Let's begin by analyzing the data from the histogram to solve the questions:

Data from the Histogram:

The mass intervals and their corresponding frequencies are represented as follows:

• $40 \leq m < 50$: Frequency = 6
• $50 \leq m < 60$: Frequency = 8
• $60 \leq m < 70$: Frequency = 12
• $70 \leq m < 80$: Frequency = 9
• $80 \leq m < 90$: Frequency = 3
• $90 \leq m < 100$: Frequency = 2

(a) Calculating the Mean Mass:

The mean mass is calculated using the midpoint of each mass interval and the corresponding frequency.

1. Midpoint of $40 \leq m < 50$ = $\frac{40 + 50}{2} = 45$
2. Midpoint of $50 \leq m < 60$ = $\frac{50 + 60}{2} = 55$
3. Midpoint of $60 \leq m < 70$ = $\frac{60 + 70}{2} = 65$
4. Midpoint of $70 \leq m < 80$ = $\frac{70 + 80}{2} = 75$
5. Midpoint of $80 \leq m < 90$ = $\frac{80 + 90}{2} = 85$
6. Midpoint of $90 \leq m < 100$ = $\frac{90 + 100}{2} = 95$

The mean mass $\mu$ is given by the formula:

$\mu = \frac{\sum (f_i \times x_i)}{\sum f_i}$

where $f_i$ is the frequency and $x_i$ is the midpoint for each class.

$\mu = \frac{(6 \times 45) + (8 \times 55) + (12 \times 65) + (9 \times 75) + (3 \times 85) + (2 \times 95)}{6 + 8 + 12 + 9 + 3 + 2}$

Now, let's calculate the total:

$\mu = \frac{(270) + (440) + (780) + (675) + (255) + (190)}{40}$ $\mu = \frac{2610}{40} = 65.25 \, \text{kg}$

So, the mean mass is 65.25 kg.

(b) Identifying the Modal Class:

The modal class is the class with the highest frequency. From the histogram, the highest frequency is 12, which corresponds to the class interval:

$60 \leq m < 70$

Thus, the modal class is 60 kg ≤ m < 70 kg.

Would you like more details or have any other questions about the problem?

Here are 5 additional questions related to this topic:

1. How would you calculate the median mass from this data?
2. What is the range of the masses in this distribution?
3. How would you calculate the standard deviation of the mass data?
4. Can you determine the skewness of this distribution based on the histogram?
5. How can cumulative frequency be used to estimate the median from grouped data?

Tip: The midpoint method is a useful approximation for calculating averages when dealing with grouped data!