1. **State the problem:** We need to estimate the mean amount of time spent on stage by actors, given grouped frequency data. 2. **Recall the formula for the mean of grouped data:** $$\text{Mean} = \frac{\sum (f \times x)}{\sum f}$$ where $f$ is the frequency of each group and $x$ is the midpoint of each time interval. 3. **Find the midpoints of each time interval:** - For $0 < t \leq 20$, midpoint $x_1 = \frac{0 + 20}{2} = 10$ - For $20 < t \leq 40$, midpoint $x_2 = \frac{20 + 40}{2} = 30$ - For $40 < t \leq 60$, midpoint $x_3 = \frac{40 + 60}{2} = 50$ 4. **Multiply each midpoint by its frequency:** - $f_1 \times x_1 = 3 \times 10 = 30$ - $f_2 \times x_2 = 8 \times 30 = 240$ - $f_3 \times x_3 = 5 \times 50 = 250$ 5. **Calculate the sum of frequencies and the sum of $f \times x$:** - $\sum f = 3 + 8 + 5 = 16$ - $\sum (f \times x) = 30 + 240 + 250 = 520$ 6. **Calculate the mean:** $$\text{Mean} = \frac{520}{16}$$ Show cancellation: $$\text{Mean} = \frac{\cancel{520}}{\cancel{16}} = 32.5$$ 7. **Interpretation:** The estimated mean time spent on stage is 32.5 minutes.