Question

A random sample is selected from a population with mean \(\mu=100\) and standard deviation \(\sigma=10 .\) Determine the mean and standard deviation of the \(\bar{x}\) sampling distribution for each of the following sample sizes: a. \(n=9\) d. \(n=50\) b. \(n=15\) e. \(n=100\) c. \(n=36\) f. \(n=400\)

Short answer

Various sample sizes have the same mean of the sampling distribution (\(\mu_{\bar{x}}\)) which is 100. However, they have different standard deviations (\(\sigma_{\bar{x}}\)), following this pattern: n=9 (\(\sigma_{\bar{x}} = 3.33\)), n=15 (\(\sigma_{\bar{x}} = 2.58\)), n=36 (\(\sigma_{\bar{x}} = 1.67\)), n=50 (\(\sigma_{\bar{x}} = 1.41\)), n=100 (\(\sigma_{\bar{x}} = 1\)), n=400 (\(\sigma_{\bar{x}} = 0.5\)).

Step-by-step solution

Calculate Mean of the Sampling Distribution

The mean of the sampling distribution is equal to the mean of the population. So, for each sample size (n=9, n=15, n=36, n=50, n=100, n=400), the mean of the sampling distribution of \(\bar{x}\) (\(\mu_{\bar{x}}\)) is \(\mu = 100\).

Calculate Standard Deviation of the Sampling Distribution

Calculate the standard deviation (standard error) of the sampling distribution, which is the population standard deviation divided by the square root of the sample size. The formula is \(\sigma_{\bar{x}} = \sigma / \sqrt{n}\). Apply this formula for each sample size to find their respective standard deviations.For n=9, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{9} = 10/3 = 3.33\).For n=15, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{15} = 2.58\).For n=36, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{36} = 10/6 = 1.67\).For n=50, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{50} = 1.41\).For n=100, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{100} = 10/10 = 1\).For n=400, the standard deviation is \(\sigma_{\bar{x}} = 10 / \sqrt{400} = 10/20 = 0.5\). These are the standard errors for sampling distributions with various sample sizes.

Key concepts

Mean of Sampling Distribution

Understanding the mean of the sampling distribution is crucial in statistics. The mean, often denoted as \(\mu_{\bar{x}}\), of a sampling distribution is the average of the sample means over all possible samples. It's a pivotal concept because it remains constant no matter how large the sample size; it's always equal to the population mean \(\mu\).

In our exercise, regardless of the sample size (whether it's 9 or 400), the mean of the sampling distribution stays at 100, the same as the population mean. This equality is key, as it reassures us that, on average, our sample means are accurate representations of the center of the population.

Standard Deviation of Sampling Distribution

The standard deviation of the sampling distribution also known as the standard error, measures how much variation or dispersion exists from the mean of the sampling distribution. A lower standard deviation indicates that the sample means are clustered closely around the population mean, which is preferred.

Using the provided formula \(\sigma_{\bar{x}} = \sigma / \sqrt{n}\), we can calculate the standard deviation of the sample means for different sample sizes. For instance, a sample size of 9 results in a standard deviation of 3.33. As we increase the sample size, the standard deviation decreases, which implies that the sample means are becoming a more precise estimate of the population mean. It's important to note that this variability decreases at a decreasing rate as the sample size increases.

Central Limit Theorem

The Central Limit Theorem (CLT) is a statistical theory that explains the shape of the sampling distribution. It asserts that, assuming the sample size is sufficiently large, the sampling distribution of the sample mean will be approximately normally distributed, no matter the shape of the population distribution.

Under the CLT, even if the original population is not normally distributed, the sampling distribution of the mean tends to become more normal as the sample size \(n\) increases. This is especially significant for our current exercise, where we see that as the sample size grows (e.g., from 9 to 400), the standard deviation gets smaller, and the distribution of sample means would become more tightly packed and more symmetrical around the mean, satisfying the conditions of normality assumed by the theorem.

Standard Error

The standard error is essentially the standard deviation of the sampling distribution of a statistic, most commonly the mean. It signifies the average distance that the observed values fall from the actual population mean. A smaller standard error implies that the observed data points are closer to the true population mean.

In the exercise, for varying sample sizes, we calculated the standard error, which decreased as the sample size increased. For example, a sample size of 9 yielded a standard error of 3.33, whereas a sample size of 400 had a much smaller standard error of 0.5. This reduction in standard error with increased sample size indicates that larger samples lead to more reliable estimates of the population parameters.