Inferential Statistics 1: Basic Concepts Fortrose Academy Geography Department Advanced Higher

Sampling There are three standard sampling techniques: There are three standard sampling techniques: Random Random Systematic Systematic Stratified Stratified

Random Sampling This is where each item from a population has an equal chance of being selected. This is where each item from a population has an equal chance of being selected. To select from this population each member is assigned a number using computer generated random number tables. To select from this population each member is assigned a number using computer generated random number tables. Once a number has been chosen it can be related to a grid reference, a angular direction, a distance or whatever else we are sampling. Once a number has been chosen it can be related to a grid reference, a angular direction, a distance or whatever else we are sampling.

Systematic Sampling This is where samples are selected in a regular manner. For example: This is where samples are selected in a regular manner. For example: taking a vegetation sample every 10 metres (linear sample), taking a vegetation sample every 10 metres (linear sample), taking a sample at a serise of points located at the intersection of a 10 metre grid (point sampling), taking a sample at a serise of points located at the intersection of a 10 metre grid (point sampling), selecting every tenth customer at the supermarket selecting every tenth customer at the supermarket etc… etc…

Stratified Sampling This takes into account the relative proportion of different groups within the sample. This takes into account the relative proportion of different groups within the sample. For example in a sand dune investigation two thirds of the dune may be managed and the rest unmanaged. For example in a sand dune investigation two thirds of the dune may be managed and the rest unmanaged. Stratified sampling will select a reprehensive sample from each of the two areas. Stratified sampling will select a reprehensive sample from each of the two areas. If nine transects are to be selected then the correct balance is six managed and three unmanaged. If nine transects are to be selected then the correct balance is six managed and three unmanaged.

Sample Size and Bias Many investigations fail because the size of the sample is too small and this leads to unreliable results. Many investigations fail because the size of the sample is too small and this leads to unreliable results. When collecting a sample the main concern is to remove bias (eg: obtain a representative sample) When collecting a sample the main concern is to remove bias (eg: obtain a representative sample)

Hypothesis A hypothesis is a statement or a hunch. A hypothesis is a statement or a hunch. To test a hypothesis the first thing we do is write down a statement – called the null hypothesis (written NH). To test a hypothesis the first thing we do is write down a statement – called the null hypothesis (written NH). The null hypothesis is the opposite of what the researcher is trying to prove. The null hypothesis is the opposite of what the researcher is trying to prove.

Example We are interested in finding out if there is any difference between the average number of Higher’s passed by S5&S6 pupils in Fortrose Academy and Dingwall Academy. We are interested in finding out if there is any difference between the average number of Higher’s passed by S5&S6 pupils in Fortrose Academy and Dingwall Academy. There may be a difference. BUT how big does are difference have to be before we can say that it is a ‘real’ or ‘significant’ difference. There may be a difference. BUT how big does are difference have to be before we can say that it is a ‘real’ or ‘significant’ difference.

Test Statistics To answer this we calculate a figure known as a test statistic, which is based in data from our samples. To answer this we calculate a figure known as a test statistic, which is based in data from our samples. Different types of problems require different test statistics these have all been put into statistical tables. Different types of problems require different test statistics these have all been put into statistical tables. All we need to do is to calculate our value and compare it with the value in the table to get our answer. All we need to do is to calculate our value and compare it with the value in the table to get our answer.

Null Hypothesis ‘There is no difference between the average number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘ ‘There is no difference between the average number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘ If you are proved correct then you reject the null hypothesis and accept the alternative hypothesis (AH) which would be: ‘There is a difference between the average number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘ ‘There is a difference between the average number of Higher’s passed at Fortrose Academy and Dingwall Academy. ‘

Significance (1) Before carrying out the test we have to decide on a significance level which lets us determine at what point to reject the null hypothesis and accept the alternative hypothesis. Before carrying out the test we have to decide on a significance level which lets us determine at what point to reject the null hypothesis and accept the alternative hypothesis.

Significance (2) Significance is based on the probability of chance. Significance is based on the probability of chance. Difference between the average number of Higher’s passed by S5&S6 pupils and Fortrose Academy and Dingwall Academy.

Probability of chance Statisticians have calculated the probability of ‘chance’ events occurring that may affect our results. Statisticians have calculated the probability of ‘chance’ events occurring that may affect our results. They have come to the conclusion that, if the probability that an event could occur by chance is less than 1 in 20, they say the result is significant. ie: The result is not just a chance event. They have come to the conclusion that, if the probability that an event could occur by chance is less than 1 in 20, they say the result is significant. ie: The result is not just a chance event.

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