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Week 2 Sampling distributions and testing hypotheses handout available at http://homepages.gold.ac.uk/aphome Trevor Thompson 8-10-2007 1 Review of following topics: 1) How individual scores are distributed 2) How mean scores are distributed 3) One-sample z-test testing the difference between a sample mean and a known population mean - Howell (2002) Chap 4 & 7. ‘Statistical Methods for Psychology’ 2 Distribution of individual scores Uniform distribution Dice scores are uniformly distributed (each score has an equal probability of occurrence) Normal distribution Scores on many variables are normally distributed (e.g. IQ) Sampling distribution is not always identical to population distribution because of ‘sampling error’ sample of 600 die throws How individual values are distributed depends on the nature of these values sample of 5000 IQ scores 3 Distribution of individual scores What is probability (p-value) of randomly sampling one person with an ≈ 50% IQ of 100 or more? What is probability of randomly selecting an IQ score of 130+? (We can calculate this from what we know about the properties of the normal curve) ≈ 2.5% The probability for any IQ can be calculated* – calculate the z-score (i.e. the number of SD’s above or below the mean), then look up corresponding pvalue in table (or use SPSS CDF function) (*assuming population parameters of M=100, SD=15 and normal distribution) 4 1) How individual scores are distributed 2) How mean scores are distributed 3) One-sample z-test 5 Sampling distribution of means Q: What is the probability of a group of 36 having a mean IQ of 106 ? We need to know how means are distributed to answer this question Specifically, we need to know (as with individual scores): 1) Mean same as mean of individual scores (100) 2) Shape of distribution next slide 3) Standard deviation 6 Shape of distribution of means I repeatedly sampled 36 scores and calculated the mean. I then repeated this several thousand times and plotted these means: Sample 1: Random sample of 36 scores produced M=103.5 Sample 2: Random sample of 36 scores produced M=100 Sample 3: Random sample of 36 scores produced M=96 Sample 4: Random sample of 36 scores produced M=102 Sample 5: Random sample of 36 scores produced M=100 95 97.5 100 102.5 105 The distribution of means appears to be the same as individual scores! – i.e. normal 7 Sampling distribution of means 1) Mean – equal to the population mean (100) 2) Shape - normal 3) Standard deviation – how widely are the means spread? Mean scores are spread more closely around the centre than individual scores this makes intuitive sense – while individual scores of 130 are not exceptionally rare (p=2.5%), mean IQs of 130 would be extremely rare when group size is 1,000! 8 Sampling distribution of means In fact, we can calculate precisely the spread of mean scores around the centre: SEM= Sx √N SEM (the standard error of the mean) is the standard deviation of the mean (rather than standard deviation of individual scores) The above formula shows that the bigger the sample size (N), the smaller the SEM – i.e. the more closely scores are clustered around the population mean 9 Sampling distribution of means Q: What is the probability of N=36 having a mean IQ of at least 106 ? We can now plot the sampling distribution of the means. We know the shape is normal, M=100 & SEM =15 = 2.5 √36 So, if we know how individual scores are distributed (i.e. shape, M & SD) we also know how means are distributed and can test hypotheses about groups 10 Confidence Limits The 95% confidence limits is the range within which 95% of the sample means will fall If a sample mean lies within these limits then we cannot reject the null hypothesis Our value of 106 lies outside these limits –we reject the null hypothesis (p<.05!) 11 1) How individual scores are distributed 2) How mean scores are distributed 3) One-sample z-test 12 One sample z-test One sample z-test: Compares the mean score of one group against a population mean. This can only be performed when we know the population mean and the population standard deviation To perform a one-sample z-test, calculate how many SEMs above/below the population mean your sample mean is. Expressed as a formula: z= X–μ (where SEM=σ/√N) SEM A one-sample z-test is what we have previously performed! z= (106 – 100)/2.5 z=2.4, which gives p<.05 –significant! 13 One sample t-test A one sample z-test is used when we already know the population mean and SD A one sample t-test is used when we know the population mean (μ) but not the population SD (σ) As we do not know σ, we estimate it from s (sample SD). But, as s is often too small, the p-value is inaccurate when using z-distribution tables. Use t-distribution tables for more conservative p-values To perform a one-sample t-test: t =X–μ (where SEM=s/√N) SEM but look up p-value from a t not a z-distribution table 14 Central Limit Theorem Everything we have done so far is explained by central limit theorem ‘Given a population with mean, , and standard deviation, , the sampling distribution of the mean will have’: (i) a mean equal to (ii) a standard deviation equal to /√N, where N is the sample size (iii) a distribution which will approach the normal distribution as N increases 15 Central Limit Theorem The approximation of the sampling distribution of the mean to a normal distribution is true -whatever the shape of the distribution of individual values distribution of single die scores distribution of mean dice scores Populati on Samp. Dist. of Mean Normal Symmetric and unimodal Highly skewed Normal for all N Normal for s mall N Normal for N>30 16 One sample z-tests - examples 1) The mean IQ of a group of 16 people was measured as 103. Is this significantly different from the population using the population parameters previous specified? No, SEM=3.75 (15/√16) z=0.8 (103-100/3.75) p>0.5 – non-significant 2) Is a sample of 25 dice throws, with a mean score of 3.85, sampled from a fair die? [=3.5, =1.7] No, SEM=0.34 (1.7/√25) z≈1 (3.85-3.5/0.34) p>0.5 – non-significant 17 Summary How to perform a one sample z-test How to perform a one sample t-test Underlying logic behind one sample tests Rules of central limit theorem 18

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