Part a) Estimate the mean height

To estimate the mean from grouped data:

1. Use the midpoint of each class interval as a representative value.
2. Multiply each midpoint by its corresponding frequency to get the total "weighted heights."
3. Divide the total weighted heights by the sum of the frequencies.

Given Data:

Step-by-step calculation:

• Total frequency = $5 + 13 + 9 + 4 = 31$
• Total weighted heights = $775 + 2145 + 1575 + 740 = 5235$

Estimated Mean Height: $\text{Mean} = \frac{\text{Total Weighted Heights}}{\text{Total Frequency}} = \frac{5235}{31} \approx 168.9 \, \text{cm (1 d.p.)}$

Part b) Explanation for the difference in means

The difference arises because:

• The estimate in part (a) uses midpoints of each group, which assumes the data is evenly spread within each interval.
• The actual mean is based on the raw data, which gives the precise distribution of values.

Thus, the estimated mean (168.9 cm) is slightly higher than the actual mean (164.7 cm) because the assumption of evenly distributed data within the groups does not perfectly reflect the actual measurements.

Let me know if you need more clarification! Here are related questions to explore further:

1. How would the mean change if there were additional frequency intervals?
2. What happens to the estimate if the frequencies are altered?
3. How can we estimate the median height from this grouped data?
4. What is the range of the height data, and why is it important?
5. How would you determine the mode for this frequency distribution?

Tip: When estimating the mean from grouped data, always consider that the result is approximate due to the use of midpoints.