To assess whether the plotted data distribution for these 1,000 means approximates a Normal distribution and to determine if this confirms the first part of the Central Limit Theorem (CLT), let’s break down the key points:

1. Normal Distribution Approximation: The Central Limit Theorem states that the sampling distribution of the mean of a sufficiently large number of independent samples from any population with a finite mean and variance will approximate a Normal distribution, regardless of the shape of the original population distribution.

2. Visual Inspection: From the provided plot, we look for characteristics of a Normal distribution, such as symmetry around the mean, a bell-shaped curve, and concentration of values near the mean with fewer values in the tails.

3. Sample Size (n): If the plot shows an approximate Normal distribution, it would suggest that the sample size per mean calculation (Step 6 in your Python script) was likely large enough to invoke the CLT.

Confirming the Central Limit Theorem

If this distribution of sample means is approximately Normal, this confirms the first part of the CLT:

• The distribution of the means of a large number of samples tends towards Normality, given a sufficient sample size per mean calculation.
• This happens even if the original data distribution (from which individual samples were drawn) was not Normal.

The presence of an approximate Normal shape in this distribution indicates that:

• The sample size used per mean calculation was likely adequate for the CLT to apply.
• This result aligns with the expectation that means of large samples will form a Normal distribution.

If you'd like a deeper analysis or further questions, here are some directions:

1. Would you like to test the normality formally using a statistical test (e.g., Shapiro-Wilk test)?
2. What was the sample size per mean calculation? This can help confirm the adequacy based on CLT criteria.
3. How would changing the sample size per calculation affect the normality of the mean distribution?
4. Was the original population distribution Normal? If not, this adds value to confirming the CLT.
5. Are there other moments (e.g., skewness, kurtosis) that could be used to quantify the normality?

Tip: When working with the Central Limit Theorem, remember that larger sample sizes improve the approximation of the Normal distribution, especially when the original data is skewed.