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

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2 Review of following topics: 1) How individual scores are distributed 1) How individual scores are distributed 2) How mean scores are distributed 2) How mean scores are distributed 3) One-sample z-test 3) One-sample z-test testing the difference between a sample mean and a known population mean testing the difference between a sample mean and a known population mean - Howell (2002) Chap 4 & 7. Statistical Methods for Psychology 1) How individual scores are distributed 1) How individual scores are distributed

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3 Distribution of individual scores How individual values are distributed depends on the nature of these values How individual values are distributed depends on the nature of these values Uniform distribution Dice scores are uniformly distributed (each score has an equal probability of occurrence) 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) Normal distribution Scores on many variables are normally distributed (e.g. IQ) sample of 600 die throws sample of 5000 IQ scores Sampling distribution is not always identical to population distribution because of sampling error Sampling distribution is not always identical to population distribution because of sampling error

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4 Distribution of individual scores What is probability (p-value) of randomly sampling one person with an IQ of 100 or more? What is probability (p-value) of randomly sampling one person with an IQ of 100 or more? 50% 50% 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) 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) The probability for any IQ can be calculated* – calculate the z-score (i.e. the number of SDs above or below the mean), then look up corresponding p- value in table (or use SPSS CDF function) The probability for any IQ can be calculated* – calculate the z-score (i.e. the number of SDs 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) 2.5% 2.5%

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

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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 We need to know how means are distributed to answer this question Specifically, we need to know (as with individual scores): Specifically, we need to know (as with individual scores): 1) Mean 1) Mean 2) Shape of distribution 2) Shape of distribution 3) Standard deviation 3) Standard deviation same as mean of individual scores (100) next slide

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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

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8 1) Mean – equal to the population mean (100) 1) Mean – equal to the population mean (100) 2) Shape - normal 2) Shape - normal 3) Standard deviation – how widely are the means spread? 3) Standard deviation – how widely are the means spread? Sampling distribution of means Mean scores are spread more closely around the centre than individual scores 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! 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!

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9 Sampling distribution of means In fact, we can calculate precisely the spread of mean scores around the centre: In fact, we can calculate precisely the spread of mean scores around the centre: SEM= Sx N SEM= Sx N SEM (the standard error of the mean) is the standard deviation of the mean (rather than standard deviation of individual scores) 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 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

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10 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 & We can now plot the sampling distribution of the means. We know the shape is normal, M=100 & SEM SEM 152.5 =15 = 2.536 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

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11 Confidence Limits The 95% confidence limits is the range within which 95% of the sample means will fall 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 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!) Our value of 106 lies outside these limits –we reject the null hypothesis (p<.05!)

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

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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 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: 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 z= X – μ (where SEM=σ/N) SEM A one-sample z-test is what we have previously performed! z= (106 – 100)/2.5 A one-sample z-test is what we have previously performed! z= (106 – 100)/2.5 z=2.4, which gives p<.05 –significant!

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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 t = X – μ (where SEM=s/N) SEM but look up p-value from a t not a z -distribution table

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15 Central Limit Theorem Everything we have done so far is explained by 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 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

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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 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

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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? 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) 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] 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) z1 (3.85-3.5/0.34) p>0.5 – non-significant

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18 Summary How to perform a one sample z-test How to perform a one sample z-test How to perform a one sample t-test How to perform a one sample t-test Underlying logic behind one sample tests Underlying logic behind one sample tests Rules of central limit theorem Rules of central limit theorem

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