Sampling, central limit theorem, standard error

Topic 5

Sampling

Process of selecting items from a population - saves time, money etc. and we want to generalise from the sample to the population to allow for the use of inferential techniques

Probability sample

A sample selected such that each item or person in the population being studied has a known likelihood of being included in the sample

Simple random sampling

Sample selected such that each item/person has the same known probability of being included e.g. random number generator

Systematic random sampling

A random starting point is selected and then every kth member of the population is selected e.g. every 10th customer entering a supermarket

Stratified random sampling

A population is divided into subgroups called strata in the appropriate proportions and a sample is randomly selected from each stratum

Cluster sampling

Population divided into clusters using naturally occurring geographic/other boundaries e.g. local authorities. Then clusters are randomly selected AND a sample is collected by randomly selecting from each cluster

Sample statistics

Data collected about sampled objects e.g. sample mean/sample standard deviation

Sample stats are random variables that can be described with distributions (sampling distributions)

Sampling distribution

Probability distribution of a statistic for all possible samples of a given size n

E.g. finding sampling distribution of the sample mean for samples of size 2

where the population is 7

Do 7 choose 2 - on calculator - = 21 which is the number of possible samples

Then find mean of sample means \mu_{\overline{x}} = Sum of sample means/Number of samples

Central Limit Theorem

If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution - especially with larger sample sizes

Although if the WHOLE population follows a normal probability distribution then the sampling distribution of the sample mean will also be normal, irrespective of the size of the sample

Standard error of the mean

The standard deviation of the sampling distribution of the sample mean

\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}

As sample size n increases the dispersion of the sampling distribution decreases

Why is the central limit theorem useful?

Even if a populations follows a non-normal distribution as long as n is large the central limit theorem guarantees that the SAMPLING distribution of the mean follows a normal distribution

z value of sample mean when variance is known

z=\frac{\left(\overline{x}-\mu\right)}{\frac{\sigma}{\sqrt{n}}}

Numerator = sampling error (difference between a sample mean and a population mean)

Denominator = standard error of the sampling distribution of the sample mean