Math Problem Statement

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.

Solution

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): p^=Number of children in sampleTotal sample size=4389\hat{p} = \frac{\text{Number of children in sample}}{\text{Total sample size}} = \frac{43}{89} Calculating this: p^0.48\hat{p} \approx 0.48 So, the sample proportion (rounded to two decimal places) is 0.48.

  2. (b) Population Proportion (p): p=Number of children in populationTotal population size=2556p = \frac{\text{Number of children in population}}{\text{Total population size}} = \frac{25}{56} Calculating this: p0.45p \approx 0.45 So, the population proportion (rounded to two decimal places) is 0.45.

Answer:

  • (a) Sample Proportion: p^=0.48\hat{p} = 0.48
  • (b) Population Proportion: p=0.45p = 0.45

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


Follow-up Questions:

  1. How does the sample proportion differ from the population proportion?
  2. Why do we use different symbols (p^\hat{p} and pp) for sample and population proportions?
  3. What would be the effect of a larger sample size on the accuracy of p^\hat{p} in estimating pp?
  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?

Tip:

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

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Math Problem Analysis

Mathematical Concepts

Statistics
Proportion
Sample Proportion
Population Proportion

Formulas

Sample Proportion (p̂) = (Number of children in sample) / (Total sample size)
Population Proportion (p) = (Number of children in population) / (Total population size)

Theorems

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Suitable Grade Level

Grades 9-10