Lecture 2 Estimating the population mean

Transcription

1 Lecture 2 Estimating the population mean 1

2 Estimating the mean of a population Estimator = function of a sample of data drawn randomly from the population. Estimate = numerical value of the estimator, given a particular sample. 2

3 Estimating the mean of a population Population mean μ Y = 1 σ N i=1 N Y i Sample mean തY = 1 σ n i=1 n Y i, n < N തY is a natural estimate of μ Y. 3

4 Estimating the mean of a population 2 questions: 1. What are the properties of തY? 2. Why use തY and not some other estimator? 4

5 Properties of തY തY is a random variable. Its properties are determined by the sampling distribution ( otantajakauma ). The individual observations which are used to calculate തY were chosen randomly. 5

6 Properties of തY തY is random. Q: what happens if you take a different random sample? The distribution of തY over different samples of the same size (n) is called the sampling distribution. 6

7 Properties of തY Sampling distribution: all the values that തY can take given n + The probability of each of these values. The mean and variance of തY are the mean and variance of its sampling distribution. The sampling distribution is very important. 7

8 Properties of തY If E തY = μ Y, then തY is an unbiased (harhaton) estimate of μ Y. (Note any estimator μ Y ). If തY μ Y when n, then തY is a consistent (tarkentuva) estimate of μ Y. This is the case, due to the Law of Large Numbers ( suurten lukujen laki ), under certain conditions. 8

9 Law of Large Numbers: conditions Y i are independently and identically distributed. E Y i = μ Y No large outliers / var Y i < 9

10 Properties of തY How precise is തY, and how does this depend on n? In other words, how large is the variance of തY? Central Limit Theorem ( Keskeinen raja-arvolause ). 10

11 Central Limit Theorem Suppose the sample is random and i.i.d. E Y i = μ Y var Y i = σ Y 2, 0 < σ Y 2 <. Then, as n, distribution of ( തY μ Y )/ σ ത Y 2 becomes arbitrarily well approximated by the standard normal distribution. 11

12 Central Limit Theorem CLT is about the distribution of the estimate of the mean. CLT applies no matter what the underlying distribution is. Examples: coin tosses (binary), age (only positive values / integers observed), 12

13 Properties of തY Result 1. E തY = μ Y 2. Var തY = σ 2 /n 13

14 Height of students in class height ind/group mean mean

15 Data in a histogram.05 Density height sum height global mean_h = r(mean) histogram height, width(1) start(150) xline($mean_h) graph save "${root}\histo_heightv", replace 15

16 same with some statistics height ind/group mean sd mean sd

17 0.05 Density times sample of mean_group5 17

18 . tab mean_group5 mean_group5 Freq. Percent Cum , , , , , Total 12,

19 0 0 Density Density times sample of times sample of mean_group_10_ mean_group_100_4 19

20 തY as a least squares estimator തY minimizes the sum of squared residuals. Optimizing (see App. 3.2) yields N min m (Y i m) 2 i=1 m = N 1 n i=1 Y i = തY 20

21 തY as a least squares estimator തY has smaller variance than all other linear unbiased estimators. തY is more efficient than other (linear) estimators. തY is BLUE (best linear unbiased estimator). 21

22 Choosing an objective / loss function Least squares Absolute deviations Min / max. May depend on context: 1. think of a basket ball team. 2. think of #incubators relative to need. 22

23 Comparing means Two means, തY 1 and തY 2. (height of male / female students). Are they (not) different? is തY 1 - തY 2 = 0? What else do you know? You have an estimate of the variances of the means. 23

24 Comparing means തY 1 and തY 2 are independently distributed. their difference is normally distributed. variance of തY 1 തY 2 is σ n 1 + σ 2 n 2 24

25 Density Density.1.15 Female height distr Male height distr height height histogram height if gender == 1, width(1) start(150) xline($mean_h) graph save "${root}\histo_heightf", replace 25

26 . tabstat height, stat(mean sd n) by(gender) Summary for variables: height by categories of: gender gender mean sd N Total

27 . ttest height, by(gender) unequal Two-sample t test with unequal variances Group Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] combined diff diff = mean(0) - mean(1) t = Ho: diff = 0 Satterthwaite's degrees of freedom = Ha: diff < 0 Ha: diff!= 0 Ha: diff > 0 Pr(T < t) = Pr( T > t ) = Pr(T > t) =