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MIT 24 910 - Basic statistics

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24.910 Laboratory Phonology Basic statisticsWriting up an experimentWriting up an experimentWriting up an experimentSome StatisticsDescriptive statisticsHypothesis TestingHypothesis Testing: t-testHypothesis Testing: t-testHypothesis TestingHypothesis TestingHypothesis TestingHypothesis TestingHypothesis Testingt test for independent meansHypothesis testingHypothesis testingFitting modelsFitting modelsMIT OpenCourseWare http://ocw.mit.edu24.910 Topics in Linguistic Theory: Laboratory PhonologySpring 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.24.910Laboratory PhonologyBasic statisticsReading: • Fowler & Housum 1987.Assignments:• Write up voicing perception experiment (due in two weeks 5/8).• Progress report on your project (5/1).• Project draft + presentation 5/15.Topics: • Statistics• The lexicon and context in speech perception.• The lexicon and context in speech production.• Phonology in speech perception.Writing up an experimentThe report on an experiment usually consists of four basic parts:1. Introduction2. Procedure3. Results4. DiscussionWriting up an experiment1. Introduction• Outline of the purpose of the experiment• state hypotheses tested etc• provide background information (possibly including descriptions of relevant previous results, theoretical issues etc).2. Procedure - what was done and how.• instructions for replication, e.g.– Experimental materials– Subjects– Recording procedure– Measurement procedures (especially measurement criteria).Writing up an experiment3. Results• Presentation of results, including descriptive statistics (means etc) and statistical tests of hypotheses.4. Discussion• Discuss the interpretation and significance of the resultsSome StatisticsTwo uses of statistics in experiments:• Summarize properties of the results (descriptive statistics).• Test the significance of results (hypothesis testing).Descriptive statisticsA measure of central tendency:•Mean:– M is used for sample mean, μ for population mean.A measure of dispersion:• Variance: mean of the squared deviations from the meanM = ΣxiN σ2 = Σ(xi-μ)2 N • Standard deviation: σ (square root of the variance).bl(ow)Time (s)0.3506780.4700905000Time (s)0.3125150.48722205000Time (s)0.3223870.52260705000Time (s)0.3371070.53741105000Time (s)0.3525440.55242705000gl(ow)br(ew) dr(ew) gr(ew)Hypothesis Testing• F2 onset (Hz)• Are these differences in means significant?• Could the apparent differences have arisen by chance, although the true (population) means of F2 onsets are the same?• I.e. given that F2 onsets vary, we might happen to sample most of our [br] onsets from the low end of the distribution, and most of our [gr] onsets from the high end.• Statistical tests allow us to assess the probability that this is the case. br dr gr mean 1225 1641 1272 s.d. 150 177 215Hypothesis Testing: t-test•The t-test allows us to test hypotheses concerning means and differences between means.1. ‘Mean F2 onset in [br] differs from mean F2 onset in [gr] in English’.2. ‘Mean F2 onset in [br] is 1250 Hz’ (unlikely, but a simpler case - cf. [afva] is identified as [afa] > 50%).• We actually evaluate two exhaustive and mutually exclusive hypotheses, a null hypothesis that the mean has a particular value, and the alternative hypothesis that the mean does not have that value.1. The mean F2 onset in [br] is the same as the mean F2 onset in [gr] (Null).2. The mean F2 onset in [bt] ≠ 1250 Hz (Alternative).• Statistical tests allow us to assess the probability of obtaining the observed data if the null hypothesis were true.Hypothesis Testing: t-test• Basic concept: If we know what the distribution of sample means would be if the null hypothesis were true, then we can calculate the probability of obtaining the observed mean, given the null hypothesis.• We arrive at the parameters of the distribution of sample means through assumptions and estimation.Hypothesis Testing110 120 130 140 150 160 170 180 190σ = 10 msif this were the population mean……it is unlikely that we would get a sample mean of this valueDistribution of sample meansHypothesis Testing• Basic assumption: The samples are drawn from normal populations.+3S2.5%68%95%99.7%+2S+S-S-2S-3S XFigure by MIT OpenCourseWare. Adapted from Kachigan, S. K. Multivariate Statistical Analysis. 2nd ed. New York, NY: Radius, 1991.Hypothesis Testing• Basic assumption: The samples are drawn from normal populations.• Properties of distribution of means of samples of size N drawn from a normal population:– The sample means are normally distributed.– Mean is the same as the population mean.– The variance is less than the population variance:σM2 = σ2N300 350 400 450 500 550 600x300 350 400 450 500 550 600x300 350 400 450 500 550 600300 350 400 450 500 550 600(a) Parent population(b) Sampling distribution ofthe mean based on samplesof size n = 4(b) Sampling distribution ofthe mean based on samplesof size n = 16(b) Sampling distribution ofthe mean based on samplesof size n = 64σ = 50xxσ1 = σn504= =25σ1 = σn5016= =12.5σ1 = σn5064= =6.25Figure by MIT OpenCourseWare. Adapted from Kachigan, S. K . 2nd ed. New York, NY: Radius, 1991.Multivariate Statistical AnalysisHypothesis Testing• The mean of the distribution is determined by hypothesis.– E.g. mean = 1250 Hz or mean difference = 0.• Population variance is estimated from the sample variance. Unbiased estimate of the population variance:– N-1 is the number of degrees of freedom of the sample.• So estimated variance of distribution of sample means, SM2= S2/N• t score:S2 = Σ(xi-Μ)2 N-1 t = M-µSMHypothesis Testing• t scores follow a t-distribution - similar to a normal distribution, but with slightly fatter tails (more extreme values) because S may underestimate σ.• t-distribution is actually a family of distributions, one for eachnumber of degrees of freedom.• Calculate t-score then consult relevant t distribution to determine the probability of obtaining that t-score or greater (more extreme).Figure by MIT OpenCourseWare. Normalt (df = 12)t (df = 5)t test for independent means• When we compare means, we are actually sampling a population of differences (e.g. differences in durations of vowels in open andclosed syllables).• If the null hypothesis is correct, then the mean difference is 0.• Variance of the distribution of mean differences is


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