Partly based on material by Sherry O’Sullivan Research Methods Part 4 T- Statistics Partly based on material by Sherry O’Sullivan

Revision General terms Population Sample Parameter Statistic Measures of central tendency Mean Median Mode Measures of spread Range Inter-quartile range Variance Standard deviation Population Sample

Revision of notation Numbers describing a population are called parameters Notation uses Greek letters Population mean = μ Population standard deviation = σ Numbers describing a sample are called statistics Notation uses ordinary letters Sample mean = Sample standard deviation = s

Revision: Z - Scores A specific method for describing a specific location within a distribution Used to determine precise location of an in individual score within the distribution Used to compare relative positions of 2 or more scores

Revision: Standard Deviation Measures the spread of scores within the data set Population standard deviation is used when you are only interested in your own data Sample standard deviation is used when you want to generalise from your sample to the rest of the population

Normal Distribution (Bell shaped)

Normal distribution Many data sets follow a Normal distribution Defined mathematically by its mean and standard deviation Many statistical tests assume that data follows the Normal distribution Strictly, you can’t use these tests unless you can show that your data follows a Normal distribution

Other possible distributions Poisson distribution – for very rare events e.g. number of BSCs (blue screen crashes) per hour of computer use Mean is small, often less than 1 Mode and median often zero Binomial distribution Very similar to the Normal distribution, but a discrete distribution (as opposed to a continuous distribution) There are lots of others…

Poisson distribution (μ = mean = variance)

Distribution of the sample Means (simple example) Frequency Distribution of 4 scores (2, 4, 6, 8) X 1 2 3 4 5 6 7 8 9 Distribution looks flat and not bell shaped (actually not enough data to decide what the distribution might be) Mean of population is (2+4+6+8)/4 = 5 It is clear that this distribution is not normal (it’s flat and not “bell-shaped”).

Distribution of the sample means Take all possible samples of two scores Calculate average for each sample (2+2)/2 = 2 (2+4)/2 = 3 (2+6)/2 = 4 (2+8)/2 = 5 (4+2)/2 = 3 (4+4)/2 = 4 (4+6)/2 = 5 (4+8)/2 = 6 (6+2)/2 = 4 (6+4)/2 = 5 (6+6)/2 = 6 (6+8)/2 = 7 (8+2)/2 = 5 (8+4)/2 = 6 (8+6)/2 = 7 (8+8)/2 = 8 Now let’s take all possible samples with n=2 in other words all possible samples of pairs of scores. We also agree to use random sampling where each individual sample is replaced into the data set. We compute the averages of all sample pairs. So, for example, we get average(2 + 4) = 3 and average(4 + 2) = 3. We get average(2 + 2) = 2 and so on. X X X X X X X X X X X X X X X X 0 1 2 3 4 5 6 7 8 9

Central Limit Theorem “For any population with a mean μ and standard deviation σ , the distribution of sample means for sample size n will have a mean of μ and standard deviation of σ/√n and will approach a normal distribution as n gets very large.” How big should the sample size be? n=30 X 1 2 3 4 5 6 7 8 9 So it looks as though, through the process of sampling, we are able to discover the population mean. This is a very important result, and is the bed-rock of statistical analysis. It also has a mathematical description, which is summarized in the “Central Limit Theorem” Colin’s notes So this means that the z-test approach is applicable to data which is not normally distributed, provided that we take samples, and calculate their means, when the sample size is big enough. How big? Well n=30 will usually do.

Standard Error σ/√n is used to calculate the Standard Error of the sample mean Sample data = x The mean of each sample = Then the standard error becomes It identifies how much the observed sample mean is likely to differ from the un-measurable population mean μ. So to be more confident that our sample mean is a good measure of the population mean, then the standard error should be small. One way we can ensure this is to take large samples (large n).

Example The population of SATs scores is normal with μ= 500, σ =100. What is the chance that a sample of n=25 students has a mean score = 540? Since the distribution is normal, we can use the z-score. First calculate the Standard Error: = 100/5 = 20 Then the Z-Score: = (540-500)/20 =2 The z-value is 2, therefore around 98% of the sample means are below this and only 2% are above. So we conclude that the chance of getting a sample mean of 540 or more is about 2%, so we are about 98% confident that this sample mean (if recorded in an experiment) is not due to random variation, but that the 25 students are (on average) brighter than average.

t - Statistics So far we’ve looked at mean and sd of populations and our calculations have had parameters But how do we deduce something about the population using our sample? We can use the t-Statistic

t - Statistics Remember SD from last week? Great for population of N but not for sample of n Why n -1? Because we can only freely choose n-1 (Degree of freedom = df) Show example

t - Statistics Standard Error t statistic is z-score redone using the above: And for the t-statistic, we substitute σ (SD of population) with s (SD of sample) But what about μ ? An example…

Hypothesis Testing Sample of computer game players n =16 Intervention = inclusion of rich graphical elements Level has 2 rooms Room A = lots of visuals Room B = very bland Put them in level 60 minutes Record how long they spend in B

Results Average time spent in B = 39 minutes Observed “sum of squares” for the sample is SS = 540. H0: Here we formulate the “null” hypothesis, that the visuals have no effect on the behaviour. H1: Here we formulate the “alternative” hypothesis, that the visuals do have an effect on the players’ behaviour. A B

Stage1: Formulation of Hypothesis H0: “null hypothesis”, that the visuals have no effect on the behaviour. H1: “alternate hypothesis”, that the visuals do have an effect on the players’ behaviour. If visuals have no effect, how long on average should they be in room B? Null hypothesis is crucial; here we can infer that μ = 30 and get rid of the population mean The null hypothesis is crucial, since it helps us to “get rid” of the population parameter . Think about it. If visuals have no effect on the population then what is the average time the player will spend in room B? Clearly half the time, so we have inferred from the null hypothesis that u =30.

Stage 2: Locate the critical region We use the t-table to help us locate this, enabling us to reject or accept the null hypothesis. To get we need: Number of degree of freedom (df) 16 -1 =15 We choose a significance or a level of confidence: α = 0.05 (95% confidence) Locate in t-table (2 tails): critical value of t=2.131,

Stage 3: Calculate statistics Calculate sample sd = 6 Sample Standard Error = 6 / 4 =1.5 The μ = 30 came from the null hypothesis: if visuals had no effect, then the player would spend 30 minutes in both rooms A and B. t-Statistic = 6

Stage 4: Decision Can we reject the H0, that the visuals have no effect on the behaviour? t = 6 which is well beyond the value of 2.131 which indicates where chance kicks in. So “yes”, we can safely reject it and say it does affect behaviour Which room do they prefer? They spent on average 39 minutes in Room B which is bland

Another use of t (part 1) Our example was a comparison of what was observed with what was expected Our analysis gave a confidence with which the observations were different from the expected Note: cannot be used to confirm similarity… Another use of t: comparison of two samples e.g. male and female performance on a game, or opinion of a website….

Another use of t (part 2) In this case, we have two sample means, and we are testing for a difference Recall: before, we had This time, it gets messy because the two standard deviations might not be the same, and we finish up with

Yet another complication: When looking for a difference, we may have no reason to suppose that one sample gives higher values than the other We don’t know which mean might be higher This is a two-tailed test: we test both tails of the distribution If we have a good reason to suppose that one set of results has to be higher than the other e.g. game scores before and after a practice session Then we have a one-tailed test

Fortunately, it can all be done on a computer…

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