Question: The table gives information about the time taken by each student in Year 11 to complete a homework task. Time taken ($t$ minutes) Frequency $10 < t \leq 25$ 15 $25 < t \leq 30$ 18 $30 < t \leq 50$ 32 $50 < t \leq 60$ 4 (a) On the grid, draw a histogram for this information. Histogram axes: x-axis is Time taken ($t$ minutes) from 10 to 60, y-axis is Frequency density from 0 to 4, with four bars over the intervals $10 < t \leq 25$, $25 < t \leq 30$, $30 < t \leq 50$, and $50 < t \leq 60$. position_hint: center One of these students who took 50 minutes or less and more than 25 minutes to complete this homework task is chosen at random. (b) Find an estimate for the probability that this student took 45 minutes or less to complete this homework task.

1. **State the problem:** We have frequency data for time intervals and need to draw a histogram and find a probability estimate. 2. **Calculate class widths and frequency densities for the histogram:** - Interval $10 < t \leq 25$: width = $25 - 10 = 15$, frequency = 15 - Interval $25 < t \leq 30$: width = $30 - 25 = 5$, frequency = 18 - Interval $30 < t \leq 50$: width = $50 - 30 = 20$, frequency = 32 - Interval $50 < t \leq 60$: width = $60 - 50 = 10$, frequency = 4 Frequency density = $\frac{\text{frequency}}{\text{class width}}$ Calculate each: $$\text{Density}_1 = \frac{15}{15} = 1$$ $$\text{Density}_2 = \frac{18}{5} = 3.6$$ $$\text{Density}_3 = \frac{32}{20} = 1.6$$ $$\text{Density}_4 = \frac{4}{10} = 0.4$$ 3. **Draw the histogram:** - X-axis: time intervals $10$ to $60$ - Y-axis: frequency density from $0$ to $4$ - Bars heights: 1, 3.6, 1.6, 0.4 over respective intervals 4. **Find the probability that a student who took $25 < t \leq 50$ minutes took $45$ minutes or less:** Total students in $25 < t \leq 50$: $$18 + 32 = 50$$ We want the number of students who took $25 < t \leq 45$ minutes. Assuming uniform distribution in $30 < t \leq 50$ interval: - Length of $30 < t \leq 50$ is 20 minutes - Length of $30 < t \leq 45$ is 15 minutes Number of students in $30 < t \leq 45$: $$32 \times \frac{15}{20} = 24$$ Number of students in $25 < t \leq 30$ is 18. Total students in $25 < t \leq 45$: $$18 + 24 = 42$$ Probability: $$P = \frac{42}{50} = 0.84$$ **Final answer:** The estimated probability is $0.84$ or $84\%$.