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SAMPLE MEAN and its distribution
1015201825
1x
152628232830
2x
1830112726
3x
1729351612
4x
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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
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SAMPLE MEAN and its distribution
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
ESTIMATORS
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• point
• interval
Properties of Point Estimators
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• UNBIASEDNESS• CONSISTENCY• EFFICIENCY
Properties of Point Estimators
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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
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UNBIASEDNESS
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Properties of Point Estimators
individual sample
estimates
true value of population parameter
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ZÁKLADNÍ VLASTNOSTI BODOVÝCH ODHADŮ
y – sample estimatesM - „average“ of sample estimates
bias of estimatestrue value of population parameter
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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.
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Properties of Point Estimators
CONSISTENCY
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Properties of Point Estimators
EFFICIENCYAn unbiased estimator is efficient if its standard error is lower than that of other unbiased estimators
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Properties of Point Estimators
unbiased estimator with large variability (unefficient)
unbiased estimator with
small variability (efficient)
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POINT ESTIMATES
E X = μ
Point estimate of population mean:
Point estimate of population variance:
2 2nS = σ
n -1
bias correction
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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
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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
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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
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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
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ONE-SIDED INTERVAL ESTIMATES
LEFT-SIDED ESTIMATE
1P(τ > T ) = 1 - α 2P(τ < T ) = 1 - α
RIGHT-SIDED ESTIMATE
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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
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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
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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
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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
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CONFIDENCE INTERVAL (CI) OF POPULATION STAND. DEVIATION
for large samples
α/2
Sσ = S ± z .
2n
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