Interpreting the Central Limit Theorem In Exercises 19–26, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.


Salaries The annual salaries for web software development managers are normally distributed, with a mean of about \$136,000 and a standard deviation of about \$11,500. Random samples of 40 are drawn from this population, and the mean of each sample is determined.

Verified step by step guidance

1. Identify the population mean (\$\(\mu\)\$) and population standard deviation (\$\(\sigma\)\$). Here, \$\(\mu\) = 136,000\$ and \$\(\sigma\) = 11,500\$.
2. Note the sample size \$n = 40\$ and recognize that the sampling distribution of the sample mean will have its own mean and standard deviation.
3. Calculate the mean of the sampling distribution of the sample mean, which is the same as the population mean: \$\(\mu\)_{\(\bar{x}\)} = \(\mu\) = 136,000\$.
4. Calculate the standard deviation of the sampling distribution of the sample mean (also called the standard error) using the formula: \$\(\sigma\)_{\(\bar{x}\)} = \(\frac{\sigma}{\sqrt{n}\)} = \(\frac{11,500}{\sqrt{40}\)}\$.
5. Since the population distribution is normal, the sampling distribution of the sample mean will also be normal. Sketch a normal curve centered at \$136,000\$ with spread determined by \$\(\sigma\)_{\(\bar{x}\)}\$.

Key Concepts

Central Limit Theorem (CLT)

The Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as the sample size becomes large, regardless of the population's distribution. For sample sizes typically greater than 30, the sample means will be approximately normally distributed, enabling inference about the population mean.

Sampling Distribution of the Sample Mean

The sampling distribution of the sample mean is the probability distribution of all possible sample means from samples of a fixed size. It has a mean equal to the population mean and a standard deviation (standard error) equal to the population standard deviation divided by the square root of the sample size.

Standard Error of the Mean

The standard error measures the variability of the sample mean from the population mean. It is calculated as the population standard deviation divided by the square root of the sample size, reflecting how much sample means fluctuate around the true mean in repeated sampling.