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The question you provided is about the distribution of heights of 400 female applicants for the Police Service. It requires you to draw a cumulative frequency curve and estimate values based on it. Let's break it down.

### Part (a): 
You need to draw a cumulative frequency curve using the given data. The scale is:
- **Horizontal scale**: 2 cm to represent 5 cm of height.
- **Vertical scale**: 2 cm to represent 50 applicants.

The cumulative frequency is already provided in the last column of the table.

**Steps to draw the graph**:
1. Plot height (in cm) on the horizontal axis starting from 150 cm, as the question indicates.
2. Plot the cumulative frequency on the vertical axis. The highest frequency is 400, so your vertical axis should go up to 400.
3. Plot the points corresponding to each cumulative frequency at the upper boundary of the height intervals:
   - 155 cm → 10 applicants
   - 160 cm → 65 applicants
   - 165 cm → 170 applicants
   - 170 cm → 280 applicants
   - 175 cm → 360 applicants
   - 180 cm → 390 applicants
   - 185 cm → 400 applicants
4. Draw a smooth curve through the plotted points.

### Part (b): 
Now using your cumulative frequency curve, you'll estimate the following:

**(i) The number of applicants whose heights are less than 170 cm**:
- Look at the cumulative frequency at 170 cm. From the table, it's 280 applicants.

**(ii) The median height of the applicants**:
- The median corresponds to the height where the cumulative frequency is 200 (since there are 400 applicants, the median is at the 50th percentile).
- You will estimate this by drawing a horizontal line from 200 applicants to the curve and then dropping it to the x-axis to get the height.

**(iii) The height that 25% of the applicants are less than**:
- 25% of 400 applicants is 100. You need to estimate the height at the cumulative frequency of 100 by reading off the graph.

**(iv) The probability that an applicant selected at random has a height that is no more than 162 cm**:
- First, find the number of applicants who have a height less than or equal to 162 cm from the graph (or estimate between 160 and 165 cm).
- The probability is this number divided by 400.

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Would you like assistance with drawing the graph or further calculations for part (b)?

Here are some additional related questions:
1. What is the importance of a cumulative frequency curve?
2. How do you interpret the shape of the cumulative frequency curve?
3. Can the cumulative frequency curve be used to estimate the mode of the data set?
4. What is the significance of the median in this type of data?
5. How could you adjust the scales on the graph to make plotting easier?

**Tip**: When drawing cumulative frequency curves, ensure smooth, consistent lines to avoid inaccuracies when making estimates.

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 and estimate the following: (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 a cumulative frequency curve based on the distribution of heights of 400 female applicants for the Police Service. Use the curve to estimate the number of applicants below a certain height, the median height, and calculate probabilities. This exercise covers important statistical concepts such as cumulative frequency and probability estimation, ideal for students in Grades 10-12.

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