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14.3. Confidence interval Confidence probability

Lecture



Previous   14.3.  Confidence interval  Confidence probability we addressed the issue of estimating an unknown parameter   14.3.  Confidence interval  Confidence probability one number. Such an assessment is called "point". In a number of tasks it is required not only to find for the parameter   14.3.  Confidence interval  Confidence probability suitable numerical value, but also to assess its accuracy and reliability. It is required to know - to what errors the replacement of the parameter can lead   14.3.  Confidence interval  Confidence probability its point estimate   14.3.  Confidence interval  Confidence probability and with what degree of confidence can we expect that these errors will not go beyond certain limits?

Such tasks are especially relevant with a small number of observations, when the point estimate   14.3.  Confidence interval  Confidence probability largely random and approximate replacement   14.3.  Confidence interval  Confidence probability on   14.3.  Confidence interval  Confidence probability can lead to serious errors.

To give an idea of ​​the accuracy and reliability of the assessment   14.3.  Confidence interval  Confidence probability , in mathematical statistics use the so-called confidence intervals and confidence probabilities.

Let for parameter   14.3.  Confidence interval  Confidence probability unbiased estimate obtained from experience   14.3.  Confidence interval  Confidence probability . We want to evaluate the possible error. Assign some fairly large probability   14.3.  Confidence interval  Confidence probability (eg,   14.3.  Confidence interval  Confidence probability   14.3.  Confidence interval  Confidence probability or   14.3.  Confidence interval  Confidence probability ) such that an event with probability   14.3.  Confidence interval  Confidence probability can be considered almost reliable, and find such a value   14.3.  Confidence interval  Confidence probability , for which

  14.3.  Confidence interval  Confidence probability . (14.3.1)

Then the range of practically possible values ​​of the error that occurs when replacing   14.3.  Confidence interval  Confidence probability on   14.3.  Confidence interval  Confidence probability , will be   14.3.  Confidence interval  Confidence probability ; large errors in absolute magnitude will appear only with a small probability   14.3.  Confidence interval  Confidence probability .

Rewrite (14.3.1) in the form:

  14.3.  Confidence interval  Confidence probability . (14.3.2)

Equality (14.3.2) means that with probability   14.3.  Confidence interval  Confidence probability unknown parameter value   14.3.  Confidence interval  Confidence probability falls into the interval

  14.3.  Confidence interval  Confidence probability . (14.3.3)

It should be noted one thing. Previously, we have repeatedly considered the probability of a random variable falling into a given non-random interval. Here it is different: the value   14.3.  Confidence interval  Confidence probability not random, but random   14.3.  Confidence interval  Confidence probability . Randomly its position on the x-axis, determined by its center   14.3.  Confidence interval  Confidence probability ; the length of the interval is also random   14.3.  Confidence interval  Confidence probability because the magnitude   14.3.  Confidence interval  Confidence probability It is calculated, as a rule, by experimental data. Therefore, in this case it is better to interpret the value   14.3.  Confidence interval  Confidence probability not like the probability of "hitting" the point   14.3.  Confidence interval  Confidence probability in the interval   14.3.  Confidence interval  Confidence probability , but as the probability that a random interval   14.3.  Confidence interval  Confidence probability will cover the point   14.3.  Confidence interval  Confidence probability (fig. 14.3.1).

  14.3.  Confidence interval  Confidence probability

Fig. 14.3.1.

Probability   14.3.  Confidence interval  Confidence probability it is accepted to call confidence probability, and the interval   14.3.  Confidence interval  Confidence probability - confidence interval. Interval boundaries   14.3.  Confidence interval  Confidence probability :   14.3.  Confidence interval  Confidence probability and   14.3.  Confidence interval  Confidence probability are called confidential boundaries.

We give another interpretation of the concept of a confidence interval: it can be considered as an interval of parameter values   14.3.  Confidence interval  Confidence probability compatible with the experimental data and do not contradict them. Indeed, if we agree to consider an event with probability   14.3.  Confidence interval  Confidence probability almost impossible then those parameter values   14.3.  Confidence interval  Confidence probability for which   14.3.  Confidence interval  Confidence probability , it is necessary to recognize contradicting experimental data, and those for which   14.3.  Confidence interval  Confidence probability compatible with them.

Let us turn to the question of finding confidential boundaries.   14.3.  Confidence interval  Confidence probability and   14.3.  Confidence interval  Confidence probability .

Let for parameter   14.3.  Confidence interval  Confidence probability there is an unbiased estimate   14.3.  Confidence interval  Confidence probability . If we knew the distribution law   14.3.  Confidence interval  Confidence probability , the task of finding a confidence interval would be quite simple: it would be enough to find such a value   14.3.  Confidence interval  Confidence probability , for which

  14.3.  Confidence interval  Confidence probability .

The difficulty is that the distribution law estimates   14.3.  Confidence interval  Confidence probability depends on the law of distribution of magnitude   14.3.  Confidence interval  Confidence probability and, therefore, from its unknown parameters (in particular, from the parameter itself).   14.3.  Confidence interval  Confidence probability ).

To circumvent this difficulty, you can apply the following roughly approximate method: replace in the expression for   14.3.  Confidence interval  Confidence probability unknown parameters of their point estimates. With a relatively large number of experiments   14.3.  Confidence interval  Confidence probability (order   14.3.  Confidence interval  Confidence probability a) this method usually gives results with satisfactory accuracy.

As an example, consider the confidence interval problem for the expectation.

Let produced   14.3.  Confidence interval  Confidence probability independent experiments on a random variable   14.3.  Confidence interval  Confidence probability whose characteristics are mathematical expectation   14.3.  Confidence interval  Confidence probability and variance   14.3.  Confidence interval  Confidence probability - unknown. Estimates for these parameters are obtained:

  14.3.  Confidence interval  Confidence probability ;   14.3.  Confidence interval  Confidence probability . (14.3.4)

Required to build a confidence interval   14.3.  Confidence interval  Confidence probability corresponding to the confidence level   14.3.  Confidence interval  Confidence probability for mathematical expectation   14.3.  Confidence interval  Confidence probability magnitudes   14.3.  Confidence interval  Confidence probability .

In solving this problem, we use the fact that   14.3.  Confidence interval  Confidence probability represents the sum   14.3.  Confidence interval  Confidence probability independent identically distributed random variables   14.3.  Confidence interval  Confidence probability and, according to the central limit theorem, with a sufficiently large   14.3.  Confidence interval  Confidence probability its distribution law is close to normal. In practice, even with a relatively small number of terms (about   14.3.  Confidence interval  Confidence probability a) the distribution law of the sum can be approximately considered normal. We will proceed from the fact that   14.3.  Confidence interval  Confidence probability distributed according to normal law. The characteristics of this law — expectation and variance — are equal, respectively.   14.3.  Confidence interval  Confidence probability and   14.3.  Confidence interval  Confidence probability (see ch. 13   14.3.  Confidence interval  Confidence probability 13.3). Suppose the value   14.3.  Confidence interval  Confidence probability we know and find the value   14.3.  Confidence interval  Confidence probability for which

  14.3.  Confidence interval  Confidence probability . (14.3.5)

Applying the formula (6.3.5) of Chapter 6, we express the probability on the left side (14.3.5) through the normal distribution function

  14.3.  Confidence interval  Confidence probability . (14.3.6)

Where   14.3.  Confidence interval  Confidence probability - standard deviation of assessment   14.3.  Confidence interval  Confidence probability .

From the equation

  14.3.  Confidence interval  Confidence probability

find the value   14.3.  Confidence interval  Confidence probability :

  14.3.  Confidence interval  Confidence probability , (14.3.7)

Where   14.3.  Confidence interval  Confidence probability - inverse function   14.3.  Confidence interval  Confidence probability , i.e., the value of the argument for which the normal distribution function is   14.3.  Confidence interval  Confidence probability .

Dispersion   14.3.  Confidence interval  Confidence probability through which the value is expressed   14.3.  Confidence interval  Confidence probability , we are not exactly known; as its approximate value, you can use the estimate   14.3.  Confidence interval  Confidence probability (14.3.4) and put approximately:

  14.3.  Confidence interval  Confidence probability . (14.3.8)

Thus, the problem of constructing a confidence interval, which is equal to:

  14.3.  Confidence interval  Confidence probability , (14.3.9)

Where   14.3.  Confidence interval  Confidence probability determined by the formula (14.3.7).

To avoid when calculating   14.3.  Confidence interval  Confidence probability inverse interpolation in function tables   14.3.  Confidence interval  Confidence probability , it is convenient to make a special table (see table. 14.3.1), where the values ​​of

  14.3.  Confidence interval  Confidence probability

depending on the   14.3.  Confidence interval  Confidence probability . Magnitude   14.3.  Confidence interval  Confidence probability determines for the normal law the number of standard quadratic deviations that need to be postponed to the right and left of the center of dispersion in order that the probability of getting into the resulting section is equal to   14.3.  Confidence interval  Confidence probability .

Through value   14.3.  Confidence interval  Confidence probability confidence interval is expressed as:

  14.3.  Confidence interval  Confidence probability .

Table 14.3.1

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

0.80

1.282

0.86

1,475

0.91

1.694

0.97

2,169

0.81

1,310

0.87

1.513

0.92

1,750

0.98

2,325

0.82

1,340

0.88

1.554

0.93

1.810

0.99

2.576

0.83

1,371

0.89

1,597

0.94

1,880

0.9973

3,000

0.84

1,404

0.90

1,643

0.95

1,960

0.999

3.290

0.85

1,439

0.96

2,053

Example 1. Produced 20 experiments on the value   14.3.  Confidence interval  Confidence probability ; the results are shown in table 14.3.2.

Table 14.3.2

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

  14.3.  Confidence interval  Confidence probability

one

10.5

6

10.6

eleven

10.6

sixteen

10.9

2

10.8

7

10.9

12

11.3

17

10.8

3

11.2

eight

11.0

13

10.5

18

10.7

four

10.9

9

10.3

14

10.7

nineteen

10.9

five

10.4

ten

10.8

15

10.8

20

11.0

Required to find a rating   14.3.  Confidence interval  Confidence probability for mathematical expectation   14.3.  Confidence interval  Confidence probability magnitudes   14.3.  Confidence interval  Confidence probability and build a confidence interval corresponding to the confidence probability   14.3.  Confidence interval  Confidence probability .

Decision. We have:

  14.3.  Confidence interval  Confidence probability .

Choosing a starting point   14.3.  Confidence interval  Confidence probability , using the third formula (14.2.14) we find the unbiased estimate   14.3.  Confidence interval  Confidence probability :

  14.3.  Confidence interval  Confidence probability ;

  14.3.  Confidence interval  Confidence probability .

According to table 14.3.1 we find   14.3.  Confidence interval  Confidence probability ;

  14.3.  Confidence interval  Confidence probability .

Confidence limits:

  14.3.  Confidence interval  Confidence probability ;

  14.3.  Confidence interval  Confidence probability .

Confidence interval:

  14.3.  Confidence interval  Confidence probability .

Parameter values   14.3.  Confidence interval  Confidence probability lying in this interval are consistent with the experimental data given in table 14.3.2.

In a similar way, a confidence interval can also be constructed for dispersion.

Let produced   14.3.  Confidence interval  Confidence probability independent experiments on a random variable   14.3.  Confidence interval  Confidence probability with unknown parameters   14.3.  Confidence interval  Confidence probability and   14.3.  Confidence interval  Confidence probability and for dispersion   14.3.  Confidence interval  Confidence probability unbiased estimate received:

  14.3.  Confidence interval  Confidence probability , (14.3.11)

Where

  14.3.  Confidence interval  Confidence probability .

It is required to approximately build a confidence interval for the variance.

From the formula (14.3.11) it can be seen that   14.3.  Confidence interval  Confidence probability represents the sum   14.3.  Confidence interval  Confidence probability random variables of the form   14.3.  Confidence interval  Confidence probability . These values ​​are not independent, since any of them includes the value   14.3.  Confidence interval  Confidence probability depending on everyone else. However, it can be shown that by increasing   14.3.  Confidence interval  Confidence probability the distribution law of their sum also approaches normal. Practically at   14.3.  Confidence interval  Confidence probability it can already be considered normal.

Suppose that this is so, and we find the characteristics of this law: expectation and variance. Since the evaluation   14.3.  Confidence interval  Confidence probability - unbiased, then

  14.3.  Confidence interval  Confidence probability .

Variance calculation   14.3.  Confidence interval  Confidence probability associated with relatively complex calculations, so we give its expression without output:

  14.3.  Confidence interval  Confidence probability , (14.3.12)

Where   14.3.  Confidence interval  Confidence probability - the fourth central moment of magnitude   14.3.  Confidence interval  Confidence probability .

To use this expression, you need to substitute the values ​​in it   14.3.  Confidence interval  Confidence probability and   14.3.  Confidence interval  Confidence probability (at least approximate). Instead   14.3.  Confidence interval  Confidence probability you can use his assessment   14.3.  Confidence interval  Confidence probability . In principle, the fourth central point   14.3.  Confidence interval  Confidence probability You can also replace it with an estimate, for example, a value of the form:

  14.3.  Confidence interval  Confidence probability (14.3.13)

but such a replacement will give extremely low accuracy, since in general, with a limited number of experiments, moments of high order will be determined with large errors. However, in practice it often happens that the type of distribution law   14.3.  Confidence interval  Confidence probability known in advance: only its parameters are unknown. Then you can try to express   14.3.  Confidence interval  Confidence probability through   14.3.  Confidence interval  Confidence probability .

Take the most frequent case when   14.3.  Confidence interval  Confidence probability distributed according to normal law. Then its fourth central moment is expressed in terms of variance (see Ch. 6   14.3.  Confidence interval  Confidence probability 6.2):

  14.3.  Confidence interval  Confidence probability ,

and the formula (14.3.12) gives

  14.3.  Confidence interval  Confidence probability

or

  14.3.  Confidence interval  Confidence probability . (14.3.14)

Replacing in (14.3.14) the unknown   14.3.  Confidence interval  Confidence probability his assessment   14.3.  Confidence interval  Confidence probability , we get:

  14.3.  Confidence interval  Confidence probability (14.3.15)

from where

  14.3.  Confidence interval  Confidence probability . (14.3.16)

Moment   14.3.  Confidence interval  Confidence probability can be expressed through   14.3.  Confidence interval  Confidence probability also in some other cases where the distribution of the magnitude   14.3.  Confidence interval  Confidence probability It is not normal, but its appearance is known. For example, for the law of uniform density (see Chapter 5) we have:

  14.3.  Confidence interval  Confidence probability ;   14.3.  Confidence interval  Confidence probability ,

Where   14.3.  Confidence interval  Confidence probability - the interval at which the law is given. Consequently,

  14.3.  Confidence interval  Confidence probability .

According to the formula (14.3.12) we get:

  14.3.  Confidence interval  Confidence probability ,

where we find approximately

  14.3.  Confidence interval  Confidence probability . (14.3.17)

In cases where the type of distribution law   14.3.  Confidence interval  Confidence probability unknown, with estimated value   14.3.  Confidence interval  Confidence probability it is recommended to use the formula (14.3.16), if there are no special grounds for believing that this law is very different from normal (it has a noticeable positive or negative kurtosis).

If approximate value   14.3.  Confidence interval  Confidence probability obtained in one way or another, it is possible to build a confidence interval for the variance, in the same way as we built it for the expectation:

  14.3.  Confidence interval  Confidence probability , (14.3.18)

where is the value   14.3.  Confidence interval  Confidence probability depending on a given probability   14.3.  Confidence interval  Confidence probability is in table 14.3.1.

Example 2. Find approximately 80% confidence interval for the variance of a random variable   14.3.  Confidence interval  Confidence probability in the conditions of example 1, if it is known that   14.3.  Confidence interval  Confidence probability distributed according to a law close to normal.

Decision. The value remains the same as in example 1:

  14.3.  Confidence interval  Confidence probability .

According to the formula (14.3.16)

  14.3.  Confidence interval  Confidence probability .

According to the formula (14.3.18) we find the confidence interval:

  14.3.  Confidence interval  Confidence probability .

The corresponding interval of values ​​of the standard deviation:   14.3.  Confidence interval  Confidence probability .


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis