Inferential Statistics 1:Inferential Statistics 1:Basic ConceptsBasic Concepts

Fortrose Academy Geography Fortrose Academy Geography DepartmentDepartment

Advanced HigherAdvanced Higher

SamplingSampling

There are three standard sampling There are three standard sampling techniques:techniques:RandomRandomSystematicSystematicStratifiedStratified

Random SamplingRandom Sampling

This is where each item from a population This is where each item from a population has an equal chance of being selected.has an equal chance of being selected.

To select from this population each To select from this population each member is assigned a number using member is assigned a number using computer generated random number computer generated random number tables.tables.

Once a number has been chosen it can be Once a number has been chosen it can be related to a grid reference, a angular related to a grid reference, a angular direction, a distance or whatever else we direction, a distance or whatever else we are sampling.are sampling.

Systematic SamplingSystematic Sampling

This is where samples are This is where samples are selected in a regular manner. For selected in a regular manner. For example:example:taking a vegetation sample every 10 taking a vegetation sample every 10

metres (linear sample), metres (linear sample), taking a sample at a serise of points taking a sample at a serise of points

located at the intersection of a 10 metre located at the intersection of a 10 metre grid (point sampling),grid (point sampling),

selecting every tenth customer at the selecting every tenth customer at the supermarket supermarket

etc…etc…

Stratified SamplingStratified Sampling This takes into account the relative proportion This takes into account the relative proportion

of different groups within the sample.of different groups within the sample.

For example in a sand dune investigation two For example in a sand dune investigation two thirds of the dune may be managed and the thirds of the dune may be managed and the rest unmanaged.rest unmanaged.

Stratified sampling will select a reprehensive Stratified sampling will select a reprehensive sample from each of the two areas.sample from each of the two areas.

If nine transects are to be selected then the If nine transects are to be selected then the correct balance is six managed and three correct balance is six managed and three unmanaged.unmanaged.

Sample Size and BiasSample Size and Bias

Many investigations fail because the Many investigations fail because the size of the sample is too small and size of the sample is too small and this leads to unreliable results.this leads to unreliable results.

When collecting a sample the main When collecting a sample the main concern is to remove bias (eg: obtain concern is to remove bias (eg: obtain a representative sample)a representative sample)

HypothesisHypothesis

A hypothesis is a statement or a A hypothesis is a statement or a hunch.hunch.

To test a hypothesis the first thing we To test a hypothesis the first thing we do is write down a statement – called do is write down a statement – called the null hypothesis (written NH).the null hypothesis (written NH).

The null hypothesis is the The null hypothesis is the oppositeopposite of of what the researcher is trying to prove.what the researcher is trying to prove.

ExampleExample

We are interested in finding out if there is We are interested in finding out if there is any difference between the average any difference between the average number of Higher’s passed by S5&S6 number of Higher’s passed by S5&S6 pupils in Fortrose Academy and Dingwall pupils in Fortrose Academy and Dingwall Academy.Academy.

There may be a difference. BUT how big There may be a difference. BUT how big does are difference have to be before we does are difference have to be before we can say that it is a ‘real’ or ‘significant’ can say that it is a ‘real’ or ‘significant’ difference.difference.

Test StatisticsTest Statistics

To answer this we calculate a figure known To answer this we calculate a figure known as a as a test statistictest statistic, which is based in data , which is based in data from our samples.from our samples.

Different types of problems require Different types of problems require different test statistics these have all been different test statistics these have all been put into put into statistical tablesstatistical tables..

All we need to do is to calculate our value All we need to do is to calculate our value and compare it with the value in the table and compare it with the value in the table to get our answer.to get our answer.

Null HypothesisNull Hypothesis

‘‘There is There is no differenceno difference between the between the average number of Higher’s passed at average number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘Fortrose Academy and Dingwall Academy. ‘

If you are proved correct then you reject the If you are proved correct then you reject the null hypothesis and accept the alternative null hypothesis and accept the alternative

hypothesis (AH) which would be:hypothesis (AH) which would be:

‘‘There There is a differenceis a difference between the average between the average number of Higher’s passed at Fortrose number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘Academy and Dingwall Academy. ‘

Significance (1)Significance (1)

Before carrying out the test we have Before carrying out the test we have to decide on a to decide on a significance levelsignificance level which lets us determine at what which lets us determine at what point to point to rejectreject the null hypothesis the null hypothesis and accept the alternative and accept the alternative hypothesis.hypothesis.

Significance (2)Significance (2)

Significance is based on the Significance is based on the probability of chance.probability of chance.

Difference between the average number of Difference between the average number of Higher’s passed by S5&S6 pupils and Fortrose Higher’s passed by S5&S6 pupils and Fortrose

Academy and Dingwall Academy.Academy and Dingwall Academy.

Probability of chanceProbability of chance

Statisticians have calculated the Statisticians have calculated the probability of ‘chance’ events probability of ‘chance’ events occurring that may affect our results.occurring that may affect our results.

They have come to the conclusion They have come to the conclusion that, if the probability that an event that, if the probability that an event could occur by chance is less than 1 could occur by chance is less than 1 in 20, they say the result is in 20, they say the result is significantsignificant. ie: The result is . ie: The result is notnot just just a chance event.a chance event.

Critical Values In UseCritical Values In Use

From your calculations and the From your calculations and the significance tables you will find that significance tables you will find that if the value of the test statistic you if the value of the test statistic you have calculated is have calculated is greatergreater than the than the value in the table (the critical value), value in the table (the critical value), you can reject the null hypothesis you can reject the null hypothesis and accept the alternative and accept the alternative hypothesis.hypothesis.