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## 7.4  Interval Estimation

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Sometimes it is required to find not only a numerical estimation of the parameter of a population, but also to evaluate its accuracy and reliability. It is especially important for a sample with small volume. Generalization of a point estimation of the statistical parameter is an interval estimation - the numerical interval containing the estimated parameter with defined probability.

In definiting general characteristics of sampled data some error arises. In this case, it is better to define an interval with the centre equal to the point estimator. Inside such an interval with given probability there is a true required value of the estimated parameter of the general population. Such an interval is called confidence. The confidence interval is a numerical interval which, with the given probability , include the estimated parameter of a population. Such probability is known as the confidential probability. The confidential probability is a probability which is sufficient for the inference about reliability of the parameters obtained on the sample. Probability to make an error is called a significance level: (7.19)

For point (sample) estimation – of a population’s parameter – with accuracy (the limiting error) – , and a confidential probability – , the confidence interval – is determined by equality: (7.20)

The limiting error ( ) and the mean error ( ) have an interrelation: (7.21)

where: t – a confidence coefficient.

The limiting error for a small sample is calculated as follows: (7.22)

The limiting error also depends on a probability (р) which guarantees such error. On the basis of theorems of Chebyshev and Lyapunov we have the interrelation of a confidence coefficient (t) with a probability (р):

 Confidence coefficient t Probability р 1.0 0.683 1.5 0.866 2.0 0.954 2.5 0.987 3.0 0.997

If we have a probability р=0.997, a confidence coefficient t will equal 3, and for an average we will have an interval: or . This is the so-called rule of three sigma. In 997 events out of 1000 we can assert that the population mean will not fall outside the limit of sample mean ± triple mean error of sample.

Confidence coefficient t (or trust’s coefficient) has connection with confidential probability. This coefficient is defined by the Laplace integral with the confidential probability : (7.23)

The confidential probability γ enables us to form a confidential bounds ( ) of a random fluctuation of the studied parameter (Q) for the given sample.

Confidential probability has the following values and significance level (α = 1 – ) corresponding to them:

 Confidential probability Significance level, α 0,90 0,10 or 10 % 0,95 0,05 or   5 % 0,99 0,01 or   1 %

The 5% significance level means that in 5 events out of 100 there is a possibility to do an error in calculation of a parameter of a population on sampled data. In other words, in 95 events out of 100, parameters of a population calculated on the sample will lie within the limits of a confidence interval.

For example, at a shearing of 100 sheep from 1000 heads, the sample mean is 4.5 kg, the sample standard deviation is 1.3 kg. Find out a limiting error of sample with probability 0.954 and possible limits of mean shearing from one sheep in population: =0.25, , , .

The population mean (at shearing of 1000 sheep) can’t be less than 4.25 kg and can’t be more than 4.75 kg with a probability 95.4%.

Let's consider an example of an estimation of arithmetic mean on a small sample. From a herd of sheep are selected 17 sheep which were weighed (sample size is n=17). Sample mean weight is =60 kg, and . Let's calculate a probability of what the population mean is in interval , if a significance level a= 5%. Using the parity: (7.24)

and denoting a limiting error as ∆, we find confidential borders of an interval: .

The number of degrees of freedom equals 16. The trust’s coefficient t = 2.12. Therefore: Confidential interval is (60-4.7; 60+4.7). With probability 0.95 it is possible to confirm that population mean weight of sheep is in an interval (55.3; 64.7).

Using the formula of limiting error it is possible to define the minimum volume (size) of a sample for which the estimation with needed properties is provided:

for selection with replacement: (7.25)

for selection without replacement: (7.26)

Let's consider characteristics of two independent samples. There are: volume (size) of two samples - n1, n2, arithmetic mean of two samples - , , and sample variance - , . Summation/subtraction of these samples has equality: (7.27) (7.28)

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