1. **Problem statement:** Explain how the histograms of simulated samples demonstrate the Central Limit Theorem (CLT) for sample means and comment on the "rule of thumb" about sample size 30. 2. **Central Limit Theorem (CLT) formula and explanation:** The CLT states that for a sufficiently large sample size $n$, the sampling distribution of the sample mean $\bar{X}$ will be approximately Normal, regardless of the population distribution, with mean $\mu$ and standard deviation $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$. 3. **Explanation of histograms and CLT:** - As sample size $n$ increases, the histograms of sample means become more symmetric and bell-shaped, approaching a Normal distribution. - The peak of the histogram increases (distribution becomes more concentrated around the mean). - The variability (spread) of the sample means decreases because $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$ gets smaller. 4. **Answer for part b:** The simulations **demonstrate** what the Central Limit Theorem says about the sampling distribution model for sample means because, as the sample size increases, the peak of the histogram **increases**, the shape of the distribution **approaches Normal**, and the variability in the sample means **decreases**. 5. **Comment on the rule of thumb (part c):** The "rule of thumb" that "With a sample size of at least 30, the sampling distribution of the mean is Normal" is generally true because by $n=30$, the sampling distribution is close enough to Normal for most practical purposes, even if the population distribution is not Normal. **Final summary:** The histograms show that larger sample sizes produce sample means that are more Normally distributed with less variability, confirming the CLT and supporting the rule of thumb about sample size 30.