Common concepts of Statistics: t-Test, F-Test, chi-square test, p-value, degrees of freedom

Origin of 1,t test and F-Test

In general, in order to determine the probability of making a mistake from the statistical results of samples (sample), we use statistical methods developed by statisticians to perform statistical verification.

By comparing the obtained statistical calibration values with the probability distributions of some random variables established by statisticians (probability distribution), we can see how much of the chance will get the current results. If, after comparison, it is found that there is little chance of such a result, that is to say, it arises only when there are few and very few opportunities, then we can confidently say that this is not a coincidence, it is statistically significant (in statistical words, it is possible to reject null hypothesis Hypothesis,ho). On the contrary, it is not uncommon for a comparison to be found to have a high probability of occurrence, and we cannot be very confident that this is not a coincidence, perhaps a coincidence, perhaps not, but we are not sure.

The F-value and T-value are the statistical calibration values, and their corresponding probability distributions are F-distributions and T-distributions. Statistical significance (SIG) is the probability that the results of the present sample appear.

2, statistical significance (P-value or sig value)
The statistical significance of the results is an estimation method of the true extent of the results (which can represent the population). Professionally, the P-value is a diminishing indicator of the confidence of the result, the greater the P-value, the more we cannot think that the correlation of the variables in the sample is a reliable indicator of the correlations among the variables in the population. The P-value is the probability that the observed result is considered effective, which has the overall representation of the error. If the p=0.05 hints that the variables in the sample are associated with 5% may be caused by contingency. That is, assuming that there is no association between any of the variables in the population, we repeat similar experiments, we will find that about 20 experiments have an experiment, we study the variable association will be equal to or stronger than our experimental results. (This is not to say that if there is an association between the variables, we can get the same result of 5% or 95% times, when the variables in the population are correlated, the possibility of repeating the study and discovering correlations is related to the statistical effectiveness of the design.) In many research areas, the P-value of 0.05 is generally considered to be an acceptable false boundary level.

3,t inspection and F-Test

The specific content to be certified depends on which statistical procedure you are doing.

For example, if you want to test whether two independent sample mean differences can be inferred to the general, the T-test of the line.
Two samples (such as a class of boys and girls) the mean of a variable (such as height) is not the same, but whether the difference can be inferred to the general, there is a difference in the overall situation.
There will be no difference between male and female students in general, but it is your coincidence that the number of the 2 samples is different.
To do this, we perform a T-Test to calculate a T-Test value.
Compare the random variable T distributions established by statisticians with "No differences in general" to see how much of the chance (i.e., the significant sig value) will be present.
If the significant sig value is very small, such as <0.05 (less than 5% probability), that is, "if" the overall "really" no difference, then only if the opportunity is very few (5%), very rare circumstances, the current situation will occur. Although there is still a 5% chance of error (1-0.05=5%), we can still "more confident" said: The current sample of the situation (the difference between male and female students) is not a coincidence, is statistically significant, "the overall gender is not the difference" of the void hypothesis should be rejected, in short, there should be differences in general.

Each statistical method of the verification of the content is not the same, is also a T-test, it is possible that the above-mentioned verification of the overall difference, but also the same can be a single value in the overall verification is equal to 0 or equal to a certain value.

As for the F-Test, the analysis of variance (or the analysis of Variance), its principle is generally said, but it is through the variance of the view variable. It is mainly used for: the significance test of the mean difference, the separation of the relevant factors and estimate its effect on the total variation, the interaction between the analysis factors, homogeneity (equality of the variances) test.

Relationship between 3,t test and F-Test

T test process, is the difference between the two mean (mean) the significance of the test. The T-Test must know whether the variance of the two populations (variances) is equal, and the calculation of the T test value will vary depending on whether the variance is equal. In other words, the T test depends on the variance homogeneity (equality of variances) results. Therefore, SPSS in the t-test for equality of means, but also to do Levene ' s test for equality of variances.

1.
In the Levene's test for equality of variances column, the F value is 2.36, Sig. 128, indicating variance homogeneity test "no significant difference", that is, two variance qi (Equal variances), Therefore, the following T test results table to see the first row of data, that is the case of homogeneity T test results.

2.
In t-test for equality of means, the case of the first row (variances=equal): t=8.892, df=84, 2-tail sig=.000, Mean difference=22.99
Since sig=.000, that is, the two-sample mean difference has significant significance.

3.
To see which Levene's Test for equality of variances column in the sig, or see t-test for equality of means in the sig. (2-tailed) Ah?
The answer is: two to see.
First look at Levene's test for equality of variances, if the variance homogeneity test "there is no significant difference", that is, two variance qi (Equal variances), so the result table of the T test to see the first row of data, That is, the result of the T test in the case of homogeneity.
Conversely, if the variance homogeneity test "there is a significant difference", that is, two variance is not homogeneous (unequal variances), so the results of the T-Test table to see the second row of data, that is, variance is not the case of the T test results.

4.
You do a T-Test, why do you have an F-value?
It is because we want to evaluate whether the variance of two population (variances) is equal, to do Levene's test for equality of variances, to test the variance, so there is an F value.

Another explanation:

The T test had a single-sample t-Test, paired T-Test and two-sample T-Test.

Single-Sample T-Test: The difference between this group of samples and the population is observed by comparing the unknown population mean and the known population mean by the mean number of the sample.

Paired T test: The following cases were observed by pairing design method, 1, two homogeneous subjects received two different treatments, 2, the same subjects received two different treatments, 3, the same subjects were treated before and after treatment.

The F-Test is also called the homogeneity test of variance. The F-test is used in the two-sample T-Test.

In order to compare the two samples randomly, we should first determine whether the two population variances are the same, that is, the variance homogeneity. If the variance of the two populations is equal, the T-Test or variable transformation or rank and test can be used directly.

To determine whether the two population variances are equal, the F test can be used.

If a single group of design, must give a standard value or the overall mean, at the same time, provide a set of quantitative observations, the application of T test is the precondition is that the group of data must obey the normal distribution; if paired design, each pair of data must obey the normal distribution, if the group design, the individual is independent of each other, Both groups of data were taken from the general distribution and the homogeneity of variance was satisfied. These prerequisites are required because the T-Statistic must be calculated under such a premise, and T-Test is based on the T-distribution as the theoretical basis of the test method.

In short, the practical t-test is conditional, one of which is to conform to the homogeneity of variance, which requires an F-test to verify.

1, Q: What is the degree of freedom. How to determine.
A: (definition) The number of independent sample observations that comprise the sample statistics or the number of sample observations that are free to change. expressed in DF.
The setting of degrees of freedom is based on the reason that, when the population averages are unknown, the calculation of the dispersion using a sample average (small s) is constrained by the fact that the standard deviation (small s) must first be known, and the sum of the data is a constant when the average of the samples and n are known. Therefore, the "last" sample data can not be changed, because if it changes, the sum is changed, and this is not allowed. As for some degrees of freedom is n-2 or something, is the same reason.
When a statistic is calculated as an estimate, the introduction of a statistic loses one degree of freedom.
Popular Point said that there are 50 people in a class, we know that their Chinese scores on average divided into 80, now only need to know the results of 49 people can infer the results of the remaining person. You can quote 49 people, but the last one you can't lie about, because the average is fixed, the degree of freedom is one less.
The simple point is like you have 100 pieces, which is fixed, known, if you are going to buy five things, then the first four pieces you can buy whatever you want, as long as you have money, such as you can eat KFC can buy pens, can buy clothes, these flowers to the amount of money, when you left only 2 dollars, Perhaps you can only buy a bottle of Coke, of course, but also to buy a meat floss omelet, but no matter how to spend, you have only two dollars, and this you spend 98 dollars at that time has been settled. (This example is really good.) ）

2, Q: X-square test of freedom problem
Answer: In the normal distribution test, here the M (three statistics) is n (total), average and standard deviation.
Because we are doing the normal test, we need to use the mean and standard deviation to determine the normal distribution pattern, in addition, to calculate the theoretical times of each interval, we also have to use to N.
Therefore, in the normal distribution test, the degree of freedom is K-3. (This one is more special, remember.) ）
In the overall distribution of the coordination degree test, the degree of freedom is K-1.
In the independent test and homogeneity test of the cross-table, the degree of freedom is (R-1) x (c-1).

3, Q: What is the difference between T-Test and ANOVA
A: t test is suitable for the difference between the two variables, the average comparison of more than two variables to be analyzed by variance. The T-Test used to compare the mean values can be divided into three categories, the first of which is quantitative data for a single set of designs, the second is quantitative data for paired designs, and the third is quantitative data for group design. The difference between the latter two design types is whether the two groups of subjects are paired in a way that is similar in character to one or several aspects. No matter what type of T test, it is reasonable to apply the application under certain preconditions.
If a single set of design, must give a standard value or the overall mean, while providing a set of quantitative observations, the application of the T test is the precondition that the group of data must obey the normal distribution; if paired design, each pair of data must obey the normal distribution, if the group design, the individual independent, Both groups of data were taken from the general distribution and the homogeneity of variance was satisfied. These prerequisites are required because the T-Statistic must be calculated under such a premise, and T-Test is based on the T-distribution as the theoretical basis of the test method.
It is worth noting that variance analysis is the same as the precondition for group design T-Test, that is, normality and variance uniformity. The
T test is the most frequently used method in medical research, and the most common one in medical papers is the hypothesis test of quantitative data processing. T test has been so widely used, the reasons are as follows: The existing medical periodicals have made more statistical demands, the research conclusions need statistical support; The traditional medical statistics teaching has introduced T-Test as an introductory method of hypothesis testing, which makes it become the most familiar method for the general medical researchers. The T test method is simple and the result is easy to explain. Simplicity, familiarity and external requirements have led to the popularity of T-Test. However, because some people understand this method is not comprehensive, leading to a lot of problems in the application process, some even very serious errors, directly affect the reliability of the conclusion. The classification of these questions can be broadly summarized in the following two cases: not considering the application of the T test, the comparison of the two groups with T-Test, the various types of experimental design are considered as multiple single-factor two-level design, many times with the T test to 22 comparison between the mean value. In both cases, the risk of concluding a false conclusion is increased to varying degrees. Moreover, in the number of experimental factors greater than or equal to 2 o'clock, it is impossible to study the interaction between the experimental factors of the size.

Q: Statistical significance (P-value)
A: The statistical significance of the results is a method of estimating the true extent of the results (which can represent the population). Professionally, the P-value is a diminishing indicator of the confidence of the result, the greater the P-value, the more we cannot think that the correlation of the variables in the sample is a reliable indicator of the correlations among the variables in the population. The P-value is the probability that the observed result is considered effective, which has the overall representation of the error. If the p=0.05 hints that the variables in the sample are associated with 5% may be caused by contingency. That is, assuming that there is no association between any of the variables in the population, we repeat similar experiments, we will find that about 20 experiments have an experiment, we study the variable association will be equal to or stronger than our experimental results. (This is not to say that if there is an association between the variables, we can get the same result of 5% or 95% times, when the variables in the population are correlated, the possibility of repeating the study and discovering correlations is related to the statistical effectiveness of the design.) In many research areas, the P-value of 0.05 is generally considered to be an acceptable false boundary level.

4, Q: How to determine the results have real significance
A: In the final conclusion, judging what the significance of the level is statistically significant, inevitably with arbitrariness. In other words, the choice of a level that is rejected as an invalid result is arbitrary. In practice, the final decision usually depends on the data set comparison and analysis process of the result is a priori or only for the 22 > Comparison between the mean, depending on the overall data set the consensus of the number of supporting evidence, relying on previous practice in the field of research. Often, the results of P-values in many scientific fields are ≤0.05 considered to be statistically significant, but this significant level also contains a fairly high probability of error. Results 0.05≥p>0.01 was considered to be statistically significant, while 0.01≥p≥0.001 was considered highly statistically significant. However, it should be noted that this classification is only an informal judgment rule based on research.

5, Q: All the test statistics are normally distributed.
A: Not entirely, but most tests are directly or indirectly related and can be deduced from the normal distribution, such as T-Test, F-Test, or chi-square test. These tests are generally required: the analyzed variables are normally distributed in general, which satisfies the so-called normal hypothesis. Many of the observed variables are normally distributed, which is why the normal distribution is the basic feature of the real world. The problem arises when people use a test that is established on the basis of a normal distribution to analyze data from non-normal-distribution variables (see normality tests for nonparametric and variance analysis). There are two methods under this condition: one is to use an alternative nonparametric test (i.e. no distribution test), but this method is inconvenient, because from the conclusion it provides, this method is inefficient and inflexible. Another approach is to use a test based on a normal distribution if the sample size is determined to be large enough. The latter method is based on a rather important principle, which plays an important role in the general test on the basis of the normal equation. That is, as the sample size increases, the sample distribution shape tends to be normal, even though the distribution of the variables studied is not normal.

6, Q: Hypothesis test connotation and steps
A: In the hypothesis test, because of randomness we may make two kinds of errors in decision-making, one is the assumption is correct, but we reject the hypothesis that such errors are "Discard true" errors, known as the first kind of error, the class is incorrect, but we do not refuse to assume that such errors are "Pseudo" error, called the second type of error. In general, in the case of a sample determination, any decision cannot simultaneously avoid the occurrence of two types of errors, that is, to avoid the probability of the first type of error, while increasing the probability of the second type of error, or to avoid the probability of the second type of error, but also increase the probability of the first type of error occurs. People often choose to control that type of error as needed to reduce the chance of such errors. In most cases, people control the probability that the first category of errors will occur.      the probability that the first type of error occurs is called the significance level, which is generally expressed in α, and when a hypothesis test is performed, the probability of the first type of error occurring is controlled by giving a value of the significant level alpha beforehand. In this premise, the hypothesis test is carried out according to the following steps:
1), the determination hypothesis,
2), the sampling, obtains the certain data,
3), according to the hypothesis condition, constructs the examination statistic, and calculates the test statistic in this sample the concrete value according to the sampling data;
4), Based on the sampling distribution of the test statistic, and the given significance level, the reject domain and its critical value are determined;
5), comparing the value of the test statistic in this sample with the critical value, if the value of the test statistic is within the reject domain, then the hypothesis is rejected;
to this step, assume that the test has been basically completed, However, since the test is to control the probability of error by the method of pre-given significant level, we cannot know the hypothesis that the two data is more similar, we can only know the maximum probability of making the first kind of error based on this sample (that is, the given significance level). It is impossible to know exactly what the probability level of error is. The calculated P-value solves this problem effectively, and the P-value is actually a probability value calculated according to the sampling distribution, which is calculated from the test statistic. By directly comparing the P-value with the given significance level α, it is possible to know whether to reject the hypothesis, which obviously replaces the method of comparing the value of the test statistic with the size of the critical value. And by this method, we can also know the actual probability of making the first kind of error in the case of P-value less than α, p=0.03<α=0.05, then the refusal hypothesis, the probability that this decision may err is 0.03. It should be noted that if p>α, then the assumption is not rejected, in which case the first type of error does not occur.

7, Q: Chi square test results, the value is bigger the better, or the smaller the better.
A: As with other tests, the larger the calculated statistic, the smaller the probability value, the closer the distribution is to the tail end.
If the test design is reasonable, the data is correct, significant or not significant are objectively reflected. There's nothing good or bad.

8. Q: What is the difference between the T-test of paired samples and the test of related samples?
A: Paired samples have homologous pairings (such as twins in an animal experiment), conditional pairing (such as the same environment), self-pairing (e.g. before and after the drug in a medical experiment), etc. (It seems that there is no clear explanation, the same question, what is the difference?) ）

9, Q: In comparison between the two groups of data rate is the same, two distribution and chi-square test what is different.
A: Chi-square distribution is mainly used for multi-group comparison, is to examine the total number of research objects and a category group of the observed frequency and the expected frequency is significantly different, requires that the if number of each block is not less than 5, if less than 5 merge adjacent groups. Two distributions do not have this requirement.
If there are only two classes in the classification, it is better to take two Tests.
If the 2*2 table can be accurately tested with Fisher, it works better under small samples.

10, Q: How to compare the differences between the two sets of data
Answer: from four aspects,
1). Design type is completely random design two sets of data comparison, do not know whether the data is a continuous variable.
2). Comparison method: If the data is continuous data, and the two sets of data obey normal distribution & homogeneity (Variance homogeneity test), then the T test can be used, if not obey the above conditions can be used rank and test.
3). Want to know whether there is a significant difference between the two sets of data. I don't know what this obvious difference means. Is whether the difference is statistically significant (that is, the probability of the difference) or the difference in the range of the two population fluctuations. If the former, you can use the 2nd step can get the P value, if the latter, it is the mean difference between the confidence interval to complete. Of course, the results of both can be obtained in SPSS.

11. Q: The relationship and difference between regression analysis and correlation
A: Regression analysis (Regression): Dependant variable is defined and can being forecasted by independent variable. Analysis (Correlation): the RelA Tionship BTW, variables. ---A dose not define or determine B.
Regression more useful to interpret the meaning of the dependent variable, there is a little causal relationship in it, and can be linear or non-linear relationship;
Correlation is more likely to explain the relationship between the 22, but generally refers to the linear relationship, especially the correlation index, sometimes the image display particularly strong two-square image, but the correlation index is still very low, and this is only because the two are not linear relationship, does not mean that there is no relationship between the two, Therefore, in the relevant index to pay special attention to how to explain the value, especially recommended to make an image observation first.
However, regardless of regression or relevance, in the causal relationship should pay special attention, not every significant regression factor or a higher correlation index implies causality, it is possible that these factors are subject to the third, fourth factor constraints, are the cause or effect of another factor.
For the difference between the two, I would like to make it easy to understand by the following analogy:
For the relationship between the two, the relationship can only know that they are lovers relationship, as to their who is the dominant, who speaks the arithmetic, who is the follower, a sneeze, the other will have what reaction, the correlation is not competent, and regression analysis can be a good solution to the problem
Return is not necessarily a causal relationship. There are two main returns: one is explanation, one is prediction. The unknown dependent variables are predicted using known autocorrelation. The relationship is primarily about the sharing of the two variables. If there is a causal relationship, it is usually the path analysis or the linear structure relationship pattern.
I think we should look at this, we do regression analysis is based on a certain theory and intuition, through the relationship between the number of independent variables and dependent variables to explore whether there is a causal relationship. The man upstairs said, "The return is not necessarily a causal relationship ... If there is a causal relationship, usually a path analysis or linear structure relationship pattern "is a bit debatable, in fact, regression analysis can be seen as a special case of linear structural relationship patterns."
I think that the return is to explore the causal relationship is correct, because in fact, in the end we are not completely based on statistical results to determine causality, only in the statistical results and theory and reality on the basis of a more consistent with the cause we are certain of this causal relationship. Any statistical method is just a tool, but it is not entirely dependent on the tool. Even SEM, we can not say that fully determine its accuracy, because even if the method is good, but the complex relationship of variables presented in a variety of ways, perhaps statistics can only tell you a direction of the optimal solution, it is not necessarily the most realistic, not to mention the quality of the sample data will also make the results do not conform to the facts, thus Cause people to doubt the accuracy of the statistical method.
Statistics only describe statistical associations.
Does not prove a factor relationship.
Regression has a causal relationship, the correlation is not necessarily.
Regression analysis is a statistical method to deal with the linear dependence between two and more than two variables. Such problems are common, such as the content of a metal element in human hair and the content of the elements in the blood, human body surface area is related to height, weight, and so on. Regression analysis is the mathematical relationship used to illustrate this dependent change.
The existence of any thing is not isolated, but is interrelated, mutual restraint. Height and body weight, body temperature and pulse, age and blood pressure have some connections. The relationship between objective things and the relative degree of correlation with appropriate statistical indicators, the process is related to correlation analysis.the logic of random sampling and statistical inference The day before yesterday, McDull asked me how to construct the confidence interval, on the phone I do not seem to explain clearly, here re-organize a note. At the same time, I feel that when doing the subject in China, the embarrassment of peers are vivid (they do not know how to test whether a variable is in accordance with the normal distribution), so remember, in order to encourage themselves to think clearly what they learn. People who have studied statistics know that an unknown population (population) can be randomly sampled, by describing the sample (sample), by calculating (for example, calculating the mean of the samples, sample variance), and then inferring some of the general characteristics (testing some assumptions, constructing confidence intervals, etc.). Of course, many modern inference methods are "cook book" Nature, do not need to be detailed by non-professionals, for example, economists often do not know what is the F distribution can also know how to test f statistics and interpretation of the conclusions, and even do not have to hand-calculate f statistics. But if you think about the relationship carefully, you can see that this random sampling-inference contains a philosophy that, in some places, does manifest human wisdom. Essentially, this approach uses a set of data (samples) that we have 100% of the information we have, to fit in a set of data (overall) that we have little or no information about. In other words, to be aware of the unknown process. Because it is the "unknown" inference, we can not have 100% of the certainty, but at the same time with the "know" data, so we will not be a little bit sure none. Perhaps through a very simple example we can see the thinking behind it. Let's start with this exercise. [Example]: Suppose a school has 20,000 students, from which a random 1000 students, ask, this school 20,000 students of the average grade of a course is 70/100. Note that there is no hypothesis about the overall distribution. [Question 1]: Can we use samples for statistical inference? It doesn't seem to be possible yet, a logical lack of a step. It is worth noting that the 1000 students taken are "random samples". The answer is in the negative. The idea of random sampling is that the distribution of the extracted sample (sample) is consistent with the distribution of the population (population), which can be established for each observed value. But the random sample also requires each observation value (observation) to be independent (independent), where the narrow understanding is that each observation value is the same probability of being taken. But in the example above, the condition is clearly not fulfilled (many of the most well-known scholars tend to ignore this condition). If you collect 1000 students ' transcripts, the results of the 1000 students are drawn to the probability that they are taken in the order. As a result of the 1000 students, each student was drawn only once, can not be repeated sampling.Thus, the probability of the first student in the sample was 1/20000, the second student was drawn to 1/19999, the third student was 1/19998, ... The 1000th student is 1/19000. In other words, after some students are taken away, the probability of the next student being pumped is definitely not equal to 1/20000. It is only in the sample that is put back (with replacement) that we can say that the probability of each student being pumped is 1/20000, and that we are taking a random sample. But in that case, we would probably not be able to get 1000 samples, because a student was pumped more than a repetition probability is not zero. Fortunately, noting that these 1000 probability values are not significant (because the overall value is large enough), it is possible to assume that the scores of these 1000 students are random samples. After doing this approximation, we can make statistical inferences. This approximation is common in statistics. For example, the central limit theorem (theorem) says that most distributions can be approximated as normal (normal) distributions, which makes the normal distribution extremely important in statistics. An important feature of these distributions is that variables can be thought of as observations and (functions), for example, two distributions can be considered as the Bernoulli of a set of test results (Bernolli tiral). In addition, for example, the Poisson (Poisson) distribution can be considered as a two-item (binomial) distribution, and the latter can be approximated as a normal distribution. However, to complete this approximation, we also need more assumptions for statistical inference and testing (inference and test). For example, we must understand the overall distribution, even if we do not know the exact values of all the parameters. (Now suppose we only discuss the parameter (parametric) method) [Situation one]: we know for sure that the results of these 20000 students are in accordance with the (normal) distribution, the mean value is unknown (unkown mean) but the variance is known (known variance). [Question 2] for samples that conform to any distribution, the sample mean and sample variance conform to what the law. The following relationships can be obtained by using the properties of the simple mathematical expectation: (1) The expectation of the sample mean = the overall mean value. (2) Variance of the sample mean = Total variance/Number of samples (the fluctuation of the sample mean does not vary a single observation) through these variables, we can construct the statistic Z: (3) z= (sample mean-overall mean value)/Radical (total variance/number of samples). According to (1) (2) and the central limit theorem, foranyOverall, the z-statistic conforms to the standard normal distribution. It is important to note that we have a lot of information about this z statistic. For example, for any given a value, we can fully calculate the Z-value in accordance with (4) Pr (z<|z|) =a%. But because Z is a variable, we don't have 100% of the information. Note that (3) and (4) mean, since we can calculate the sample mean and sample variance, the population variance, then z is the only function that is determined by the overall mean value. Thus, we can work out the function expression of the overall mean, because the overall mean is just the inverse of Z. Given a, we know the range of Z values, and we know the range of the overall mean. This range of changes is what we call a (self) signaling interval (confidence interval), such as a PR (c1< overall mean &LT;C2) =90%,C1 is a 5% percentile (percentile) value, and C2 is a 95% percentile value. In other words, the probability of the total mean falling in the C1,c2 interval is 90%. So we can make hypothesis test: H0: Overall mean value =70VS h1:not H0. (assume:size=10%). At this time, we know the PR (c1< overall mean value <c2) =90%, so long as the overall mean <c1 or the overall mean value &GT;C2 we can overturn the H0 at 10% levels. [Case two] we don't know the overall variance and we don't know the overall mean. Looking at the formula (3), we know that we cannot infer it with normal distribution, so we have to use a new method, the T distribution. By definition, the sample variance =sum (observation value i-sample mean) ^2;i=1,2,1000. Sample mean value =sum (observation i)/sample number i=1,2, ... 1000. It can be proved that (the process is complex, orthogonal matrix operations are required), (sample variance/population variance) conforms to the Chi-square (chi-squared) distribution of degrees of freedom (sample Value-1). At the same time, the sample variance and the sample mean are independent variables. Then construct a new T variable: t=z/radical (CHI-square/DOF). It is worth noting that the numerator denominator is a fraction, each with an unknown denominator, the population variance. But fortunately, the two are sold to each other. Thus, T is only a function that is determined by the overall mean value. Then we can do exercises to construct confidence intervals. It is important to note that (i) for a population that conforms to an arbitrary distribution, z conforms to the standard normal distribution, since the sample mean is the "sum" of all observations (multiplied by a constant) and, as long as the number of samples is large enough, the central limit theorem guarantees its approximation to the standard normal distribution. (ii) However, if the population does not conform to the normal part, then we cannot perform the T test. Because the sample variance cannot be guaranteed to match the chi-squared distribution, there is no guarantee that T is in accordance with T distribution. Summarize the philosophy here. We used a sample that mastered 100% of the information, calculated several values (sample mean, sample variance). Then we construct a statistic Z, or T, in which we have a great amount of information. Then use this information to understand that we have mastered a small number of information in general. The intriguing point is that this corresponds to the standard normal distribution of the statistics Z, and T, where some of the information comes from the sample, some from the overall. This part of our understanding of the volume is exactly the bridge of our statistical inference. Because the direct analysis of the whole, our information is not enough-we know almost nothing. and the direct analysis of the sample, although we have sufficient information, the sample is not closely related to the overall mean, we only know (1) and (2) formula. So z and T variable will play the role of "curve salvation". However, because of this, we can only say that we have the a% to believe that the overall mean falls within the (C1,C2) interval. When more information is lacking, we need to add more steps, such as constructing t variables to prove that the sample mean and sample variance are independent variables, and to understand Chi-squared distribution. But, before people have foundSuch a distribution, paving the way for this approach. It seems to me that these people really have greatly promoted the development of the history of human thought. How do these people come up with such a distribution of chi-square distribution? How to find the relationship between normal distribution and t distribution. These are, in themselves, worth striking. [Scenario three] we don't know the overall distribution, nor do we know any parameters. As I said earlier, if you don't know the overall distribution, as long as you know the population variance, the z variable conforms to the standard normal distribution. But now that we don't know the overall variance, we can't even use the parametric method, either the Nonparametric approach (Nonparametic method) or the semi-parametric approach (Semi-parametric method). But the logic is still consistent, that is, it is necessary to contact the sample and the population through an intermediate statistic, for example, the rank test, and the scale test (size test) needs to construct a new statistic.