Calculate the angles for a pie chart of university admissions and the variance for an ages distribution.

here are the solutions to the questions:

2)

• University Admissions Pie Chart: To draw a pie chart, we first calculate the total number of students and the angle for each faculty. Total students = $350 (\text{Education}) + 150 (\text{Medicine}) + 200 (\text{Engineering}) + 100 (\text{Law}) + 100 (\text{Arts}) = 900$ students.

  The angle for each faculty is calculated as:

  $$\text{Angle} = \frac{\text{Number of students in faculty}}{\text{Total number of students}} \times 360^\circ$$
  • Education:
  $$\frac{350}{900} \times 360^\circ = 140^\circ$$
  • Medicine:
  $$\frac{150}{900} \times 360^\circ = 60^\circ$$
  • Engineering:
  $$\frac{200}{900} \times 360^\circ = 80^\circ$$
  • Law:
  $$\frac{100}{900} \times 360^\circ = 40^\circ$$
  • Arts:
  $$\frac{100}{900} \times 360^\circ = 40^\circ$$

  To draw the pie chart, use a compass to draw a circle with a radius of 5 cm. Then, use a protractor to mark out the calculated angles from the center of the circle for each faculty. Label each sector accordingly.

• Variance of Ages Distribution: To calculate the variance for grouped data, we first find the midpoint ($x$) for each class interval, then calculate $fx$ and $fx^2$.

  Step 1: Create a table with midpoints, $fx$, and $fx^2$.

  Ages (yrs)	Frequency ($f$)	Midpoint ($x$)	$fx$	$x^2$	$fx^2$
  21 – 30	3	25.5	76.5	650.25	1950.75
  31 – 40	10	35.5	355	1260.25	12602.5
  41 – 50	14	45.5	637	2070.25	28983.5
  51 – 60	10	55.5	555	3080.25	30802.5
  61 – 70	8	65.5	524	4290.25	34322
  Total	$\sum f = 45$		$\sum fx = 2147.5$		$\sum fx^2 = 108661.25$

  Step 2: Calculate the mean ($\bar{x}$).

  $$\bar{x} = \frac{\sum fx}{\sum f} = \frac{2147.5}{45} \approx 47.7222$$

  Step 3: Calculate the variance ($\sigma^2$).

  $$\sigma^2 = \frac{\sum fx^2}{\sum f} - \bar{x}^2$$ $$\sigma^2 = \frac{108661.25}{45} - (47.7222)^2$$ $$\sigma^2 = 2414.6944 - 2277.4081$$ $$\sigma^2 = 137.2863$$

  Rounding to two decimal places:

  $$\boxed{\text{Variance} = 137.29}$$

3)
The table shows the height of teak trees harvested by a farmer:

Height (m) ($x$)	Number of trees ($f$)
3	4
4	6
5	4
6	5
7	6
8	2

a) Find the median height.

Step 1: Find the total number of trees ($\sum f$).

$$\sum f = 4 + 6 + 4 + 5 + 6 + 2 = 27$$

Step 2: Determine the position of the median.

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