# week 6.pdf - 21/11/2018 Introduction to t‐distributions;...

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21/11/20181Introduction totdistributions;Estimation; Confidence intervalsWeek 61Distributions so far…Binomial distributionDiscrete random variableXis count of successes out ofntrialsܺ~ܤ݅݊ሺ݊, ݌ሻNormal distributionContinuous random variableXis (usually) a measurementܺ~ܰሺߤ, ߪܼ~ܰሺ0,1ሻfor standard Normal distributionNormal dist is symmetric, very few outliersNew:tdistributionContinuous random variableܶ~ݐυ(υ= degrees of freedom, pronounced ‘nu’)Symmetric, very few outliers2tdistributionstdistributions are used for continuous data (measurements)e.g.Based on Normal distribution – symmetric, very few outliersTypically used for formal inference about a population with a Normal distributionwhen:sample size is ‘small’ andσis unknownwe estimateσwith its point estimate, stdistribution gets closer to the Normal distribution as sample size increasesfor large enough sample sizes,tdistribution and Normal distribution give almost identicalresults3
21/11/20182tdistributions are the same basic ‘bell shape’ as the Normal distributionThe smaller the sample size, the flatter the curveand the ‘heavier’ the tailssample sizesT݊ ൌ 11݊ ൌ 5݊ ൌ 3݊ ൌ 24sample sizesT݊ ൌ 11݊ ൌ 5݊ ൌ 3݊ ൌ 2The diagram shows that withsmall values ofnit is importantto use thecorrecttdistributionas they differ substantiallyneed the assumption of Normallydistributed data in order to use atdistributionEachtdistribution ischaracterised by itsdegrees offreedom– this is thesample sizeminus one݂݀ ൌ ݊ െ 1Notice that the mean of atdistribution is zero5Degrees of freedomT݊ ൌ 11݊ ൌ 5݊ ൌ 3݊ ൌ 2T6Sample sizeሺ݊ሻand degrees of freedomሺ݂݀orυ)
21/11/20183Estimation in Statistics7Point estimates and interval estimatesEstimatesNumerical summaries of sample data (sample statistics) give usestimatesof population parametersa)Point estimatesb) Interval estimatesWe generally get ‘good’ estimates ifsample is randomly chosenn’ is large8Point estimatesApoint estimateof a population parameter (such asߤ, the population mean)is a single number that is our best guess of the value of that parameterThis estimate will usually be ‘wrong’, as it will not generally be the true valueof the population median/mean, butit is a very good starting point.In fact, itis the most sensible starting point in our quest to find information about thewhole population9
21/11/20184Some population parameters and their point estimatesPopulation parameter (words)Population parameter (symbol)Point estimate of the parameterPopulation meanߤ(mu)ݔ̅Population standard deviationߪ(sigma)ݏPopulation proportion݌݌̂10Confidence intervals(Interval estimates)11Interval estimatesAninterval estimateof a population parameter is a range of plausible valuesthat we think (with some specified level of confidence) contains the (unknown)population parameter

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