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The image shows a question regarding the distribution of heights for 400 female applicants to the police service. Here's a breakdown of the problem:

### Given Table:
| Height (cm) | Number of Applicants | Cumulative Frequency |
|-------------|----------------------|----------------------|
| 151 - 155   | 10                   | 10                   |
| 156 - 160   | 55                   | 65                   |
| 161 - 165   | 105                  | 170                  |
| 166 - 170   | 110                  | 280                  |
| 171 - 175   | 80                   | 360                  |
| 176 - 180   | 30                   | 390                  |
| 181 - 185   | 10                   | 400                  |

#### Questions:
**(a)** Using the following scales, draw a cumulative frequency curve:
- **Horizontal scale**: 2 cm represents 5 cm of height.
- **Vertical scale**: 2 cm represents 50 applicants.
- Start the horizontal scale at 150 cm.

**(b)** Use the graph to estimate:
1. The number of applicants whose heights are less than 170 cm.
2. The median height of applicants.
3. The height that 25% of applicants are less than.
4. The probability that a randomly selected applicant has a height that is no more than 162 cm.

---

### Explanation and Steps:

1. **Cumulative Frequency Curve**:
   To plot the cumulative frequency curve, use the table's height intervals and cumulative frequency values:
   - Horizontal axis represents height (starting at 150 cm), with each 2 cm representing 5 cm of height.
   - Vertical axis represents cumulative frequency, with each 2 cm representing 50 applicants.
   - Plot points like (155, 10), (160, 65), (165, 170), etc., and draw a smooth curve through them.

2. **Estimations from the graph**:
   - **(i) Number of applicants below 170 cm**: From the cumulative frequency table, this is directly 280 applicants.
   - **(ii) Median height**: The median height is the 50th percentile, so locate the height corresponding to the cumulative frequency of 200 (since half of 400 = 200).
   - **(iii) Height that 25% of applicants are less than**: 25% of 400 is 100 applicants, so find the height corresponding to a cumulative frequency of 100.
   - **(iv) Probability that height ≤ 162 cm**: The cumulative frequency for heights up to 162 cm is needed. From the graph, you can estimate this frequency and calculate the probability by dividing the number of applicants with height ≤ 162 cm by 400.

---

Do you want detailed instructions on drawing the curve or on specific parts of the solution? 

### 5 Related Questions:
1. How do you estimate the median from a cumulative frequency curve?
2. What is the significance of a cumulative frequency curve in statistics?
3. How can you calculate the interquartile range using a cumulative frequency graph?
4. How does probability relate to cumulative frequency in such distributions?
5. What other types of graphs can represent similar data distributions?

### Tip:
Always label your axes clearly when drawing cumulative frequency graphs, as it helps in reading the estimates accurately!

The table below gives the distribution of heights of 400 female applicants for the Police Service. Using a horizontal scale of 2 cm to represent a height of 5 cm and a vertical scale of 2 cm to represent 50 applicants, draw a cumulative frequency curve of the heights. Start your horizontal scale at 150 cm. Then use your graph to estimate (i) the number of applicants whose heights are less than 170 cm, (ii) the median height of applicants, (iii) the height that 25% of the applicants are less than, (iv) the probability that an applicant selected at random has a height that is no more than 162 cm.

Learn how to create a cumulative frequency curve and use it to estimate values such as the median height, number of applicants below a certain height, and probability calculations based on a given distribution of heights for 400 female police service applicants.

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