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The Law of Large Numbers in Children's Education

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
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Received: 09 Feb 2022
Accepted: 09 Apr 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

Big data technology provides data and “evidence” support for education management and decision-making and provides a new research paradigm for education research. However, education research, education management, and education decision-making emphasize both “regularity” and “purposes.” Therefore, the application of big data technology for educational research and educational decision-making still has limitations. For this reason, big data technology cannot be mechanically applied to solve the problems faced by education [1]. We should rationally look at the “empirical science” of educational research and educational decision-making. At the same time, we must rationally realize the scientific nation of educational research and educational decision-making. Research on children's social attention is helpful to grasp the more serious and urgent problems faced by children. This is conducive to solving children's problems and strengthening precise services for the care and protection of children. “Probability theory and mathematical statistics is a mathematical subject that studies the statistical laws of random phenomena.” “Probability theory is to study a large number of random phenomena from a quantitative perspective and obtain the laws that these random phenomena obey.”

For this reason, we apply the law of large numbers to children's education [2]. At the same time, through correlation or correlation analysis, we can find the leading internal factors and highly correlated external factors of children's educational activities. This provides technical support for educational decision-making, educational management, and educational research.

Law of Large Numbers

In statistics, the law of large numbers is as follows:

If ξ1, ξ2, … ξn, … is a random variable sequence, if there is a constant column a1, a2, … an, … so that for any ε > 0, the formula (1) holds, limnP(|i=1nξinan|<ε)=1 \mathop {\lim}\limits_{n \to \infty} \,P\left({\left| {{{\sum\limits_{i = 1}^n {{\xi _i}}} \over n} - {a_n}} \right| < \varepsilon} \right) = 1

Then we believe that the random variable sequence {ξn} obeys the law of large numbers.

Bernoulli's theorem is one of the famous laws of large numbers. Let μn be the number of occurrences of an event A in the n weighted Bernoulli trial and the probability of occurrence of A in each trial is p (0 < p < 1). For any ε >0 limnP(|μnnp|<ε)=1 \mathop {\lim}\limits_{n \to \infty} \,P\left({\left| {{{{\mu _n}} \over n} - p} \right| < \varepsilon} \right) = 1

This law of large numbers does not tell us why the probability of A appearing in each experiment is p and when P(A) = p is. Why does μAn {{{\mu _A}} \over n} obey the binomial distribution? The law of large numbers here is only a mathematics deduction, not an empirical law [3]. As long as we give the sequence of random variables, we can prove whether they have the above definitions and theorems. But in empirical research, we prove the existence of statistical laws through analyzing actual data.

At the same time, the textbook describes the law of statistics like this: When a certain condition group is realized, there are multiple possible results. People can’t predict what kind of result will appear beforehand, but the result is obtained when many repeated observations show a certain pattern. We call it the statistical regularity of random phenomena [4]. This description does not conform to scientific norms. Below we define statistical laws based on the definition of probability and hypothesis testing. In this way, statistical laws are scientifically standardized and can be tested in practice. To this end, first, discuss the interpretation of hypothesis testing.

Hypothesis test explanation

Hypothesis testing in mathematical statistics includes two parts: parametric and non-parametric. Only some discussions on parameter testing are given below [5]. The available parameter inspection system can be described as follows:

Suppose that the distribution function F(x: θ) of the population ξ contains an unknown parameter θ, and the parameter space is denoted as Ω. θ ∈ Ω consider the following hypothesis testing problem H0:θΩ0,H1:θΩΩ {H_0}:\theta\in {\Omega _0},\,\,{H_1}:\theta\in \Omega- \Omega

We divide the sample space X into two disjoint parts X and X − X0. ∀θ* ∈ Ω. Hypothesis M(X0,θ*)=P{(ξ1,,ξn)X0|θ=θ*}=Pθ*{X0} M\left({{{\rm{X}}_0},{\theta ^*}} \right) = P\left\{{\left({{\xi _1}, \cdots,{\xi _n}} \right) \in {{\rm{X}}_0}|\theta= {\theta ^*}} \right\} = {P_{{\theta ^*}}}\left\{{{{\rm{X}}_0}} \right\}

If X0 is the negation domain of negating hypothesis H0 (also called rejection domain), two types of errors may be made in the choice of hypothesis testing. The first type of error: the observation value (x1, …, xn) ∈ X0 of (ξ1, …, ξn) when θ ∈ Ω0 is. At this time, we believe that it made a mistake of rejecting the truth [6]. The second type of error: when θ¯Ω0 \theta \bar\in {\Omega _0} is the observed value (x1, …, xn) ∈ X − X0 of (ξ1, …, ξn). At this time, we believe that he has made a false recognition error. Usually, the probability of making the first type of error is α: M(X0,θ)H0=P{(ξ1,,ξn)X0|θΩ0}=α M{\left({{{\rm{X}}_0},\theta} \right)_{{H_0}}} = P\left\{{\left({{\xi _1}, \cdots,{\xi _n}} \right) \in {{\rm{X}}_0}|\theta\in {\Omega _0}} \right\} = \alpha

And the probability of making the second type of error is P{(ξ1,,ξn)XX0|θ¯Ω0}=1P{(ξ1,,ξn)X|θ¯Ω0}=1M(X0,θ)H1 P\left\{{\left({{\xi _1}, \cdots,{\xi _n}} \right) \in {\rm{X}} - {{\rm{X}}_0}|\theta \bar\in {\Omega _0}} \right\} = 1 - P\left\{{\left({{\xi _1}, \cdots,{\xi _n}} \right) \in {\rm{X}}|\theta \bar\in {\Omega _0}} \right\} = 1 - M{\left({{{\rm{X}}_0},\theta} \right)_{{H_1}}}

It is generally believed that the basic idea is the so-called small probability event principle. Small probability events (or events with very low probability) are almost impossible to occur in an experiment or observation [7]. However, suppose you think carefully about the above logical process of determining the negative domain. In that case, you will find that it has nothing to do with the small probability event principle at all. In the significance test, its basic idea is to make the probability of making the second type of error as small as possible under the control of making the first type of error.

It is still questionable when we consider the empirical significance of the logic of determining the negative domain. Because when in fact θ ∈ Ω0, why do we need to consider the probability of making a mistake in θ ∈ Ω − Ω0? We cannot make two types of mistakes at the same time. The two types of errors are incompatible independent events. So the questions with practical empirical significance are as follows: What kind of negation domain is the most likely negation domain when H0 is true? Is it the negation field that minimizes α? This doesn’t seem right. Because for the same α, there will be infinitely many corresponding intervals in the normal population test. And the smaller the rejection domain, the smaller α. The answer to this question cannot rely solely on mathematical deduction but on empirical judgment [8]. Take the two-tailed test of the normal population mean as an example: H0:μ=μ0H1:μμ1 {H_0}:\,\mu \, = \,{\mu _0}\,\,{H_1}:\,\mu \, \ne \,{\mu _1}

This kind of inspection is used to check whether a batch of products is qualified, and the other is to check whether the production process technology system is in a normal state. Under the assumption that the population is normally distributed, these two tests must take μ0 as the central value of the acceptance interval, and the critical point of negation of H0 is usually within the allowable error value. In the T test, the larger the sample standard deviation s, the easier it is to be accepted [9]. This is contrary to the purpose of quality inspection. Therefore, the standard deviation σ must be limited in the quality specification, so the standard deviation or variance test must be carried out first.

Let's talk about the empirical significance of significance level α. Negated domain X0, P{(ξ1, …, ξn) ∈ X0|θ = θ0} = α. This shows that when θ = θ0 is true, the observation value of (ξ1, …, ξn) falls within X0 and the probability of H0 being rejected is α. And the probability of being accepted when H0 is true is 1 − α. So as long as H0 is indeed true, it will generally not be negated when α is a decimal. Thus a large 1 − α is good for accepting H0. From this we can also find that α is not as small as possible but depends on the needs of the actual work. So in general, hypothesis testing can only tell us the probability of a choice, and how to choose depends on experience [10]. For example, in the two-tailed test of educational and biological survey statistics α is set to be larger. This will make the probability of accepting H0 smaller when F is true. Once accepted, the probability of making mistakes is smaller, which makes the research results more credible. This obviously cannot be explained by the idea of the principle of small probability.

Definition of statistical laws

Suppose A is a possible outcome of a randomized trial E. μA is the number of occurrences of A in a n weight Bernoulli trial, if (1) the probability of A exists in the trial. The existence of 0 ≤ p ≤ 1 makes formula (8) hold. limnμAnc__p \mathop {\lim}\limits_{n \to \infty} {{{\mu _A}} \over n}\underline{\underline c} p

(2) In the N group n weighted Bernoulli test, assuming that H: fA=μAn=p {f_A} = {{{\mu _A}} \over n} = p passes the hypothesis test at the significance level α, we say that the result of the random test E is that the occurrence of A obeys the statistical law. The proposition “The probability of A occurring in a random trial E is p” is a statistical law.

c__ \underline{\underline c} ” is referred to here as “recognized equal to”. That is, “a recognized conclusion of a certain range of experts.” This kind of logic is exactly the manifestation of real social logic (including scientific activities).

There are two types of statistical laws in practice. For example, if y = f (x) is a statistical law, it may appear in one of two ways in the experiment: (1) As soon as y = f (x) occurs in the experiment, it is completely accurate. (2) y = f (x) is only an approximate average relationship. From this we distinguish the following two types of statistical laws. Among them, yf (x) means that there is a certain logical relationship between y and f (x).

In the result of an experiment, if there are the number of (exactly) that relationship yf (x) holds in n times of the same experiment, B obeys (1) or (2) in Definition 2, then yf (x) is called the first type of statistical law.

When yf (x) is true, variable η = 1, and when yf (x) is not true, variable η = 0, then the first type of statistical law says that η is a random variable that obeys a certain two-point distribution [11]. The proposition about the specific distribution of “η obeys a two-point distribution (p, q)” is the second type of statistical law below. To describe the second type of statistical law, we first define “statistics holds.”

Suppose that there are variables y and x in experiment E, and if hypothesis H0: yf (x) is valid under the significance level α, then it is said that yf (x) in experiment E is statistically valid. The abbreviation for statistics is established.

In the result of an experiment, if there is a statistically valid number μA of relation yf (x) in N times of the same experiment, it obeys the formula (1) or (2) in Definition 5, then yf (x) is said to be the second type of statistical law.

Such as “the English vocabulary in children's education is 6000”. There are many ways to measure vocabulary first, and the results of each will be slightly different. Secondly, even with the same method, the result of each measurement will be different in the case of random sampling, and the result of one time rarely is exactly 6000. So this proposition about vocabulary can only be the second type of statistical law.

Illustrate:

When μAN or p → 1 is in the above definition, people can abstract a decisive law: yf (x). Of course, this kind of abstraction is not strictly mathematical logic, so that the result may be under a certain degree of accuracy.

In the second statistical law, each trial is generally a n fold Bernoulli trial.

In the practice of science and production society, there are many statistical laws of the second type [12]. When the difference between the test result and yf(x) is only interpreted as measurement error, yf(x) is regarded as a deterministic law or scientific law. When the test E in Definition 2 is to toss a uniform coin, then “the probability of heads facing upwards in a toss is 12 {1 \over 2} ” is the second type of statistical law. Many physics laws are only the second type of statistical laws with a certain accuracy in the practical sense.

The relationship between the law of large numbers and the law of statistics

The law of large numbers in mathematical statistics is an abstract reflection of statistical laws [13]. The definition of the law of large numbers and Bernoulli's law of large numbers are explained below.

About definition

The definition of the law of large numbers refers to such a statistical law:

Assuming Sn=1ni=1nξi {S_n} = {1 \over n}\sum\limits_{i = 1}^n {{\xi _i}} , when the conclusion in Definition 1 is true, there should be sequences {Sn} and { an} having the same limit. But this statistical law cannot be tested by strict mathematical logic, only by experiment [14]. It can be expressed as the following hypothesis.

Axiom 1: If the random variable sequence {ξn } obeys the law of large numbers, that is, there is a constant sequence {an} so that for any ε > 0, the formula (9) holds. limnP(|i=1nξinan|<ε)=1 \mathop {\lim}\limits_{n \to \infty} \,P\left({\left| {{{\sum\limits_{i = 1}^n {{\xi _i}}} \over n} - {a_n}} \right| < \varepsilon} \right) = 1

There is a statistical law “ {Sn=1ni=1nxi}t \left\{{{S_n} = {1 \over n}\sum\limits_{i = 1}^n {{x_i}}} \right\}t ” the same limit as {an}”. Where {xn} is the observed value of {ξn}.

An application of Axiom 1 is “mean estimation,” that is, there is the following inference: Suppose Ω is a statistical population, ξ is a single-valued function defined on Ω, and ξ represents the population mean of ξ, if D is a capacity for Ω Simple random sampling of n < ∞. Then the sample mean Xn of ξ has the following statistical law: “The limit of {Xn}is ξ ”.

This inference is the theoretical basis for using the sample mean to estimate the population mean, and the estimate's accuracy is not only related to n. And it is related to the standard deviation or variance of the distribution of ξ on Ω. For the same n, the smaller the overall variance, the higher the estimation accuracy [15]. This is the basic nature of interval estimation. Compared with the law of large numbers, the above hypotheses can be called the axioms of large numbers.

About Bernoulli's Law of Large Numbers

Construct the variable δi so that δi = 1 is when A appears in the i trial. When A does not appear in the i test, δi = 0. Then Bernoulli's law of large numbers says that the sequence of random variables {δi}obeys the law of large numbers, and its corresponding statistical law is “the limit of {1ni=1nδi} \left\{{{1 \over n}\sum\limits_{i = 1}^n {{\delta _i}}} \right\} is p ”.

This statistical law can be used as a theoretical basis for estimating the overall distribution with sample distribution [16]. For example, estimate the distribution ratio of each color ball in the bag. Similarly, the accuracy of this estimation is related to n and the overall distribution p. The accuracy is the worst when n, p=12 p = {1 \over 2} is given, and the smaller the p, the higher the accuracy.

In summary, it can be inferred that the law of large numbers reflects a limit statistical law. This limit can be regarded as a practical limit. As a statistical law, it can be tested directly by increasing the number of trials, or the following proposition (N, ε), namely: “There is a sequence of {Sn}. ∀ ε >0, ∃ N > 0, so that when n > N is |Sn− a| < ε.”

Conclusion

We further analyzed the various statistical laws mentioned above and found that they can all be attributed to the “law of averages.” For each statistical law, an average variable can be designed. And this mean variable has a limit. The meaning of the second type of statistical law “yf (x)” can be interpreted as “there is a certain logical relationship between a certain limit average variable of y and f (x). “This means that all statistical laws can be expressed as an axiom of large numbers. General scientific theories are relatively truthful. The practice or experimental test of scientific truth is an infinite process.

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