Let's start with part (a) by drawing a frequency diagram to represent the data.
### Part (a): Drawing a Frequency Diagram
To draw a frequency diagram (also known as a bar chart) for the given data:
1. Label the Axes:
- The horizontal axis ([tex]\(x\)[/tex]-axis) will represent the intervals of the number of breakfasts sold.
- The vertical axis ([tex]\(y\)[/tex]-axis) will represent the frequency.
2. Plot the Data:
- For each interval on the [tex]\(x\)[/tex]-axis, draw a bar corresponding to its frequency. The height of the bar should match the frequency value.
Here is how you can interpret and graph the data given in the table:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of breakfasts sold} & \text{Frequency} \\
\hline
0-9 & 1 \\
\hline
10-19 & 3 \\
\hline
20-29 & 7 \\
\hline
30-39 & 11 \\
\hline
40-49 & 5 \\
\hline
\end{array}
\][/tex]
The corresponding frequency diagram would look like this:
```
| |
| | x
| | x
| | x x
| | x x
| | x x
| | x x
| | x x
| | x x
| | x x x
| x | x x x
+------------+----+-----+---+-----+---+---+---+---+---+---+---+
0-9 10-19 20-29 30-39 40-49
```
### Part (b): Identifying the Mistake in the Frequency Table
To determine if there is a mistake in the frequency table, follow these steps:
1. Sum the Frequencies:
Calculate the total frequency by summing all the individual frequencies:
[tex]\[
1 + 3 + 7 + 11 + 5 = 27
\][/tex]
2. Compare with the Number of Days in a Month:
Typically, a month has around 30 or 31 days, but it could also have 28 or 29 days in the case of February.
3. Identify the Mistake:
Since the table represents the number of breakfasts sold each day in one month, the total frequency should match the number of days in the month.
Given that common months have:
- 30 days (April, June, September, November)
- 31 days (January, March, May, July, August, October, December)
- Even if it were February in a non-leap year, it should have 28 days.
The total frequency is 27, which does not correspond to any common month.
Thus, the mistake can be explained as follows:
- The total frequency should be equal to the number of days in the month. In this case, the calculated total frequency is only 27, which means there is an inconsistency because a typical month has either 28, 30, or 31 days. Therefore, the person who made the frequency table likely made an error in recording the frequencies or missed considering a few days.
