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Solve these problems and get the explanation 

Let's break down and solve the questions based on the image provided. The tasks revolve around interpreting a cumulative frequency diagram for two sports centers, A and B.

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### Question (a)
The cumulative frequency diagram shows the distribution of the ages of 1000 people using sports centers A and B.

#### Part (a)(i)
Estimate the **median age** for sports center A.

- The median is the value at the 50th percentile (the middle value), which corresponds to a cumulative frequency of 500 (since there are 1000 people in total).
- Locate the point on the cumulative frequency graph for sports center A where the frequency is 500, then drop a line down to the x-axis to find the age.
  
#### Answer:
- From the graph, the median age for sports center A appears to be approximately **42 years**.

#### Part (a)(ii)
Estimate the **number of people over the age of 50** for sports center B.

- To find the number of people over 50, locate the cumulative frequency at 50 years for sports center B and subtract it from 1000 (the total population for B).
  
#### Answer:
- The cumulative frequency for age 50 in sports center B is approximately **750**.
- So, the number of people over 50 years in sports center B is: \( 1000 - 750 = 250 \).

#### Part (a)(iii)
Estimate the **interquartile range** (IQR) for sports center A.

- The IQR is the difference between the 75th percentile (upper quartile) and the 25th percentile (lower quartile).
- For the lower quartile, locate the cumulative frequency of 250 (25% of 1000).
- For the upper quartile, locate the cumulative frequency of 750 (75% of 1000).
  
#### Answer:
- From the graph:
  - The lower quartile (25th percentile) for sports center A is approximately **30 years**.
  - The upper quartile (75th percentile) is approximately **55 years**.
- So, the interquartile range (IQR) is: \( 55 - 30 = 25 \) years.

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### Question (b)
Compare the age distribution between sports centers A and B using the graph.

#### Answer:
1. **Median age comparison**: The median age for sports center A (42 years) is higher than that for sports center B, which appears to be around 28 years. This suggests that the participants at sports center A tend to be older.
2. **Spread of ages**: The interquartile range for sports center A (25 years) is larger than that for sports center B (which appears to be around 15 years), indicating a wider spread of ages at sports center A.

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### Question (c)
The table shows a breakdown of people using the sports center on a given day.

|             | Aged under 30 | Aged 30 or over |
|-------------|---------------|-----------------|
| **Male**    | 25            | 15              |
| **Female**  | 35            | 25              |

#### Part (c)(i)
One person is selected at random. Find the probability that this person is:

**Male and aged under 30.**

- Total number of people = \( 25 +

Solve these problems and get the explanation based on the cumulative frequency diagram for sports centers A and B.

Learn how to analyze cumulative frequency diagrams by estimating the median, interquartile range, and probability based on the ages of people using sports centers A and B. This solution includes key statistical concepts and detailed explanations. Suitable for students in Grades 9-12.

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