(i) To find the mean age of the whole group, use the formula for the combined mean:

\(\bar{x} = \frac{(\bar{x}_w \times 10) + (\bar{x}_m \times 13)}{23} = \frac{(41.2 \times 10) + (46.3 \times 13)}{23} = 44.1\)

(ii) First, find \(\Sigma x_w^2\) and \(\Sigma x_m^2\) using the given standard deviations:

\(15.1^2 = \frac{\Sigma x_w^2}{10} - 41.2^2\)

\(\Sigma x_w^2 = 19254.5\)

\(12.7^2 = \frac{\Sigma x_m^2}{13} - 46.3^2\)

\(\Sigma x_m^2 = 29964.74\)

Calculate the total sum of squares:

\(\Sigma = 19254.5 + 29964.74 = 49219.24\)

Finally, find the standard deviation for the whole group:

\(\text{sd} = \sqrt{\frac{49219.24}{23} - 44.1^2} = 14.0\)

To illustrate the data diagrammatically, follow these steps:

1. Choose either pie charts or bar charts for the representation.
2. If using pie charts, create two separate pie charts: one for males and one for females. Divide each pie chart into three sectors representing 'Young', 'Middle-aged', and 'Elderly' with angles proportional to the percentages given (e.g., for males: Young = 40%, Middle-aged = 35%, Elderly = 25%).
3. If using bar charts, create two sets of bars: one set for males and one set for females. Each set should have three bars representing 'Young', 'Middle-aged', and 'Elderly'. Ensure the height of each bar is proportional to the percentage given.
4. Label each chart or set of bars with the appropriate categories and percentages. Include labels for 'Males' and 'Females', as well as 'Young', 'Middle-aged', and 'Elderly'.

To estimate the mean number of hours of sunshine, we first find the midpoints of each class interval:

• For \(30 \leq x < 60\), midpoint = 45
• For \(60 \leq x < 90\), midpoint = 75
• For \(90 \leq x < 110\), midpoint = 100
• For \(110 \leq x < 140\), midpoint = 125
• For \(140 \leq x < 180\), midpoint = 160
• For \(180 \leq x < 240\), midpoint = 210

Next, we calculate the weighted sum of the midpoints:

\(4 \times 45 + 8 \times 75 + 14 \times 100 + 25 \times 125 + 7 \times 160 + 2 \times 210 = 6845\)

The total number of years is 60. Therefore, the estimated mean is:

\(\text{Mean} = \frac{6845}{60} = 114.08\overline{3}\)

To find \(f\) and the total number of children:

The midpoints of the intervals are: 5, 15, 30, and 60.

The equation for the mean is:

\(\frac{5 \times 2 + 15f + 30 \times 11 + 60 \times 4}{17 + f} = 27.5\)

\(10 + 15f + 330 + 240 = 27.5(17 + f)\)

\(15f + 580 = 467.5 + 27.5f\)

\(580 - 467.5 = 27.5f - 15f\)

\(112.5 = 12.5f\)

\(f = 9\)

Total number of children = \(17 + f = 26\).

To find the standard deviation:

Calculate \(\sigma\) using the formula:

\(\sigma = \sqrt{\frac{\sum f(x - \bar{x})^2}{n}}\)

\(\sigma = 16.1\)

(i) To find the mean:

\(\bar{x} = \frac{\Sigma x}{n} = \frac{745}{18} = 41.4\)

To find the standard deviation:

\(s = \sqrt{\frac{\Sigma x^2}{n} - \left(\frac{\Sigma x}{n}\right)^2} = \sqrt{\frac{33951}{18} - \left(\frac{745}{18}\right)^2} = 13.2\)

(ii) Let the age of the person who left be \(x\).

The sum of the ages of the remaining 17 people is:

\(17 \times 41 = 697\)

Therefore, \(745 - x = 697\), giving \(x = 48\).

To find the standard deviation of the remaining 17 people:

\(s = \sqrt{\frac{\Sigma x^2 - x^2}{17} - (41)^2} = \sqrt{\frac{33951 - 48^2}{17} - 41^2} = 13.4\)