Business Statistics CONTENTS Sampling The central limit theorem - PowerPoint PPT Presentation

SAMPLING, THE CLT, AND THE STANDARD ERROR Business Statistics

CONTENTS Sampling The central limit theorem Point and interval estimates for 𝜈 Confidence intervals for 𝜈 Old exam question Further study

SAMPLING Suppose you’re a scissors manufacturer in the UK ▪ What proportion of your production should be left-handed? ▪ Three strategies ▪ look at Wikipedia (“ Studies suggest that 70 – 90% of the world population is right-handed.[4][5] ”) ▪ ask all persons in the UK (~63 million) ▪ ask a sample of persons (100?) in the UK

SAMPLING Sampling is the process of collecting data about a sample (a subset of the population), with the aim of representing the entire population ▪ Arguments pro sampling ▪ too costly to probe entire population ▪ too time-consuming ▪ too dangerous ▪ too destructive ▪ etc. ▪ Arguments against sampling ▪ limited accuracy  confidence intervals (later in this course) ▪ not representative  design of experiments (not in this course)

SAMPLING A sample should be representative ▪ e.g., don’t ask people at Schiphol if they’re afraid of flying A sample should be large enough ▪ cf. the “ 𝑜 ” law later on Choice in sampling ▪ with replacement or without replacement ▪ this has consequences for the probability model

SAMPLING Population Sample unknown known we would like to know irrelevant parameter statistic mostly Greek letters ( 𝜌 , 𝜏 ) mostly Roman letters ( 𝑞 , 𝑡 ) some deviating notations ( 𝑂 ) some deviating notations ( ҧ 𝑦 , 𝑜 )

THE CENTRAL LIMIT THEOREM ▪ Let 𝑌 1 , 𝑌 2 , … , 𝑌 𝑜 be a random sample from a population 2 𝑌 with mean 𝜈 𝑌 and variance 𝜏 𝑌 ▪ e.g., body heights of 𝑜 persons Capital 𝑌 , because it is a ▪ waiting times of 𝑜 customers random variable! ▪ failure rates of 𝑜 cars, ... 𝑌 1 +𝑌 2 +⋯+𝑌 𝑜 ▪ Then, for 𝑜 sufficiently large, the mean ത 𝑌 = 𝑜 1. is normally distributed 2. with mean 𝜈 ത 𝑌 = 𝜈 𝑌 2 2 = 𝜏 𝑌 3. and variance 𝜏 ത Capital ത 𝑌 , because this is 𝑌 𝑜 also a random variable!

THE CENTRAL LIMIT THEOREM So for large 𝑜 : 2 2 = 𝜏 𝑌 ത 𝑌~𝑂 𝜈 ത 𝑌 = 𝜈 𝑌 , 𝜏 ത 𝑌 𝑜 ▪ or for short 2 𝑌~𝑂 𝜈 𝑌 , 𝜏 𝑌 ത 𝑜 ▪ This holds regardless of the distribution of 𝑌 ! ▪ so that’s why the normal distribution is called “normal” ▪ this fact is called the central limit theorem (CLT) ▪ it is one of the most important results of statistics ▪ it holds for “sufficiently large” 𝑜

THE CENTRAL LIMIT THEOREM The CLT for a fair die Distribution of ത 𝑌 for ▪ 𝑜 = 1 ▪ 𝑜 = 2 ▪ 𝑜 = 5 ▪ 𝑜 = 20

THE CENTRAL LIMIT THEOREM The CLT for a loaded (unfair) die Distribution of ത 𝑌 for ▪ 𝑜 = 1 ▪ 𝑜 = 2 ▪ 𝑜 = 5 ▪ 𝑜 = 20

EXERCISE 1 We roll with a die 100 times. The outcomes are 𝑌 = 𝑌 1 , 𝑌 2 , … , 𝑌 100 . How is ത 𝑌 distributed?

THE CENTRAL LIMIT THEOREM A “proof” of the theorem (for normal populations) ▪ Recall the additive property of the normal distribution: 2 and 𝑌 2 ~𝑂 𝜈 𝑌 , 𝜏 𝑌 2 , then 𝑌 1 + ▪ if 𝑌 1 ~𝑂 𝜈 𝑌 , 𝜏 𝑌 2 (provided 𝑌 1 and 𝑌 2 are independent) 𝑌 2 ~𝑂 2𝜈 𝑌 , 2𝜏 𝑌 2 then 𝑏𝑌~𝑂 𝑏𝜈 𝑌 , 𝑏 2 𝜏 𝑌 2 ▪ Also recal that if 𝑌~𝑂 𝜈 𝑌 , 𝜏 𝑌 2 2 then 𝑌 1 +𝑌 2 𝜏 𝑌 ▪ So, if 𝑌 1 + 𝑌 2 ~𝑂 2𝜈 𝑌 , 2𝜏 𝑌 ~𝑂 𝜈 𝑌 , 2 2 2 𝑌 1 +⋯+𝑌 𝑜 𝜏 𝑌 ▪ and more general: ~𝑂 𝜈 𝑌 , 𝑜 𝑜 You don’t need to reproduce 2 𝜏 𝑌 ▪ or equivalently: ത such proofs, but it may help 𝑌~𝑂 𝜈 𝑌 , 𝑜 ▪ This proof works for normal populations and all 𝑜 , but the CLT is valid for all populations and “large” 𝑜

THE CENTRAL LIMIT THEOREM Some consequences of the CLT ▪ ത 𝑌 is an estimator of 𝜈 𝑌 ▪ and ҧ 𝑦 is the best estimate of 𝜈 𝑌 ▪ ത 𝑌 will be a better estimator for large 𝑜 ▪ because 𝜏 ത 𝑌 decreases with 𝑜 ▪ we can use the distribution of ത 𝑌 to construct a confidence interval for 𝜈

THE CENTRAL LIMIT THEOREM The CLT holds for 𝑜 “sufficiently” large ▪ More specifically: ▪ if 𝑌 is normally distributed, the CLT holds for all sample sizes 𝑜 ▪ if the distribution of 𝑌 is fairly symmetric without extreme outliers, for sample sizes 𝑜 ≥ 15 the CLT gives a pretty good approximation of the distribution of ത 𝑌 ▪ for any distribution of ത 𝑌 and a sample size 𝑜 ≥ 30 , the CLT gives a pretty good approximation of the distribution of ത 𝑌

THE CENTRAL LIMIT THEOREM The effect of asymmetry vs. sample size

POINT AND INTERVAL ESTIMATES FOR 𝜈 A statistic is a function of the (randomly sampled) data ▪ important example: the statistic ത 𝑌 1 ▪ defined by ത 𝑜 𝑜 σ 𝑗=1 𝑌 = 𝑌 𝑗 1 𝑜 ▪ in a concrete case, ҧ 𝑦 𝑗 is the best possible 𝑜 σ 𝑗=1 𝑦 = estimate of the parameter 𝜈 ▪ so the sample mean ҧ 𝑦 is the best possible estimate of the population mean 𝜈 ▪ because it is just one value, it is a point estimate

ҧ POINT AND INTERVAL ESTIMATES FOR 𝜈 Due to sampling variation, ҧ 𝑦 will be different in each sample ▪ and there will be a distribution of ҧ 𝑦 -values, the distribution ത 𝑌 ▪ the true value of 𝜈 may be different from the value of 𝑦 obtained ▪ however, keep in mind that the value of ҧ 𝑦 obtained cannot be “too” wrong 2 , so it follows that a specific ▪ we know that ത 𝑌~𝑂 𝜈 ത 𝑌 , 𝜏 ത 𝑌 value ҧ 𝑦 must be within 𝜈 ത 𝑌 − 1.96𝜏 ത 𝑌 , 𝜈 ത 𝑌 + 1.96𝜏 ത 𝑌 with 95% probability

ҧ ҧ ҧ ҧ ҧ ҧ POINT AND INTERVAL ESTIMATES FOR 𝜈 Conversely, the population value 𝜈 ത 𝑌 must be within 𝑌 with 95% probability 𝑦 − 1.96𝜏 ത 𝑌 , 𝑦 + 1.96𝜏 ത ▪ and because 𝜈 ത 𝑌 = 𝜈 𝑌 , the population value 𝜈 𝑌 must be within 𝑌 with 95% probability 𝑦 − 1.96𝜏 ത 𝑌 , 𝑦 + 1.96𝜏 ത ▪ this is an interval estimate for 𝜈 𝑌 ▪ we say that 𝑌 is a 95% 𝑦 − 1.96𝜏 ത 𝑌 , 𝑦 + 1.96𝜏 ത confidence interval for 𝜈 𝑌

POINT AND INTERVAL ESTIMATES FOR 𝜈 So: ▪ we estimate 𝜈 𝑌 by ҧ 𝑦 ▪ and we know with 95% probability that ҧ 𝑦 − 1.96𝜏 ത 𝑌 ≤ 𝜈 𝑌 ≤ ҧ 𝑦 + 1.96𝜏 ത 𝑌 𝜏 𝑌 ▪ the quantity 𝜏 ത 𝑜 is the standard error of the 𝑌 = distribution of the mean ത 𝑌 ▪ it is so important that we give it a special name: the standard error of the mean ▪ sometimes (unfortunately!) abbreviated as the standard error

EXERCISE 2 We sample ( 𝑜 = 25 ) from a normal population 𝑌 with 2 = 4 . We find ҧ unknown 𝜈 𝑌 and known 𝜏 𝑌 𝑦 = 3 . a. Give a point estimate for 𝜈 𝑌 . b. Find the standard error of the mean, 𝑡 ത 𝑌 . b. Give a 95% -confidence interval for 𝜈 𝑌 .

ҧ CONCEPTS AND SYMBOLS ▪ Carefully distinguish: ▪ 𝜈 𝑌 (a value, often unknown) 𝑦 (a value from observations) ▪ 𝑌 (a distribution, not a value) ത ▪ 2 (both are values, often ▪ and its two parameters 𝜈 ത 𝑌 and 𝜏 ത 𝑌 unknown) ▪ Later on, we will follow a similar logic, e.g. 2 ▪ 𝜏 𝑌 2 ▪ 𝑡 𝑌 and the CLT claims that 2 𝜈 ത 𝑌 = 𝜈 𝑌 ▪ 𝑇 𝑌 2 2 = 𝜏 𝑌 ▪ and its two parameters 𝜏 ത 𝑌 𝑜

OLD EXAM QUESTION 23 March 2015, Q1h

FURTHER STUDY Doane & Seward 5/E 8.1-8.3 Tutorial exercises week 2 sampling distribution central limit theorem standard error