Let's solve the question step by step based on the image provided:

Part (a): Find the midpoint of each group of marks and calculate an estimate of the mean mark.

To find the midpoint of each group of marks, use the formula: $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

For the provided dataLet's solve the question step by step based on the image provided:

Part (a): Find the midpoint of each group of marks and calculate an estimate of the mean mark.

To find the midpoint of each group of marks, use the formula: $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

For the provided

Now, calculate the mean using the formula for the estimated mean:

$\text{Mean} = \frac{\sum (\text{Midpoint} \times \text{Frequency})}{\sum \text{Frequency}}$

Calculations:

$(92 \times 4) + (77 \times 7) + (62 \times 8) + (47 \times 5) = 368 + 539 + 496 + 235 = 1638$

$\text{Total Frequency} = 4 + 7 + 8 + 5 = 24$

So, the estimated mean is:

$\text{Mean} = \frac{1638}{24} \approx 68.25$

Part (b): Explain why the answer is an estimate.

The answer is an estimate because we are using the midpoints of the groups to represent all the values within each group. Since we don't know the exact marks of each student, we assume that each student's mark is approximately the midpoint of their respective group.

Would you like further details or explanations? Here are some related questions:

1. How do you calculate the median in grouped data like this?
2. What is the range of this data set?
3. How would the estimated mean change if one student's score were known exactly?
4. How can we calculate the variance or standard deviation from this data?
5. What is the impact of the frequency distribution on the mean?

Tip: When estimating from grouped data, using the midpoints of each group gives a practical but not exact average value. Always keep this approximation in mind.