SUMMARY
The sampling distribution of the sample mean is defined by the formula $$\bar{x}\sim N\left(\mu,\frac{\sigma}{\sqrt{n}}\right)$$ when the population standard deviation ($\sigma$) is known. In cases where $\sigma$ is unknown, the student-$t$ distribution should be utilized instead of the normal distribution. This distinction is crucial for accurately determining the sampling distribution and solving related statistical problems.
PREREQUISITES
- Understanding of normal distribution and its properties
- Knowledge of the Central Limit Theorem
- Familiarity with student-$t$ distribution
- Basic statistical concepts such as population mean ($\mu$) and standard deviation ($\sigma$)
NEXT STEPS
- Study the Central Limit Theorem in depth
- Learn about the properties and applications of the student-$t$ distribution
- Explore practical examples of sampling distributions in real-world scenarios
- Review statistical software tools for calculating sampling distributions, such as R or Python's SciPy library
USEFUL FOR
Students, educators, and professionals in statistics or data analysis who need a clear understanding of sampling distributions and their applications in hypothesis testing and confidence interval estimation.
