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The sampling distribution is a critical concept in statistics. It refers to the probability distribution of a given statistic based on a random sample. For Jack and Diane, the statistic of interest is the sample mean.
When Jack and Diane draw multiple samples, they each calculate the sample mean for those samples. These sample means form their own distributions. This is what we call the 'sampling distribution of the sample mean' (or sometimes just 'sampling distribution' for brevity).
For Jack, who has a sample size of 3, his sampling distribution will likely reflect the skewness present in the population because smaller sample sizes tend to retain more characteristics of the original population. Hence, Jack’s sampling distribution is expected to be skewed left, similar to the original skewed-left population distribution.
In contrast, Diane’s larger sample size of 30 means that, according to the Central Limit Theorem, her sampling distribution will be closer to a normal distribution, even if the original population is skewed. The larger sample size helps in averaging out the skewness, leading to a more symmetric, bell-shaped distribution.

The mean of the sample means (also known as the expected value) is a fascinating property. No matter how skewed the population is, the mean of the sample means is always equal to the population mean, as long as the samples are random and unbiased.
For both Jack and Diane, irrespective of their sample sizes, the mean of the sample means will always be the same as the population mean. Since the population mean is given as 50, we can confidently say that the mean of Jack’s distribution of sample means and Diane’s distribution of sample means will both be 50.

The standard error (SE) is the term we use for the standard deviation of the sampling distribution. It gives us an idea of how much the sample means are expected to vary around the population mean.
The formula for the standard error of the sample mean is: \ \( \text{SE} = \frac{\text{Population Standard Deviation}}{\text{Square Root of Sample Size}} \)
For Jack, with a sample size of 3:
\ \( \text{SE}_{Jack} = \frac{10}{\text{sqrt}(3)} \approx 5.77 \)
For Diane, with a sample size of 30:
\ \( \text{SE}_{Diane} = \frac{10}{\text{sqrt}(30)} \approx 1.83 \)
Notice how Diane’s standard error is much smaller. This is because a larger sample size reduces the variability of the sample means. Hence, Diane’s sample means will be clustered more tightly around the population mean, while Jack’s will be more spread out.