21/11/20181Introduction tot‐distributions;Estimation; Confidence intervalsWeek 61Distributions so far…•Binomial distributionDiscrete random variable•Xis count of successes out ofntrials•ܺ~ܤ݅݊ሺ݊, ሻ•Normal distributionContinuous random variable•Xis (usually) a measurement•ܺ~ܰሺߤ, ߪଶሻ•ܼ~ܰሺ0,1ሻfor standard Normal distribution•Normal dist is symmetric, very few outliersNew:•t‐distributionContinuous random variable•ܶ~ݐυ(υ= degrees of freedom, pronounced ‘nu’)•Symmetric, very few outliers2t‐distributions•t‐distributions are used for continuous data (measurements)e.g.•Based on Normal distribution – symmetric, very few outliers•Typically used for formal inference about a population with a Normal distributionwhen:sample size is ‘small’ andσis unknownwe estimateσwith its point estimate, s•t‐distribution gets closer to the Normal distribution as sample size increases•for large enough sample sizes,t‐distribution and Normal distribution give almost identicalresults3
21/11/20182t‐distributions 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 ൌ 2•The diagram shows that withsmall values ofnit is importantto use thecorrectt‐distributionas they differ substantially•need the assumption of Normallydistributed data in order to use at‐distribution•Eacht‐distribution ischaracterised by itsdegrees offreedom– this is thesample sizeminus one•݂݀ ൌ ݊ െ 1•Notice that the mean of at‐distribution is zero5Degrees of freedomT݊ ൌ 11݊ ൌ 5݊ ൌ 3݊ ൌ 2T6Sample sizeሺ݊ሻand degrees of freedomሺ݂݀orυ)
21/11/20183Estimation in Statistics7Point estimates and interval estimatesEstimates•Numerical summaries of sample data (sample statistics) give usestimatesof population parametersa)Point estimatesb) Interval estimates•We generally get ‘good’ estimates if•sample is randomly chosen•‘n’ 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 proportion̂10Confidence intervals(Interval estimates)11Interval estimates•Aninterval 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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