Revise Sampling and estimation for Cambridge 9709 (6.4)

When we take a sample, we can calculate statistics like the sample mean and sample variance. Our goal is to use these sample statistics to estimate the corresponding population parameters (the population mean $\mu$ and population variance $\sigma^2$). For our estimate to be useful, it should be 'unbiased', meaning that on average, it gives the correct value. The sample mean $\bar{x}$ is an unbiased estimator for the population mean $\mu$.

Unbiased estimate of population mean $\mu$: $\hat{\mu} = \bar{x} = \frac{\sum x}{n}$

Slightly more complex is the estimate for population variance. If we simply calculated the variance of our sample, it would tend to underestimate the true population variance. To correct this, we use a slightly different formula with a denominator of $n-1$ instead of $n$. This is known as Bessel's correction.

Unbiased estimate of population variance $\sigma^2$: $s^2 = \frac{n}{n-1} \times (\text{sample variance}) = \frac{1}{n-1} \sum (x - \bar{x})^2 = \frac{1}{n-1} \left( \sum x^2 - \frac{(\sum x)^2}{n} \right)$

The Central Limit Theorem is a cornerstone of statistics. It tells us something remarkable about the distribution of sample means. Even if the original population is not normally distributed (e.g., it could be skewed or bimodal), the distribution of the means of samples taken from that population will be approximately normal, provided the sample size is large enough (usually $n > 30$ is a good rule of thumb). This allows us to use the normal distribution for calculations involving sample means, which is incredibly powerful.

If $X$ is any distribution with mean $\mu$ and variance $\sigma^2$, and $n$ is large, then the sample mean $\bar{X}$ has the approximate distribution: $\bar{X} \sim N \left( \mu, \frac{\sigma^2}{n} \right)$

An unbiased estimate like $\bar{x}$ gives us a single point value for $\mu$. A confidence interval provides a range of plausible values for $\mu$. A 95% confidence interval, for example, is a range constructed by a method that, in the long run, will capture the true population mean 95% of the time. For this part of the syllabus, we focus on the case where the population variance $\sigma^2$ is known, or where the sample size is large enough to use the unbiased estimate $s^2$ as a good approximation for $\sigma^2$.

A $C\%$ confidence interval for the population mean $\mu$ is given by: $\bar{x} \pm z \times \frac{\sigma}{\sqrt{n}}$ where $\bar{x}$ is the sample mean, $\sigma$ is the known population standard deviation, $n$ is the sample size, and $z$ is the critical value from the standard normal distribution for the given confidence level.