Statistical Hypothesis Testing

Simple random sampling is a procedure where every possible sample (of a given size) has an equal chance of being selected.

Most common sampling techniques do not produce a simple random sample. E.g. is is common to include equal numbers of each gender when selecting a sample for a social science study.  Common methods for generating a simple random sample include lottery machines and random number generators.

Worked Example: A student wants to take a sample of students frome his college. He has a list of all students, numbered 1 to 478. He uses a random number generator on his calculator, which can generate three-digit random numbers between 001 and 999, inclusive. The first ten numbers he obtains are:

237, 155, 623, 078, 523, 078, 003, 554, 263

Suggest which could be the first four students in his sample.

237, 155, skip 623 as there is no student with is number, 078, skip 523 as there is no student with this number and skip 078 as it has already been selected, 003

Opportunity sampling involves chosing respondents based upon their availability and convienience.

This does not produce a random sample but it may produce a good estimate of the population parameters that you are investigating. However, in some situations it can introduce bias if the group consists of very similar members.

Worked Example: Del wishes to take a sample of residents from her neighbourhood. She decides to ask some people waiting at the bus stop

Is the sample likely to be representative if the question is aboout

a) attitudes to the enivironment - The sample may not be representative because people wwwho use puplic transport are arguably more likely to have 'green' attitudes

b) their favourite football team - The sample could be representative as there is no obvious link between the use of public transport and football 

A simple random sample might just happen to include only people from London, or only people with the first name John. If these outcomes would be problematic, an alternative is systematic sampling. This requires a list of all participants ordered in some way.

Systematic sampling means taking participants at regular intervals from a list of the population, with the starting point chosen at random.

Worked Example: A sample is formed by taking a telephone book and calling the person at the top of each page. The calls are made at between 10 a.m. and 2 p.m. on a Wednesday to enquire about the number of children in the household.

Explain why this is not simple random sampling - Not all samples are equally likely e.g. the sample with all the people at the bottom of each page has zero probability of being selected.

Suggest a reason why the mean value calculated will be biased - People without children may be working at this time, so they may not answer. This would mean that the mean is higher than it should be.

You need to decide in advance which factors you thinkn might be important. You seperate the population by these factors and within each group you take a simple random sample. the size of each sample is in proportion to the size of the group.

Stratified sampling is splitting the population into groups based on factors relevant to the research, then random sampling from each group in proportion to the size of that group.

Worked Example: A school is made up of 250 girls and 150 boys. A sample size of 80 is to be chosen, stratified between boys and girls. How many girls must be included in the sample?

The proportion of the school which is girls is 250/(250+150) = 5/8

5/8 of 80 is 50. 50 girls should be included

Stratified sampling isn't always practical as you need to have access to every member of the population to make a random sample. A common alternative is to use opportunity sampling instead of simple random sampling within each group.

Quota sampling is splitting the population inot groups based on factors relevant to the research, then opportunity sampling from each group until a required number of participants are found.

Worked Example: A market researcher is required to sample 100 men and 100 women in a supermarket to find out how much they are spending on that day.

Explain why this method is used rather than stratified sampling - The researcher would have to know in advance who was going to be shopping on that day to create a random sample

State one disadvantage of this method - The people who stop to talk to the researcher might not be representative

One of the main concerns in real-world sampling is cost. Creating a list of all members of a population and travelling or contacting the sample may be very difficult and expensive. One method that tries to make the process more efficient is cluster sampling. Like stratified sampling, this involves splitting the population into groups, called clusters. Unlike stratified sampling, these clusters do not have to be based on factors relevant to the research. In cluster sampling only some of the clusters are chosen to be studied. This makes it less accurate than stratified sampling, as choosing an unrepresentative cluster can have a large effect on the outcome.

Cluster sampling is splitting the population into clusters based on convenience, then randomly choosing some clusters to study further.

Worked Example: Jacob wants to estimate the percentage of people in the UK who travel to work by train. He selects five local authorities at random and uses their information to work out the mean.

Describe the difference between this sampling and a stratified sample across local authorities - Only some local authorities are chosen in this sample. If it were a stratified sample, values would be chosen from all local authorities and combined in proportion to the size of the local authority.