1. **State the problem:** We have a frequency distribution of the number of sick leave days per year taken by employees and need to determine if the data is discrete or continuous, find the unknown frequency $k$ given the mean, and identify the sampling technique. 2. **Part (a): Discrete or continuous data?** The number of sick leave days is counted in whole numbers (2, 3, 4, ...), so the data is **discrete** because it consists of distinct, separate values. 3. **Part (b): Find $k$ given the mean.** The mean number of sick leave days is given as 4. The formula for the mean of grouped data is: $$\text{Mean} = \frac{\sum (x_i \times f_i)}{\sum f_i}$$ where $x_i$ is the number of sick leave days and $f_i$ is the frequency (number of employees). From the table: \begin{align*} \sum f_i &= 12 + 22 + 24 + 15 + k + 3 + 1 = 77 + k \\ \sum (x_i f_i) &= 2\times12 + 3\times22 + 4\times24 + 5\times15 + 6\times k + 7\times3 + 8\times1 \\ &= 24 + 66 + 96 + 75 + 6k + 21 + 8 = 290 + 6k \end{align*} Using the mean formula: $$4 = \frac{290 + 6k}{77 + k}$$ Multiply both sides by $(77 + k)$: $$4(77 + k) = 290 + 6k$$ $$308 + 4k = 290 + 6k$$ Bring terms involving $k$ to one side: $$308 - 290 = 6k - 4k$$ $$18 = 2k$$ $$k = 9$$ 4. **Part (c): Identify the sampling technique.** Employees were arranged alphabetically and every 5th person was selected. This is an example of **systematic sampling**. **Final answers:** - (a) Discrete data - (b) $k = 9$ - (c) Systematic sampling