53. Single Sample: Estimating the Variance.

If a sample of size n is drown from normal population with variance and the Sample variance is computed the obtain a value of statistics . This computed sample variance is used as a point estimate of hence, statistics is called Estimator of . If we use the definition of chi-square distribution with (n-1) degrees of freedom we may right

P( < )=1-α there and is leaving areas of to the right. If we substituting = (*) formula with (*)we may write the next formula P[ ]= 1-α

Def2. Confidence interval for . If the variance of random sample of size n from a normal population 100(1-α)% confidence interval for is equal where - is value is leaving the area to the right

And v=(n-1)degree of freedom.

54. Sampling Distribution of s2.

Def1. Sampling distribution of important statistics allow as to learn information about parameters. If is the variance of random sample of size n taken from a normal population having the variance then statistics (ши/хи) . = =

And the statistics has a chi-squared distribution with v=(n-1)-degrees of freedom.

Remark1.

The values of are calculate from each sample by the formula

=

The probability that a random sample produce a value greater than some specified value is equal to the area under the curve to the right of this area and this figure

Example: =2,167

V=7

α = 0,95

55. Statistical Hypotheses: General Concepts.

Null and alternative Hypotheses

Type of Alternative Hypotheses(for initial simple parameter hypotheses)

13.4.1

Alternative Hypotheses is : and (gamma)ϓ+α=1 and we consider the critical areas

The double-side critical area(domain)

Remark.3 The double-side critical area is area there null hypotheses is not allowed. 13.4.2

: >

13.4.3 : <

56. Prove the formula of Poisson distribution:

m=0,1,2,… where λ is the average number of outcomes np-λ, for n->

= =1… |p | = =1|