https://file.aiquickdraw.com/mathbot/file/1726510058397oad3sids.jpg

Based on the image you've uploaded, here is an explanation of the key components:

1. **Symbols and Definitions:**
   - \( p = 11\% \) : This is the population proportion, representing the percentage of left-handed college students in the population.
   - \( n = 225 \) : This is the sample size, indicating the number of randomly selected college students.
   - \( N = 5625 \) : This is the population size, representing the total number of college students.

2. **Identifying Individual, Variable, Random Variable X, and \( \hat{p} \):**
   - **Individual:** A randomly selected college student.
   - **Variable collected from each individual:** The response you need here likely pertains to whether the student is left-handed or not. 
   - **Random Variable X:** Defined as the number of left-handed students in a sample of 225.
   - **Random Variable \( \hat{p} \):** Defined as the percentage of left-handed students in the sample of 225.

3. **Normality of Sampling Distribution:**
   - The condition for normality is checked using:
     - \( np = 24.75 \), which is greater than 10.
     - \( n(1 - p) = 200.25 \), which is also greater than 10.
     - Since both conditions are met, the sampling distribution of \( \hat{p} \) is approximately normal.

Would you like more details on how to interpret these symbols, or would you like me to clarify any steps?

Here are 5 related questions that could expand on this concept:
1. What is the role of the sample size \( n \) in determining the sampling distribution of \( \hat{p} \)?
2. Why is it important for \( np \) and \( n(1 - p) \) to be greater than 10 for the normal approximation to hold?
3. How would you calculate the standard error for the sample proportion \( \hat{p} \)?
4. How would the results change if the population size \( N \) were unknown or significantly larger than \( n \)?
5. How could you use the sampling distribution of \( \hat{p} \) to construct a confidence interval for the population proportion?

**Tip:** The larger the sample size, the closer the sampling distribution of \( \hat{p} \) will be to a normal distribution, due to the Central Limit Theorem.

List the given information with the correct symbols: p = 11%, n = 225, N = 5625. Identify the individual, variable, random variable X, and the random variable p̂ in the context of the problem. Can we say that the sampling distribution of p̂ is approximately normal?

This problem involves determining whether the sampling distribution of the sample proportion p̂ is approximately normal. It covers key statistical concepts like population proportion, sample size, and checks for normality using conditions np > 10 and n(1 - p) > 10. The scenario involves 225 randomly selected college students, with a population proportion of 11% and a total population of 5625.

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