulative frequency = $41 + 6 = 47$
- For $35 < x \leq 60$: cumulative frequency = $47 + 3 = 50$
4. **Final cumulative frequency table:**
| Time ($x$ seconds) | Cumulative Frequency |
|--------------------|----------------------|
| $35 < x \leq 40$ | 4 |
| $35 < x \leq 45$ | 11 |
| $35 < x \leq 50$ | 41 |
| $35 < x \leq 55$ | 47 |
| $35 < x \leq 60$ | 50 |
This completes the cumulative frequency table correctly.