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To solve this problem, we need to find both the sample proportion and the population proportion of attendees who were children at the Gazelles' match. Let's go through each part:

### Given Information:
1. **Population Proportion**: Out of every 56 attendees at the match, 25 were children.
2. **Sample Proportion**: In a sample taken by the facilities manager, 43 out of 89 attendees were children.

### Solution:
1. **(a) Sample Proportion (p-hat)**: 
   \[
   \hat{p} = \frac{\text{Number of children in sample}}{\text{Total sample size}} = \frac{43}{89}
   \]
   Calculating this:
   \[
   \hat{p} \approx 0.48
   \]
   So, the sample proportion (rounded to two decimal places) is **0.48**.

2. **(b) Population Proportion (p)**:
   \[
   p = \frac{\text{Number of children in population}}{\text{Total population size}} = \frac{25}{56}
   \]
   Calculating this:
   \[
   p \approx 0.45
   \]
   So, the population proportion (rounded to two decimal places) is **0.45**.

### Answer:
- **(a) Sample Proportion**: \( \hat{p} = 0.48 \)
- **(b) Population Proportion**: \( p = 0.45 \)

Would you like further explanation on any step or more details?

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#### Follow-up Questions:
1. How does the sample proportion differ from the population proportion?
2. Why do we use different symbols (\( \hat{p} \) and \( p \)) for sample and population proportions?
3. What would be the effect of a larger sample size on the accuracy of \( \hat{p} \) in estimating \( p \)?
4. How would we calculate a confidence interval for this sample proportion?
5. In what situations might the sample proportion differ significantly from the population proportion?

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#### Tip:
Always ensure to label sample and population proportions correctly, as they represent data from different groups and serve different purposes in statistical analysis.

The Gazelles are a professional soccer team based in a large city. At the team's home matches, children's admission is half-off. At the last home match, 25 out of every 56 attendees were children. At that match, the facilities manager took a sample of the attendees. He found that 43 of the 89 attendees in his sample were children. For the facilities manager's sample, find and write with proper notation the sample proportion and population proportion of attendees who were children. Write the proportions as decimals (not percentages) rounded to two decimal places.

Learn how to find the sample and population proportion of children attendees at a soccer match for the Gazelles. This example calculates the sample proportion from a group of 89 attendees and the population proportion from a ratio of 25 children per 56 total attendees. Suitable for students in Grades 9-10.

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