Let's break down the three parts of the problem, analyzing the changes in shape, center, and spread of the sampling distribution model for sample proportions as sample size increases.

Part 1: Shape

• Correct answer: B. The distributions for small samples are right-skewed but are more symmetric as n increases.

  Explanation: For small sample sizes, the distribution of sample proportions tends to be right-skewed. This occurs because when there are fewer observations, extreme proportions (like very few or none responding) are more likely, skewing the distribution to the right. As the sample size increases, the distribution becomes more symmetric and approaches the shape of a normal distribution due to the Central Limit Theorem.

Part 2: Center

• Correct answer: D. The center does not change with sample size; all the histograms are centered near 0.05.

  Explanation: The sample proportion's expected value is the population proportion, which is 0.05 in this case. Regardless of the sample size, the center of the sampling distribution of sample proportions remains close to the population proportion. Hence, all histograms are centered around 0.05, as the center is not significantly impacted by changes in sample size.

Part 3: Spread (Variability)

• Correct answer: C. As n gets larger, the histograms approach the Normal shape, and the variability in the sample proportions decreases.

  Explanation: As sample size increases, variability in the sample proportions decreases. This is because larger samples tend to provide more stable estimates of the population proportion. Smaller sample sizes lead to more variability (spread), whereas larger samples result in less variability. Additionally, the shape becomes more normal as $n$ increases, again due to the Central Limit Theorem.

Summary

• Shape: The distribution is right-skewed for small sample sizes but becomes more symmetric (and normal-like) as $n$ increases.
• Center: The center stays around the population proportion of 0.05, regardless of sample size.
• Spread: The variability decreases with larger sample sizes, making the sample proportions more concentrated around the true population proportion.

Would you like further details or have any questions about these concepts?

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Here are five related questions to expand your understanding:

1. How does the Central Limit Theorem apply to the sampling distribution of proportions?
2. What happens to the standard error of the sample proportion as sample size increases?
3. Why does a small sample size lead to more variability in the sample proportion?
4. How can we calculate the expected spread (standard error) for a given sample size $n$ in a proportion?
5. What is the effect of changing the population proportion on the shape of the sampling distribution?

Tip: As the sample size increases, the spread of the sampling distribution narrows. You can calculate this spread (standard error) using $\sqrt{\frac{p(1 - p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size.