Lecture Outline Simple random sampling Statistics for Business Distribution of the sample average Large sample approximation to the distribution of the sample mean Sampling Distributions, Interval Estimation and Hypothesis Tests. ◮ Law of Large Numbers ◮ Central Limit Theorem Panagiotis Th. Konstantinou Estimation of the population mean ◮ Unbiasedness MSc in International Shipping, Finance and Management , ◮ Consistency ◮ Efficiency Athens University of Economics and Business Hypothesis test concerning the population mean First Draft : July 15, 2015. This Draft : September 17, 2020. Confidence intervals for the population mean ◮ Using the t -statistic when n is small Comparing means from different populations P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 1 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 2 / 61 Sampling and Sampling Distributions Sampling: Intro Sampling and Sampling Distributions Simple Random Sampling Sampling Simple Random Sampling – I A population is a collection of all the elements of interest, while a Simple random sampling means that n objects are drawn randomly from sample is a subset of the population. a population and each object is equally likely to be drawn The reason we select a sample is to collect data to answer a research Let Y 1 , Y 2 , ..., Y n denote the 1st to the n th randomly drawn object. Under question about a population. simple random sampling The sample results provide only estimates of the values of the ◮ The marginal probability distribution of Y i is the same for all i = 1 , 2 , ..., n population characteristics. With proper sampling methods , the sample and equals the population distribution of Y . results can provide “good” estimates of the population characteristics. ⋆ because Y 1 , Y 2 , ..., Y n are drawn randomly from the same population. A random sample from an infinite population is a sample selected such ◮ Y 1 is distributed independently from Y 2 , ..., Y n . knowing the value of Y i does not provide information on Y j for i � = j that the following conditions are satisfied: ◮ Each element selected comes from the population of interest. When Y 1 , Y 2 , ..., Y n are drawn from the same population and are ◮ Each element is selected independently . independently distributed, they are said to be I.I.D. random variables ⋆ If the population is finite, then we sample with replacement... P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 3 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 4 / 61
Sampling and Sampling Distributions Simple Random Sampling Sampling and Sampling Distributions Sampling Distribution of the Sample Average Simple Random Sampling – II The Sampling Distribution of the Sample Average – I The sample average ¯ Y of a randomly drawn sample is a random variable Example with a probability distribution called the sampling distribution Let G be the gender of an individual ( G = 1 if female, G = 0 if male) n Y = 1 n ( Y 1 + Y 2 + · · · + Y n ) = 1 � G is a Bernoulli r.v. with E ( G ) = µ G = Pr( G = 1 ) = 0 . 5 ¯ Y i n Suppose we take the population register and randomly draw a sample of i = 1 size n ◮ The individuals in the sample are drawn at random. ◮ The probability distribution of G i is a Bernoulli with mean 0 . 5 ◮ Thus the values of ( Y 1 , Y 2 , · · · , Y n ) are random ◮ G 1 is distributed independently from G 2 , ..., G n ◮ Thus functions of ( Y 1 , Y 2 , · · · , Y n ), such as ¯ Y , are random: had a different Suppose we draw a random sample of individuals entering the building sample been drawn, they would have taken on a different value of the accounting department ◮ The distribution of over different possible samples of size n is called the sampling distribution of ¯ ◮ This is not a sample obtained by simple random sampling and Y . G 1 , G 2 , ..., G n are not i.i.d ◮ The mean and variance of are the mean and variance of its sampling distribution, E (¯ Y ) and Var (¯ ◮ Men are more likely to enter the building of the accounting department! Y ) . ◮ The concept of the sampling distribution underpins all of statistics/econometrics. P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 5 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 6 / 61 Sampling and Sampling Distributions Sampling Distribution of the Sample Average Sampling and Sampling Distributions Sampling Distribution of the Sample Average The Sampling Distribution of the Sample Average – II The Sampling Distribution of the Sample Average – III Example n Y = 1 n ( Y 1 + Y 2 + · · · + Y n ) = 1 � ¯ Y i Let G be the gender of an individual ( G = 1 if female, G = 0 if male) n i = 1 The mean of the population distribution of G is Suppose that Y 1 , Y 2 , ..., Y n are I.I.D. and the mean & variance of the population distribution of Y are respectively µ Y and σ 2 E ( G ) = µ G = Pr( G = 1 ) = p = 0 . 5 Y ◮ The mean of (the sampling distribution of) ¯ Y is The variance of the population distribution of G is � � n n 1 = 1 E ( Y i ) = 1 � � E (¯ Y ) = E nn E ( Y ) = µ Y Y i Var ( G ) = σ 2 G = p ( 1 − p ) = 0 . 5 ( 1 − 0 . 5 ) = 0 . 25 n n i = 1 i = 1 ◮ The variance of (the sampling distribution of) ¯ The mean and variance of the average gender (proportion of women) ¯ Y is G in a random sample with n = 10 are � � n n n n 1 = 1 Var ( Y i ) + 2 1 � � � � Var (¯ Y ) = Var Cov ( Y i , Y j ) Y i n 2 n 2 n E (¯ G ) = µ G = 0 . 5 i = 1 i = 1 i = 1 j = 1 , j � = i n Var ( Y ) = σ 2 1 G = 1 n 2 n Var ( Y ) + 0 = 1 1 Var (¯ n σ 2 Y G ) = 100 . 25 = 0 . 025 = n P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 7 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 8 / 61
Sampling and Sampling Distributions Sampling Distribution of the Sample Average Sampling and Sampling Distributions Sampling Distribution of the Sample Average The Sampling Distribution of the Average Gender ¯ The Finite-Sample Distribution of the Sample Average G Suppose G takes on 0 or 1 (a Bernoulli random variable) with the probability distribution The finite sample distribution is the sampling distribution that exactly Pr( G = 0 ) = p = 0 . 5 , Pr( G = 1 ) = 1 − p = 0 . 5 describes the distribution of ¯ Y for any sample size n . In general the exact sampling distribution of ¯ Y is complicated and As we discussed above: depends on the population distribution of Y . E ( G ) = µ G = Pr( G = 1 ) = p = 0 . 5 A special case is when Y 1 , Y 2 , ..., Y n are IID draws from the N ( µ Y , σ 2 Y ) , σ 2 Var ( G ) = G = p ( 1 − p ) = 0 . 5 ( 1 − 0 . 5 ) = 0 . 25 because in this case � � µ Y , σ 2 ¯ Y Y ∼ N The sampling distribution of ¯ G depends on n . n Consider n = 2. The sampling distribution of ¯ G is G = 0 ) = 0 . 5 2 = 0 . 25 ◮ Pr(¯ ◮ Pr(¯ G = 1 / 2 ) = 2 × 0 . 5 × ( 1 − 0 . 5 ) = 0 . 5 G = 1 ) = ( 1 − 0 . 5 ) 2 = 0 . 25 ◮ Pr(¯ P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 9 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 10 / 61 Sampling and Sampling Distributions Sampling Distribution of the Sample Average Sampling and Sampling Distributions Asymptotic Approximations The Finite-Sample Distribution of the Average Gender ¯ The Asymptotic Distribution of the Sample Average ¯ G Y Suppose we draw 999 samples of n = 2: · · · Sample 1 Sample 1 Sample 3 Sample 999 Given that the exact sampling distribution of ¯ Y is complicated and given ¯ ¯ ¯ ¯ G 1 G 2 G G 1 G 2 G G 1 G 2 G G 1 G 2 G that we generally use large samples in statistics/econometrics we will 0 . 5 0 . 5 1 0 1 1 1 0 1 0 0 0 often use an approximation of the sample distribution that relies on the sample being large Sample distribution of average gender 999 samples of n=2 The asymptotic distribution or large-sample distribution is the .5 approximate sampling distribution of ¯ Y if the sample size becomes very .4 large: n → ∞ . probability We will use two concepts to approximate the large-sample distribution of .3 the sample average .2 ◮ The law of large numbers. ◮ The central limit theorem. .1 0 0 .2 .4 .5 .6 .8 1 sample average . P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 11 / 61 P. Konstantinou (AUEB) Statistics for Business – III September 17, 2020 12 / 61
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