3. Statistics Problem
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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3. Statistics Problem
a) How many candidates sat for the exam?
Step 1: Sum all the frequencies (number of candidates).
$$N = \sum f = 8+4+2+2+20+17+10+15+20+22+22+15$$
$$N = 157$$
$$\boxed{\text{157 candidates}}$$
b) Find the median.
Step 1: Calculate the total number of candidates, $N = 157$.
Step 2: Determine the position of the median.
$$\text{Median position} = \frac{N+1}{2} = \frac{157+1}{2} = \frac{158}{2} = 79^{\text{th}}$$
Step 3: Find the cumulative frequencies to locate the $79^{\text{th}}$ value.
| Subjects (x) | Candidates (f) | Cumulative Frequency (cf) |
| :------------ | :-------------- | :------------------------ |
| 0 | 8 | 8 |
| 1 | 4 | 12 |
| 2 | 2 | 14 |
| 3 | 2 | 16 |
| 4 | 20 | 36 |
| 5 | 17 | 53 |
| 6 | 10 | 63 |
| 7 | 15 | 78 |
| 8 | 20 | 98 |
| 9 | 22 | 120 |
| 10 | 22 | 142 |
| 11 | 15 | 157 |
The $79^{\text{th}}$ value falls within the group of 8 subjects (since cf 78 is for 7 subjects, and cf 98 is for 8 subjects).
$$\boxed{\text{8 subjects}}$$
c) State the mode.
Step 1: Identify the highest frequency in the table.
The highest frequency is 22, which occurs
