Notion of interval estimation
The
estimation of parameters
of a parent population by one
number
is above considered, i.e.
– by the number
p
– by the number w,
2
– by the number s2.
Such estimations of parameters are said to be pointwise.
However a pointwise estimation
is only approximated value of unknown parameter
even
in the case when it is unbiased (it coincides in the average with ),
consistent (it tends to
with a growth of n)
and effective (it possesses the least degree of random deviations
from )
and for a sample of small volume can differ essentially from .
To
receive a representation on accuracy and reliability of an estimate
of
a parameter ,
one uses an interval estimation of parameter. An
interval estimate
of a parameter
is the numerical interval
which
with a given probability
covers an unknown value of the parameter.
We pay attention that boundaries of the interval
and its size are found on sampling data and consequently are random
variables
as against the estimated parameter
– non-random
variable,
therefore more correctly to speak that the interval “covers”
instead of “contains” the value .
Such an interval is said to be confidence, and probability – confidence probability, confidence level or reliability of estimation.
Glossary
estimate – оценка; unbiased – несмещенная
unbiasedness – несмещенность
consistent – состоятельная; parent mean – генеральная средняя
sample mean – выборочная средняя
confidence interval – доверительный интервал
pointwise – точечный
Exercises for Seminar 14
14.1. To determine losses of a grain at harvest 100 measurements by random way have been carried out. The average size of losses has made 1,8 centner from one hectare of crops at the mean square deviation 0,5 centner per a hectare. Determine with confidential probability 0,95 borders in which there will be an average size of losses of a grain per a hectare and possible size of losses if the area of grain harvest has made 640 hectares.
(grain – зерно; harvest – уборка; centner – центнер; hectare – гектар; sowing – посев; crops – посевы).
14.2. By means of a random sample the time of performing the industrial operation by workers of a brigade was studied. On the basis of 60 observations it has been established that for a performing the industrial operation were spent 0,5 hours on the average at the mean square deviation 0,12 hours. Assuming that the time of performing the industrial operation is a normally distributed random variable, determine borders in which there is average time of performing the industrial operation of all the workers with confidential probability: a) 0,9; b) 0,95.
14.3. There are 10000 country sites of the population in a district. As a result of sample inspection of 300 country sites was appeared that average sample productivity of vegetables has made 250 centners per a hectare at the mean square deviation 60 centners per a hectare. It is known that 40% of the general area of crops of vegetables was occupied by tomatoes. Determine with confidential probability 0,95 borders in which there will be an average productivity of vegetables on all country sites and densities of crops of tomatoes. How many is necessary to survey country sites that the limiting mistake of sample on attributes has decreased in 1,5 times?
(country site – дачный участок; the population – население; densities – удельный вес; to survey – обследовать; limiting – предельный; attribute – признак).
14.4.
Find by the method of moments the pointwise estimation of an excess
of theoretical distribution.
14.5.
Find by the method of the greatest plausibility an estimation of the
parameter
of exponential distribution
if
as a result of trials the random variable X
distributed under the exponential law take the values x1,
x2,
…, xn.