"Answer the following questions for the sampling distribution of the sample mean shown to the right.

a. What is the value of μx̄?"

Verified step by step guidance

1. Identify that the problem is asking for the mean of the sampling distribution of the sample mean, denoted as \(\mu_{\bar{x}}\).
2. Recall the key property of the sampling distribution of the sample mean: its mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\).
3. Look at the graph provided, which shows a normal distribution centered at 500. This center point represents the mean of the sampling distribution.
4. Therefore, the value of \(\mu_{\bar{x}}\) corresponds to the center of the distribution, which is 500 in this case.
5. Summarize that \(\mu_{\bar{x}} = \mu = 500\) based on the graph and the properties of the sampling distribution.

Key Concepts

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean describes the distribution of sample means from all possible samples of a given size drawn from a population. It shows how the sample mean varies and is typically approximately normal if the sample size is large enough, regardless of the population distribution.

Mean of the Sampling Distribution (μx̄)

The mean of the sampling distribution of the sample mean, denoted μx̄, is equal to the population mean μ. This means the average of all sample means will center around the true population mean, making the sample mean an unbiased estimator of the population mean.

Interpreting the Graph of the Sampling Distribution

The graph shows a normal distribution centered at 500, with values 480 and 520 marked as reference points. The center of the curve represents the mean of the sampling distribution, which is the population mean. Understanding this helps identify μx̄ directly from the graph.