Final answer:
The standard error of the mean is a measure of the sampling variability representing the average variation of the sample mean from the population mean, while the distribution of the sample mean generally approaches a normal distribution as the sample size increases.
Step-by-step explanation:
The standard error of the mean is a measure of the sampling variability of the statistic. It represents how much the sample mean will vary from sample to sample. It is calculated as the population standard deviation (σ) divided by the square root of the sample size (n), denoted as σ/√n. The distribution of the sample mean will approximately follow a normal distribution if the sample size is large enough, according to the Central Limit Theorem.
To construct a confidence interval for the population mean, we need to define a margin of error, which is calculated using the standard error and the critical value from the relevant distribution, typically a t-distribution or a standard normal distribution, depending on whether the population standard deviation is known and if the sample size is sufficiently large.
The random variable X, in these scenarios, often represents a particular measure of interest, such as the number of letters sent home by campers or the weight of candies. The correct distribution to use for a particular problem depends on the situation, such as whether the data is normally distributed or if we are working with proportions.