To determine which of the histograms (A, B, C) represents the sampling distribution of the sample mean (\(\bar{x}\)) for sample sizes of 3, 10, and 30, it is important to consider the Central Limit Theorem (CLT) and the characteristics of sampling distributions based on sample size and population distribution shape.
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Histogram A (n=3): With a small sample size like 3, the sampling distribution can still reflect the skewness and shape of the underlying population distribution. Therefore, if the population is highly skewed, the histogram for this sample size will also likely appear skewed.
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Histogram B (n=10): At a sample size of 10, the sampling distribution will begin to approximate normality, especially if the population distribution is not too skewed. However, any pronounced skewness in the population might still be visible in the sampling distribution.
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Histogram C (n=30): With a sample size of 30 or greater, the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal regardless of the population's distribution shape. Therefore, this histogram should exhibit a normal shape, even if the original population is skewed.
Conclusion:
- Histogram for n=3 should look similar to the population and thus may be skewed (likely Histogram A).
- Histogram for n=10 will have a distribution still somewhat influenced by the population but starting to show signs of normality (likely Histogram B).
- Histogram for n=30 should be approximately normal (likely Histogram C).
To confirm which histogram corresponds to each sample size, you would need to see the specific shapes depicted in Histograms A, B, and C to match them appropriately based on the reasoning outlined above. Without the visual context of the histograms in question, one can't definitively assign the histograms to the sample sizes.