Notion of interval estimation

The estimation of parameters q of a parent population by one number is above considered, i.e. – by the number p – by the number w, s ² – by the number s ². Such estimations of parameters are said to be pointwise. However a pointwise estimation is only approximated value of unknown parameter q even in the case when it is unbiased (it coincides in the average with q), consistent (it tends to q with a growth of n) and effective (it possesses the least degree of random deviations from q) and for a sample of small volume can differ essentially from q.

To receive a representation on accuracy and reliability of an estimate of a parameter q, one uses an interval estimation of parameter. An interval estimate of a parameter q is the numerical interval which with a given probability g covers an unknown value of the parameter q. 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 q – non-random variable, therefore more correctly to speak that the interval “ covers ” instead of “contains” the value q.

Such an interval is said to be confidence, and probability g – 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 l of exponential distribution if as a result of trials the random variable X distributed under the exponential law take the values x₁, x₂, …, x_n.

 

Exercises for Homework 14

14.6. The rests of combustive-lubricating materials in a warehouse of enterprises were studied by random way without replacement. 110 enterprises of 750 have been surveyed. The average rests have made 150 tons at the mean square deviation 42 tons. Determine with confidential probability 0,95 borders in which there will be average rests of combustive-lubricating materials per one enterprise and the general rests of combustive-lubricating materials.

(rests – остатки; combustive-lubricating materials – горюче-смазочные материалы; warehouse – склад).

 

14.7. To define the humidity of a grain 25 tests have been taken by random way. The average percent of humidity of a grain has made 16 %, and the sample mean square deviation – 2,5 %. Determine: a) the unbiased estimations of the mathematical expectation, dispersion and mean square deviation; b) an interval which will cover the mathematical expectation with confidential probability 0,95.

(humidity – влажность; test – проба).

 

14.8. Find by the method of moments the pointwise estimations of the parameters a and b of the scale-distribution:

Direction: Make the substitution и, using the scale-function find firstly and equate

(scale-distribution – гамма-распределение; scale-function – гамма-функция; to equate – приравнять).

 

14.9. Find by the method of the greatest plausibility on sample x₁, x₂, …, x_n the pointwise estimation of the unknown parameter b of scale-distribution if the parameter a is known.

Direction: Use density of the scale-distribution presented in Exercise 14.8.

L E C T U R E 15