Let's work through the problem together.

Part a) Estimate for the mean height of Oliver's classmates:

We have to calculate an estimate for the mean height from the grouped data. The steps are:

1. Find the midpoint of each interval: $\text{Midpoint} = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

   • For $150 < h \leq 160$: Midpoint = $\frac{150 + 160}{2} = 155$
   • For $160 < h \leq 170$: Midpoint = $\frac{160 + 170}{2} = 165$
   • For $170 < h \leq 180$: Midpoint = $\frac{170 + 180}{2} = 175$
   • For $180 < h \leq 190$: Midpoint = $\frac{180 + 190}{2} = 185$

2. Multiply each midpoint by its frequency to get the total height for each group:

   • $155 \times 5 = 775$
   • $165 \times 13 = 2145$
   • $175 \times 9 = 1575$
   • $185 \times 4 = 740$

3. Add these totals to get the sum of the estimated heights: $775 + 2145 + 1575 + 740 = 5235$

4. Add the frequencies to get the total number of classmates: $5 + 13 + 9 + 4 = 31$

5. Estimate the mean: $\text{Mean} = \frac{\text{Total height}}{\text{Total number of classmates}} = \frac{5235}{31} = 168.87 \, \text{cm}$ Rounding to 1 decimal place: $\boxed{168.9 \, \text{cm}}$

Part b) Why is there a difference between the actual mean and the estimated mean?

The estimated mean from part (a) is $168.9 \, \text{cm}$, while the actual mean is given as $164.7 \, \text{cm}$. The difference arises because in part (a) we used midpoints to estimate the mean for each group. This introduces some approximation since the actual heights within each group are not all concentrated at the midpoint but are distributed across the interval. The original data contains the exact measurements, which provide a more accurate mean.

Would you like more details on any part? Here are some questions to expand your understanding:

1. What would happen if the intervals were narrower?
2. How would the result change if the frequencies were higher in the smaller height intervals?
3. Why do we use midpoints for estimating grouped data?
4. Can you think of a scenario where the estimated mean would be much closer to the actual mean?
5. What could be done to minimize the difference between estimated and actual mean?

Tip: When calculating the mean from grouped data, smaller intervals generally give a more accurate estimate because they reduce the potential difference between the midpoint and the actual data points.