91Ó°ÊÓ

A foundational concept in statistics involves understanding what a sampling distribution is. Imagine you are drawing many random samples from a larger population. For each of these samples, you compute a statistic, such as the mean. The sampling distribution is the probability distribution of that statistic.

In this problem, we deal with the sampling distribution of the sample mean Medicare spending. It's usually a common rule that when sample sizes are sufficiently large, the sampling distribution of the mean closely follows a normal distribution. This concept is grounded in the Central Limit Theorem, which states that irrespective of the population distribution shape, the distribution of the sample mean will approximate normality when the sample size is large, typically over 30.

In our case of 50 samples, it guarantees that the sampling distribution of the mean is approximately normal. This allows us to use normal distribution characteristics to find probabilities and make inferences about the population.

The standard error is crucial in inferential statistics, as it measures the dispersion or "spread" of a sampling distribution. While standard deviation refers to the variability of data points in a population, standard error shows how much the sample mean estimates might deviate from the actual population mean.

Mathematically, the standard error is calculated as: \[ SE = \frac{\sigma}{\sqrt{n}} \] where

• \( \sigma \) is the population standard deviation,
• \( n \) is the sample size.

In this scenario, the given population standard deviation is \( 2000 \), and the sample size \( n \) is \( 50 \). By using the formula, the standard error comes out to be approximately \( 282.84 \). This value demonstrates how much variability we can expect in the sample means due to random sampling.

Probability in the context of sampling distributions helps us estimate the likelihood of different outcomes related to sample means. More specifically, we can determine the probability of the sample mean falling within a range or exceeding a certain value.

For example, to find the probability that the sample mean is within \( \pm 300 \) of the population mean, we look at the interval of interest: \[ P(6583 \leq \bar{x} \leq 7183) \] By converting this range into z-scores using the formula: \[ z = \frac{x - \mu}{SE} \] we can find the corresponding probabilities from the standard normal distribution. The calculations show a probability of \( 0.7098 \), suggesting there's about a 71% chance that a sample mean would fall within this range.

Similarly, to determine the probability of a sample mean greater than \( 7500 \), the procedure uses the z-score and observes that such an outcome has a low probability of \( 0.0146 \). This indicates a less than 2% likelihood that the sample mean would exceed \( 7500 \) by pure chance, raising concerns about the sampling method's validity if such a result occurs.