The Central Limit Theorem (CLT) states that if we take repeated samples of size n from any population with a mean μ and standard deviation σ, then the sampling distribution of the sample means will have an approximate normal distribution, regardless of the underlying distribution of the population. This holds true as long as n is sufficiently large (usually n ≥ 30).

In this case, our population has a mean µ = 40 and a standard deviation σ = 12. If we were to draw samples of size 100 from this population, then according to the CLT, the sample means would be approximately normally distributed with an expected mean equal to 40 and a standard deviation equal to 1.2 (=12/sqrt(100)). Therefore, the bulk of our sample means would lie within approximately two standard deviations (or 24 units) away from our original mean: between 16 and 64. In other words, 95% of our sample means would be between 16 and 64 for sufficiently large samples sizes.

As far as drawing valid inferences about this population goes, these sampling distributions play an important role in hypothesis testing since they provide information on how likely it is that we could get such results based on chance alone. Generally speaking, if our results are unlikely according to these sampling distributions than it may indicate that there is something else at play causing us to see these trends in data beyond mere chance variability- or in other words that there may actually be some kind of relationship between our variables or effects present in the data even when controlling for randomness.

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