To determine the total number of students and the mean time, we need to analyze the histogram. The histogram provides frequency density and time intervals, and the total frequency is obtained using the formula:

$\text{Frequency} = \text{Frequency Density} \times \text{Class Width}$

Step 1: Extract Data from the Histogram

From the graph:

1. Class Intervals (Time in seconds):

   • $0 \leq t < 2$
   • $2 \leq t < 6$
   • $6 \leq t < 10$
   • $10 \leq t < 20$

2. Frequency Densities:

   • For $0 \leq t < 2$: 7
   • For $2 \leq t < 6$: 4
   • For $6 \leq t < 10$: 8
   • For $10 \leq t < 20$: 2

Step 2: Calculate the Frequencies

For each interval, calculate the frequency:

• $0 \leq t < 2$: $7 \times (2 - 0) = 14$
• $2 \leq t < 6$: $4 \times (6 - 2) = 16$
• $6 \leq t < 10$: $8 \times (10 - 6) = 32$
• $10 \leq t < 20$: $2 \times (20 - 10) = 20$

Total Number of Students

$\text{Total Frequency} = 14 + 16 + 32 + 20 = 82$

Step 3: Estimate the Mean Time

To estimate the mean, use the formula: $\text{Mean} = \frac{\sum (\text{Midpoint} \times \text{Frequency})}{\text{Total Frequency}}$

• Midpoints for intervals:

  • $0 \leq t < 2$: $\frac{0+2}{2} = 1$
  • $2 \leq t < 6$: $\frac{2+6}{2} = 4$
  • $6 \leq t < 10$: $\frac{6+10}{2} = 8$
  • $10 \leq t < 20$: $\frac{10+20}{2} = 15$

• Multiply midpoint by frequency:

  • $1 \times 14 = 14$
  • $4 \times 16 = 64$
  • $8 \times 32 = 256$
  • $15 \times 20 = 300$

• Sum of midpoint products: $14 + 64 + 256 + 300 = 634$

• Mean time: $\text{Mean} = \frac{634}{82} \approx 7.73 \, \text{seconds (3 sf)}$

Final Answers:

1. Number of students: 82
2. Mean time: 7.73 seconds (3 sf)

Let me know if you'd like further details or have questions!

Here are 5 related questions:

1. How does changing the class width affect the histogram and calculations?
2. Can you calculate the median time from this histogram?
3. What is the range of the times taken?
4. How would you estimate the mode from this histogram?
5. What is the importance of frequency density in constructing histograms?

Tip: Always verify that the area under the histogram bars represents the total frequency for accuracy!