CHAPTER 21 Inferential Statistical Analysis

Understanding probability The idea of probability is central to inferential statistics. It means the chance or likelihood of a particular event occurring, or the proportion of times it will occur in a long series of trials. Statistical significance levels are probabilities. Student Activity 1

Testing hypotheses Formulate null and alternative hypotheses Select statistical technique and test statistic Decide on required level of significance Collect data Use test to determine significance level of results Decide whether to reject the null hypothesis Relate this decision to your research question, and draw conclusions Student Activity 2

What statistics do for you Statistical tests don’t prove anything – they merely indicate probabilities of statements about parent populations being true. It is possible to obtain false positives and false negatives. The power of a statistical test is 1–β: ie 1–the probability of a false positive.

Many statistics assume a normal curve Two examples of normal curves

Defining a normal curve A normal curve is a theoretical distribution. It is defined by two parameters: its mean and its standard deviation. The standard deviation (σ) indicates the variability of observations about the mean. 5% of the area under a normal curve lies more than approx. 2σ from the mean. 1% of the area under a normal curve lies more than approx. 2.5σ from the mean.

Normal curves and probability Probability areas on a normal curve

What statistics do for you You can use statistics to: estimate population values from a sample test for differences between samples test whether variables are related find a line which best fits your data

Estimating population values Sampling error is the difference between the (unknown) population value and your sample statistic. This is why you can only estimate population values from sample measures. You can work out the probability that they will fall within a certain range.

Testing for differences Key questions: Are samples paired or unpaired? A 1 B 1 C 1 D 1 E 1 F 1 G 1 H 1 I 1 ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ A 2 B 2 C 2 D 2 E 2 F 2 G 2 H 2 I 2 or A B C D E F G H I J K L M N O P Q R Does the direction of difference matter? Student Activity 3

Testing for association Correlation does not mean causation! A and B might be related because A caused B B caused A both are related to C the effect is due to chance

Choosing statistical tests The appropriate statistical test will depend upon: the question you are asking the nature of your variables the size of your sample. Tests are parametric or non- parametric.

Parametric tests Parametric tests are more powerful – ie more likely to identify a difference as significant. They assume interval or ratio data, a normal distribution and random samples with similar variance. If these conditions are not met or samples are too small, they can mislead.

Guidelines for statistical testing Choose an adequate sample. Choose an appropriate statistical test. Use a one-tailed test only if justified. Check that any computer output makes sense. Ensure that your conclusions are justified by the tests, and do not mislead. Student Activity 4