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The question provides a cumulative frequency table of heights for 400 female applicants, and you are asked to perform several tasks involving drawing a cumulative frequency curve and making estimations from it.

Let's break down the steps:

### (a) Draw the cumulative frequency curve
You are asked to use:
- A horizontal scale of 2 cm to represent a height of 5 cm (starting at 150 cm).
- A vertical scale of 2 cm to represent 50 applicants.

Here’s how to approach this:
1. **Set up your axes**:
   - On the x-axis (horizontal), represent height from 150 cm upwards.
   - On the y-axis (vertical), represent the cumulative frequency, starting from 0 to 400.
   
2. **Plot the cumulative frequency points** from the table:
   - (151, 10)
   - (156, 65)
   - (161, 170)
   - (166, 280)
   - (171, 360)
   - (176, 390)
   - (181, 400)

3. **Draw a smooth curve** through these points.

### (b) Use your graph to estimate:

#### (i) Number of applicants whose heights are less than 170 cm.
- From the graph, find the cumulative frequency at 170 cm (this should be around 280 based on the table).
- Therefore, **280 applicants** have heights less than 170 cm.

#### (ii) The median height of applicants.
- The median is the 50th percentile, which means the cumulative frequency will be 200 (half of 400).
- Find the corresponding height for a cumulative frequency of 200 from the graph. This height should be **around 163 cm**.

#### (iii) The height that 25% of applicants are less than.
- The 25th percentile corresponds to a cumulative frequency of 100 (25% of 400).
- From the graph, the height at this point is **around 160 cm**.

#### (iv) Probability that a randomly selected applicant has a height no more than 162 cm.
- From the graph, find the cumulative frequency corresponding to 162 cm. This should be around **190 applicants**.
- The probability is therefore \( \frac{190}{400} = 0.475 \) or **47.5%**.

If you'd like more details on these steps or have any other questions, feel free to ask!

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**Follow-up questions:**
1. What are the key differences between a cumulative frequency curve and a bar graph?
2. How can you estimate other percentiles (like the 75th) from the graph?
3. How would the curve change if more applicants were between 171 and 175 cm?
4. What is the significance of the median height in this context?
5. How would the results differ if the number of applicants were doubled?

**Tip:** Always ensure your axes are clearly labeled when drawing graphs, as this will make interpretation much easier.

The table below gives the distribution of heights of 400 female applicants for the Police Service. (a) 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. (b) 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 draw and interpret a cumulative frequency curve for the distribution of heights of 400 female applicants. Estimate values such as the median height, 25th percentile, and probability of height ranges. Ideal for students in Grades 9-12.

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