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When we talk about the sampling distribution, we are referring to the distribution of a statistic, like the sample mean, over many samples. It helps us understand how the sample mean might vary across different random samples drawn from the same population. A crucial point to remember is that the size of the sample plays an essential role in determining this distribution's shape.
For smaller sample sizes, especially from skewed populations, the sampling distribution might not appear normal. Its shape can maintain the skewness of the original population, so if your population is skewed to the left, so may be your sampling distribution for a small sample.
However, as your sample size increases, the Central Limit Theorem tells us that this distribution will tend to be normal. So, even from a skewed population, a large enough sample size will give a normal sampling distribution.

The sample mean is simply the average of the observed values in a sample. It's an important statistic because it estimates the population mean. It acts as the center of your sampling distribution.
For any given sample, the sample mean may slightly depend on which values were chosen, making it a variable across different samples. The more samples we take, the better our understanding of how the sample mean behaves.
With smaller samples, the sample mean might more closely reflect the population's skewness, but with larger samples, the Central Limit Theorem ensures that the sample mean's distribution trends toward normality. This makes the sample mean a reliable estimator for the population mean with sufficient sample size.

Population skewness refers to how much a population distribution deviates from symmetry. A skewed distribution can be tilted left or right. In the given exercise, the population is skewed to the left, meaning it has a longer tail on the left side.
Skewness affects how sample statistics might behave, particularly in small samples. When you draw a small sample from a skewed population, that sample is more likely to maintain the skewness of the population distribution because the small sample size can’t compensate for the population’s asymmetry.
However, as the sample size increases, the effect of skewness on the sampling distribution diminishes due to the Central Limit Theorem. This means that even if the original population is skewed, the larger sample sizes result in a sampling distribution that is more normal.

The standard error is a measure of how much the sample mean can be expected to vary from the population mean. It acts as the spread in the sampling distribution. The formula for standard error is given by \( \frac{\sigma}{\sqrt{n}} \), where \( \sigma \) is the population standard deviation, and \( n \) is the sample size.
This equation tells us two important things:

• The standard error decreases as the sample size \( n \) increases. This means that with larger samples, the sample mean is likely to be closer to the population mean.
• The population standard deviation \( \sigma \) also plays a role. A larger \( \sigma \) indicates more spread in the population, and thus more potential variability in the sample mean.

In summary, a larger sample provides a more precise estimate because it reduces the variability captured by the standard error.