8-10-2007

1. Week 2 Sampling distributions and testing hypotheses handout available at http://homepages.gold.ac.uk/aphome Trevor Thompson 8-10-2007

2. Review of following topics: • 1) How individual scores are distributed • 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’

3. Distribution of individual scores • Uniform distributionDice scores are uniformlydistributed (each score has an equal probability of occurrence) • How individual values are distributed depends on the nature of these values sample of 600 die throws • Normal distributionScores on many variablesare normallydistributed (e.g. IQ) sample of 5000 IQ scores • Sampling distribution is not always identical to population distribution because of ‘sampling error’

4. Distribution of individual scores ≈ 50% • What is probability (p-value) of randomly sampling one person with an 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 p-value in table (or use SPSS CDF function) (*assuming population parameters of M=100, SD=15 and normal distribution)

5. 1) How individual scores are distributed • 2) How mean scores are distributed • 3) One-sample z-test

6. 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 • 2) Shape of distribution • 3) Standard deviation same as mean of individual scores (100) next slide

7. 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 The distribution of means appears to be the same as individual scores!– i.e. normal 95 97.5 100 102.5 105

8. 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!

9. 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 deviationofindividual 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

10. Sampling distribution of means • We can now plot the sampling distribution of the means. We know the shape is normal, M=100 & SEM Q: What is the probability of N=36 having a mean IQ of at least 106 ? =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

11. 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!)

12. 1) How individual scores are distributed • 2) How mean scores are distributed • 3) One-sample z-test

13. 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!

14. 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

15. 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

16. 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

17. 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

18. 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