1. C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview Parameters and Statistics Probabilities The Binomial Probability Test Probability Distributions Sampling Distributions Standard Errors

2. C82MCP Diploma Statistics School of Psychology University of Nottingham 2 Parameters When a population is described in terms of its mean, standard deviation, median etc. these values are known as parameters Parameters are numerical descriptors of the properties of the population Parameters can describe: The population’s central tendency The population’s variability The population’s form

3. C82MCP Diploma Statistics School of Psychology University of Nottingham 3 Statistics When a sample is described in terms of its mean, standard deviation, median etc. these values are known as statistics Statistics are numerical descriptors of the properties of the sample Statistics can describe The sample’s central tendency The sample’s variability The sample’sform

4. C82MCP Diploma Statistics School of Psychology University of Nottingham 4 Parametric Hypothesis Testing The purpose of parametric statistical testing is to make inferences about the population parameters on the basis of the sample statistics. Since we often don't know the precise population parameters we have to make judgements about how well the sample statistics estimate the population parameters. These judgements are made on the basis of probabilities.

5. C82MCP Diploma Statistics School of Psychology University of Nottingham 5 Parametric Hypothesis Testing Types of Parametric Tests We can test how representative a statistic is of its parent population’s parameter We can test whether different samples come from the same population Parametric Tests and Probabilities Both these kinds of tests rely on assuming that the population is distributed in a particular way. This assumption is necessary in order to calculate the probabilities associated with the different kinds of parametric tests

6. C82MCP Diploma Statistics School of Psychology University of Nottingham 6 Probabilities A probability is a measure of the likelihood that a particular event will occur In terms of mathematical theory a probability is defined more specifically as A number between zero and one, inclusive, that attaches to an event, and is assigned in such a way that the sum of all of the probabilities of all of the events within a given situation is one. NB. a probability cannot be less than one NB. a probability of one is a certainty

7. C82MCP Diploma Statistics School of Psychology University of Nottingham 7 Probabilities Probabilities do not necessarily tell us anything about the real world For Example: A die may be rolled six times and the six never appear even though the probability of a six on each roll of the die is 1/6 A die could be rolled six times and the six appear six times even though the probability of a six on each roll of the die remains 1/6 Probabilities reflect what can happen in the real world, they do not tell us what will happen.

8. C82MCP Diploma Statistics School of Psychology University of Nottingham 8 Probability Outcomes and Events An outcome is something that can occur in a given situation. An event is a collection of one or more outcomes The probability of a particular event is given by:

9. C82MCP Diploma Statistics School of Psychology University of Nottingham 9 Probability Rules The addition rule There's a simple rule for answering either/or questions about probabilities: p(event 1 or event 2) = p(event 1) + p(event 2) The multiplication rule The probability of two events happening is given the multiplication rule: p(event 1 and event 2) = p(event 1) x p(event 2)

10. C82MCP Diploma Statistics School of Psychology University of Nottingham 10 Binomial Probability Test Suppose that we drew a sample of two students from a psychology class The class is comprised of 75% women and 25% men We have a 0.75 probability of selecting a women and a 0.25 probability of selecting a man There are four possible outcomes when taking a sample of two from this class:female femalemale malefemalemale

11. C82MCP Diploma Statistics School of Psychology University of Nottingham 11 Binomial Probability Test For each of these outcomes we can calculate a probability using the probability multiplication rule

12. C82MCP Diploma Statistics School of Psychology University of Nottingham 12 Binomial Probability Test From these outcome possibilities we can calculate probabilities associated with three different events using the addition rule. These events are based on the number of women that could occur in our sample of two from the class.

13. C82MCP Diploma Statistics School of Psychology University of Nottingham 13 Binomial Probability Test The probability of randomly selecting a sample of two people from the class where the sample has no women in it is p=0.0625. This is not significant. Given a class distributed 75% female and 25% male a sample of two is too small to ever produce a significant result.

14. C82MCP Diploma Statistics School of Psychology University of Nottingham 14 Binomial Probability Test In general, for the binomial probability test, each of the outcomes is the result of a Bernoulli trial. In a Bernoulli trial, one of two outcomes occurs: One of the outcomes is called a success The other outcome is called a failure Sometimes, they are called a hit and a miss respectively. The binomial test gives the probability of each possible number of successes (hits) in N number of trials given that the probability of success is known on each trial.

15. C82MCP Diploma Statistics School of Psychology University of Nottingham 15 Binomial Probability Distribution The graph below represents the binomial probability distribution for 10 trials with each outcome having a probability of 1/2 or 0.5 This is the same as tossing a coin times and counting the number of heads

16. C82MCP Diploma Statistics School of Psychology University of Nottingham 16 Probability Distributions A distribution that gives the probability of all of the possible values of a set of events is called a probability distribution. We can generate many such distributions depending on the nature of the outcomes and events that are associated with the distribution. We can use such probability distributions to test inferences.

17. C82MCP Diploma Statistics School of Psychology University of Nottingham 17 Sampling Distributions Sampling distributions are probability distributions. Sampling distributions give the probabilities that a particular value of a sample statistic occurred by chance. The sampling distribution allows us to make inferences from a sample back to a population.

18. C82MCP Diploma Statistics School of Psychology University of Nottingham 18 Sampling Distributions Let's suppose that we knew the values of a particular population of scores: 1 2 3 4 5 6 7 8 9 The mean of the population is 5 If we randomly take samples of size two from the population, we could get the following: The sample means are not all the same

19. C82MCP Diploma Statistics School of Psychology University of Nottingham 19 Sampling Distributions Given that we know the values of this population we can define each sample mean for a sample of two as a probability event We can calculate the probabilities of particular means occurring

20. C82MCP Diploma Statistics School of Psychology University of Nottingham 20 Sampling Distributions We can use the sampling distribution to estimate the probability of obtaining a mean of a sample. Example 1 Mean of Sample=5 Probability=0.1111 Example 2 Mean of Sample=1.5 Probability=0.02778

21. C82MCP Diploma Statistics School of Psychology University of Nottingham 21 Sampling Distributions A sampling distribution gives the probability of a sample statistic occurring given that a population is known. When the probability of a statistic coming from a particular population is <0.05 we conclude that the sample was not taken from the population. Sampling distributions can be obtained for any population parameter e.g., the mean, the standard deviation, the skew etc. The sampling distribution is specific to the sample size.

22. C82MCP Diploma Statistics School of Psychology University of Nottingham 22 Standard Errors The standard deviation of the sampling distribution of any statistic is called the standard error of that statistic Standard errors can be calculated for any population parameter: mean median variance standard deviation skew

23. C82MCP Diploma Statistics School of Psychology University of Nottingham 23 Standard Errors The standard deviation of the sampling distribution of the mean is known as the standard error of the mean The mean of the sampling distribution of the sample mean is denoted: The standard deviation of the sampling distribution of the sample mean is denoted:

24. C82MCP Diploma Statistics School of Psychology University of Nottingham 24 Standard Errors The standard error of the mean can be calculated directly if the standard deviation of the population is known where the standard deviation of the population the sample size