1

SAMPLE MEAN and its distribution

1015201825

1x

152628232830

2x

1830112726

3x

1729351612

4x

2

SAMPLE MEAN and its distribution

E(X) =X X

σσ = SE =

nCENTRAL LIMIT THEOREM:

If sufficiently large sample is taken from population with any distribution with mean and standard deviation , then sample mean has sample normal distribution N(,2/n)

It means that:

sample mean is a good estimate of population mean

with increasing sample size n, standard error SE is lower and estimate of population mean is more reliable

3

SAMPLE MEAN and its distribution

http://onlinestatbook.com/stat_sim/sampling_dist/index.html

ESTIMATORS

4

• point

• interval

Properties of Point Estimators

5

• UNBIASEDNESS• CONSISTENCY• EFFICIENCY

Properties of Point Estimators

6

UNBIASEDNESS

An estimator is unbiased if, based on repeated sampling from the population, the average value of the estimator equals the population parameter. In other words, for an unbiased estimator, the expected value of the point estimator equals the population parameter.

Properties of Point Estimators

7

UNBIASEDNESS

8

Properties of Point Estimators

individual sample

estimates

true value of population parameter

9

ZÁKLADNÍ VLASTNOSTI BODOVÝCH ODHADŮ

y – sample estimatesM - „average“ of sample estimates

bias of estimatestrue value of population parameter

10

Properties of Point Estimators

CONSISTENCY

An estimator is consistent if it approaches the unknown population parameter being estimated as the sample size grows larger

Consistency implies that we will get the inference right if we take a large enough sample. For instance, the sample mean collapses to the population mean (X̅�  → μ) as the sample size approaches infinity (n → ∞). An unbiased estimator is consistent if its standard deviation, or its standard error, collapses to zero as the sample size increases.

11

Properties of Point Estimators

CONSISTENCY

12

Properties of Point Estimators

EFFICIENCYAn unbiased estimator is efficient if its standard error is lower than that of other unbiased estimators

13

Properties of Point Estimators

unbiased estimator with large variability (unefficient)

unbiased estimator with

small variability (efficient)

14

POINT ESTIMATES

E X = μ

Point estimate of population mean:

Point estimate of population variance:

2 2nS = σ

n -1

bias correction

15

POINT ESTIMATES

sample

population

this distance is unknown (we do not know the exact value of so we can not quatify reliability of our estimate

X

16

INTERVAL ESTIMATES

1 2P T τ T = 1- α

Confidence interval for parametr with confidence level(0,1) is limited by statistics T1 a T2:.

point estimate of unknown population mean computed from sample data– we do not know anything about his distance from real population mean

T1T2

interval estimate of unknown population mean - we

suppose, that with probability P =1- population mean is anywhere in this interval of

number line

X

17

CONFIDENCE LEVEL IN INTERVAL ESTIMATES

1x

2x

2x

these intervals include real value of population mean (they are „correct“), there will be at least (1- ).100 % these „correct“ estimatesthis interval does not

include real value of population mean (it is „incorrect“), there will be at most (100) % of these „incorrect“ estimates

18

TWO-SIDED INTERVAL ESTIMATES

T1 T2

P = 1 - = 1 – (1 + 2)1= /2

2= /2

T

1 a 2 represent statistical risk, that real population parameter is outside of interval (outside the limits T1 a T2

19

ONE-SIDED INTERVAL ESTIMATES

LEFT-SIDED ESTIMATE

1P(τ > T ) = 1 - α 2P(τ < T ) = 1 - α

RIGHT-SIDED ESTIMATE

20

COMPARISON OF TWO- AND ONE-SIDED INTERVAL ESTIMATES

T1 two-sided interval estimateP = 1 -

/2 /2

T

T2

one-sided interval estimate P = 1 - T1

21

CONFIDENCE INTERVAL (CI) OF POPULATION MEAN

small sample (less then 30 measurements)

S S

n n /2,n-1 /2,n-1x - t x + t

t/2,n-1 quantil of Student ‘s t-distribution with (n-1) degrees of freedom and /2 confidence level

lower limit of CI upper limit of CI

22

CONFIDENCE INTERVAL (CI) OF POPULATION MEAN

large sample (over 30 data points)

n n

/2 /2x - z x + z

z/2 quantile of standardised normal distribution

lower limit of CI upper limit of CI

instead of (population SD) there is possible to use sample estimate S

23

CONFIDENCE INTERVAL (CI) OF POPULATION STAND. DEVIATION

for small samples

, 1 , 1n n

2 2

2 2α α

1-2 2

(n -1) S (n -1) Sσ

χ χ

24

CONFIDENCE INTERVAL (CI) OF POPULATION STAND. DEVIATION

for large samples

α/2

Sσ = S ± z .

2n