SAMPLING, THE CLT, AND THE STANDARD ERROR. Business Statistics

Transcription

1 SAMPLING, THE CLT, AND THE STANDARD ERROR Business Statistics

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

3 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

4 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)

5 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 n law later on Choice in sampling with replacement or without replacement this has consequences for the probability model

6 SAMPLING Population unknown we would like to know parameter mostly Greek letters (π, σ) some deviating notations (N) Sample known irrelevant statistic mostly Roman letters (p, s) some deviating notations ( x, ҧ n)

7 THE CENTRAL LIMIT THEOREM Let X 1, X 2,, X n be a random sample from a population X with mean μ X and variance σ X 2 e.g., body heights of n persons waiting times of n customers failure rates of n cars,... Then, for n sufficiently large, the mean തX = X 1+X 2 + +X n n 1. is normally distributed 2. with mean μ ത X = μ X 3. and variance σ ത X 2 = σ X 2 n Capital X, because it is a random variable! Capital തX, because this is also a random variable!

8 THE CENTRAL LIMIT THEOREM So for large n: or for short തX~N μ ത X = μ X, σ ത X 2 = σ X 2 തX~N μ X, σ X 2 This holds regardless of the distribution of X! 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 n n n

9 THE CENTRAL LIMIT THEOREM The CLT for a fair die Distribution of തX for n = 1 n = 2 n = 5 n = 20

10 THE CENTRAL LIMIT THEOREM The CLT for a loaded (unfair) die Distribution of തX for n = 1 n = 2 n = 5 n = 20

11 EXERCISE 1 We roll with a die 100 times. The outcomes are X = X 1, X 2,, X 100. How is തX distributed?

12 THE CENTRAL LIMIT THEOREM A proof of the theorem (for normal populations) Recall the additive property of the normal distribution: if X 1 ~N μ X, σ 2 X and X 2 ~N μ X, σ 2 X, then X 1 + X 2 ~N 2μ X, 2σ 2 X (provided X 1 and X 2 are independent) Also recal that if X~N μ X, σ 2 X then ax~n aμ X, a 2 2 σ X So, if X 1 + X 2 ~N 2μ X, 2σ X 2 then X 1+X 2 2 and more general: X 1+ +X n n or equivalently: തX~N μ X, σ X 2 ~N μ X, σ X 2 n ~N μ X, σ X 2 n This proof works for normal populations and all n, but the CLT is valid for all populations and large n 2 You don t need to reproduce such proofs, but it may help

13 THE CENTRAL LIMIT THEOREM Some consequences of the CLT തX is an estimator of μ X and xҧ is the best estimate of μ X തX will be a better estimator for large n because σ ത X decreases with n we can use the distribution of തX to construct a confidence interval for μ

14 THE CENTRAL LIMIT THEOREM The CLT holds for n sufficiently large More specifically: if X is normally distributed, the CLT holds for all sample sizes n if the distribution of X is fairly symmetric without extreme outliers, for sample sizes n 15 the CLT gives a pretty good approximation of the distribution of തX for any distribution of തX and a sample size n 30, the CLT gives a pretty good approximation of the distribution of തX

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

16 POINT AND INTERVAL ESTIMATES FOR μ A statistic is a function of the (randomly sampled) data important example: the statistic തX defined by തX = 1 n σ i=1 n X i in a concrete case, x ҧ = 1 σ n i=1 n x i is the best possible estimate of the parameter μ so the sample mean xҧ is the best possible estimate of the population mean μ because it is just one value, it is a point estimate

17 ҧ POINT AND INTERVAL ESTIMATES FOR μ Due to sampling variation, xҧ will be different in each sample and there will be a distribution of x-values, ҧ the distribution തX the true value of μ may be different from the value of x obtained however, keep in mind that the value of xҧ obtained cannot be too wrong we know that തX~N μx, ത σx 2 ത, so it follows that a specific value xҧ must be within μx ത 1.96σX, ത μx ത σX ത with 95% probability

18 ҧ ҧ ҧ ҧ ҧ ҧ POINT AND INTERVAL ESTIMATES FOR μ Conversely, the population value μx ത must be within x 1.96σX, ത x σX ത with 95% probability and because μx ത = μ X, the population value μ X must be within x 1.96σX, ത x σX ത with 95% probability this is an interval estimate for μ X we say that x 1.96σX, ത x σX ത is a 95% confidence interval for μ X

19 ҧ POINT AND INTERVAL ESTIMATES FOR μ So: we estimate μ X by xҧ and we know with 95% probability that xҧ 1.96σX ത μ X x σX ത the quantity σx ത = σ X is the standard error of the n distribution of the mean തX it is so important that we give it a special name: the standard error of the mean sometimes (unfortunately!) abbreviated as the standard error

20 EXERCISE 2 We sample (n = 25) from a normal population X with unknown μ X and known σ 2 X = 4. We find x ҧ = 3. a. Give a point estimate for μ X. b. Find the standard error of the mean, sx. ത b. Give a 95%-confidence interval for μ X.

21 CONCEPTS AND SYMBOLS Carefully distinguish: μ X (a value, often unknown) x ҧ (a value from observations) തX (a distribution, not a value) and its two parameters μ ത X and σ ത X 2 (both are values, often unknown) Later on, we will follow a similar logic, e.g. σ X 2 s X 2 S X 2 and its two parameters and the CLT claims that μ തX = μ X σ ത X 2 = σ X 2 n

22 OLD EXAM QUESTION 23 March 2015, Q1h

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