week 6.pdf - Introduction to tu2010distributions...

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21/11/2018 1 Introduction to t distributions; Estimation; Confidence intervals Week 6 1 Distributions so far… Binomial distribution Discrete random variable X is count of successes out of n trials ܺ~ܤ݅݊ሺ݊, ݌ሻ Normal distribution Continuous random variable X is (usually) a measurement ܺ~ܰሺߤ, ߪ ܼ~ܰሺ0,1ሻ for standard Normal distribution Normal dist is symmetric, very few outliers New: t distribution Continuous random variable ܶ~ݐ υ ( υ = degrees of freedom, pronounced ‘nu’) Symmetric, very few outliers 2 t 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 distribution when: sample size is ‘small’ and σ is unknown we estimate σ with 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 identical results 3
21/11/2018 2 t distributions are the same basic ‘bell shape’ as the Normal distribution The smaller the sample size, the flatter the curve and the ‘heavier’ the tails sample sizes T ݊ ൌ 11 ݊ ൌ 5 ݊ ൌ 3 ݊ ൌ 2 4 sample sizes T ݊ ൌ 11 ݊ ൌ 5 ݊ ൌ 3 ݊ ൌ 2 The diagram shows that with small values of n it is important to use the correct t distribution as they differ substantially need the assumption of Normally distributed data in order to use a t distribution Each t distribution is characterised by its degrees of freedom – this is the sample size minus one ݂݀ ൌ ݊ െ 1 Notice that the mean of a t distribution is zero 5 Degrees of freedom T ݊ ൌ 11 ݊ ൌ 5 ݊ ൌ 3 ݊ ൌ 2 T 6 Sample size ሺ݊ሻ and degrees of freedom ሺ݂݀ or υ )
21/11/2018 3 Estimation in Statistics 7 Point estimates and interval estimates Estimates Numerical summaries of sample data (sample statistics) give us estimates of population parameters a) Point estimates b) Interval estimates We generally get ‘good’ estimates if sample is randomly chosen n ’ is large 8 Point estimates A point estimate of a population parameter (such as ߤ , the population mean) is a single number that is our best guess of the value of that parameter This estimate will usually be ‘wrong’, as it will not generally be the true value of the population median/mean, but it is a very good starting point. In fact, it is the most sensible starting point in our quest to find information about the whole population 9
21/11/2018 4 Some population parameters and their point estimates Population parameter (words) Population parameter (symbol) Point estimate of the parameter Population mean ߤ (mu) ݔ̅ Population standard deviation ߪ (sigma) ݏ Population proportion ݌ ݌̂ 10 Confidence intervals (Interval estimates) 11 Interval estimates An interval estimate of a population parameter is a range of plausible values that we think (with some specified level of confidence) contains the (unknown) population parameter

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