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Let's break down and solve this problem step-by-step.

### Part (a): Create a Histogram
To create a histogram for the data given, we use the following intervals of "Time" (in seconds) and their corresponding "Frequency":

| Time Interval (s) | Frequency |
|-------------------|-----------|
| 30 - 40           | 10        |
| 40 - 50           | 15        |
| 50 - 60           | 20        |
| 60 - 80           | 30        |
| 80 - 100          | 24        |
| 100 - 150         | 20        |

Each time interval represents the range within which the task completion time falls, and the frequency is the number of employees who took that much time. In the histogram:

- The x-axis will represent the time intervals.
- The y-axis will represent the frequency of employees.

The histogram bars will have different widths for each interval (since not all intervals have equal lengths) and heights corresponding to the frequency. You can construct this histogram by plotting bars where the base of each bar represents the time interval and the height represents the frequency.

### Part (b): Discussing Error with Frequency Instead of Density
The error that could occur by using frequency instead of density stems from the fact that the time intervals are not of equal width. For example, the time interval from 30 to 40 seconds has a width of 10 seconds, while the time interval from 100 to 150 seconds has a width of 50 seconds.

When we calculate the density, we divide the frequency by the width of the interval. For example:

- For the interval 30-40, the density would be \( \frac{10}{10} = 1 \).
- For the interval 100-150, the density would be \( \frac{20}{50} = 0.4 \).

Without this adjustment, simply plotting frequency values would distort the relative proportions, giving the misleading impression that intervals with larger widths but lower density have more occurrences. Using density instead of frequency accounts for the varying widths of intervals and gives a more accurate representation of the data's distribution.

### Do you need further clarification or assistance?

Here are five related questions to deepen your understanding:
1. How would the histogram change if all intervals had the same width?
2. How can you calculate cumulative frequency for this dataset?
3. What would the total number of employees be based on the frequencies given?
4. How would you compute the mean time to complete the task using this grouped data?
5. How can the histogram be used to estimate the median completion time?

**Tip:** When dealing with data that has unequal intervals, always check if using frequency or density gives a more accurate representation!

A supervisor of a plant kept records of the time (in seconds) that employees needed to complete a particular task. The data are summarized in a table of time intervals and frequencies. Part (a) asks to graph the data using a histogram, and Part (b) asks to discuss possible errors if frequency is used instead of density, with a hint to consider the intervals.

Learn how to graph a histogram for task completion times and explore the error that can occur when using frequency instead of density in unequal intervals. Suitable for high school students analyzing real-world data distributions.

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