Sampling and estimation — FAQ

$\sigma^2$ is a fixed, usually unknown, parameter of the entire population. $s^2$ is a statistic calculated from a sample, used to estimate $\sigma^2$. We use $s^2 = \frac{1}{n-1} \sum (x_i - \bar{x})^2$ because it provides a more accurate (unbiased) estimate of $\sigma^2$ than if we were to divide by $n$.

You can apply the CLT when you are dealing with the distribution of the sample mean, $\bar{X}$, and the sample size, $n$, is large (typically $n > 30$). It's essential when the original population distribution is unknown or not normal.

Standard deviation ($\sigma$) measures the spread of data in a population. Standard error (specifically, the standard error of the mean, $\frac{\sigma}{\sqrt{n}}$) measures the typical distance between a sample mean and the population mean. It quantifies the uncertainty in your estimate of the mean.

If the population standard deviation, $\sigma$, is given in the question, use it. If it is not known, you must first calculate an unbiased estimate, $s$, from the sample data. In P6, you will typically be given $\sigma$ or be working with a large enough sample that $s$ is a reliable substitute.

It will be wider. To be more confident (99% vs 95%) that your interval contains the true mean, you need to allow for a larger range of possibilities. This is reflected in the larger z-value for 99% confidence (2.576) compared to 95% (1.96).