Understanding Sample Means and Standard Error: A Practical Example

When working with statistical data, it is essential to understand the concepts of sample means and standard errors. This article explores a practical scenario involving a random sample with a specific sample size drawn from a population with a given mean and standard deviation.

Scenario Definitions

Imagine a scenario where a random sample of size 25 is drawn from a population. The population has a mean of 30 and a standard deviation of 8. The goal is to determine the mean and standard error of the sample means. This exercise serves to reinforce the concepts of sample statistics and their relationship to population parameters.

Calculation of the Standard Error

The standard error (SE) of the sample means is calculated using the formula:

SE σ / √n

where σ is the population standard deviation and n is the sample size.

Given Data:

Population mean (μ) 30 Population standard deviation (σ) 8 Sample size (n) 25

Calculation:

SE 8 / √25 8 / 5 1.6

Interpreting the Sample Mean in the Context of the Population

While the sample mean does not provide a definitive value for the population mean, it does offer a measure around which the population mean is likely to cluster. By constructing a confidence interval, we can estimate the range within which the true population mean lies with a certain level of confidence.

Constructing the Confidence Interval

The confidence interval (CI) is calculated using the t-distribution, where the t-value for a 95% confidence level with 24 degrees of freedom is approximately 2.064 (using the sample size minus one).

The formula to construct the CI is:

CI μ ± (t * SE)

Calculation:

CI 30 ± (2.064 * 1.6) 30 ± 3.3024

This results in:

Lower CI limit: 30 - 3.3024 26.6976

Upper CI limit: 30 3.3024 33.3024

Therefore, the 95% confidence interval for the population mean is approximately 26.6976 to 33.3024.

Key Concepts

Sample Mean: The mean of a sample, which is used as an estimate of the population mean.

Standard Error: The standard deviation of the sample mean, which measures the variability of the sample means around the population mean.

Confidence Interval: A range of values that is likely to contain the true population mean with a specified level of confidence.

Conclusion

This example illustrates the practical application of sample means and standard errors to estimate population parameters. By understanding these statistical measures, we can make more informed decisions about the data we collect and use them to draw meaningful conclusions about the larger population.

Related Keywords

Sample Mean Standard Error Population Mean Confidence Interval

Further Reading

For more detailed information and examples, consult statistical textbooks or online resources such as Statistics How To, which provides comprehensive guidance on these topics.