# Common statistical concepts: t-test, F-test, Chi-square test, P-value, degree of freedom

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Author: User

1. Origins of t-test and F-test

In general, to determine the probability of making mistakes from the statistical result of the sample to the overall result, we will use some statistical methods developed by the statistician for statistical verification.

By comparing the obtained statistical verification value with the statistician, the probability distribution of some random variables (probability distribution) is established, we can know how many % chances will get the current result. If, after comparison, we find that the probability of such a result is very small, that is, the result is only available in rare cases with few opportunities, then we can confidently say, this is not a coincidence, it is statistically significant (in statistics, it is the ability to reject null hypothesis, Ho ). On the contrary, if we find that the probability of occurrence is very high and not uncommon after comparison, then we cannot directly point in confidence that this is not a coincidence, maybe a coincidence, maybe not, but we are not sure.

The F value and the T value are the Statistical Verification values. The probability distribution corresponding to them is the f distribution and the tdistribution. Statistical significance (SIG) is the probability of the current sample.

2. statistical significance (P value or SIG value) the statistical significance of the results is an estimation method of the True Degree of the results (which can represent the overall. Professionally, the P value is a decreasing indicator of the result credibility. The larger the P value, the less we can think that the association of variables in the sample is a reliable indicator of the association of variables in the population. The P value indicates the probability of making mistakes that are regarded as valid. For example, if P = 0.05 indicates that 5% of variables in the sample are associated may be caused by chance. That is, if there is no association between any variables in the population, we repeat similar experiments and we will find that about 20 experiments have one, and the association of variables we study will be equal to or better than our experimental results. (This does not mean that if there is an association between variables, we can get the same result of 5% or 95% times. When there is an association between variables in the population, the likelihood of repeated research and discovery associations is related to the Statistical effectiveness of the design .) In many research areas, a 0.05 P value is generally considered an acceptable boundary level.

3. t-test and F-test

The specific content to be verified depends on the statistical program you are using.

For example, if you want to test whether the differences between the average numbers of two independent samples can be inferred to the population, the t-test is performed. The average number of variables (such as height) in two samples (such as boys and girls in a class) is different, but can the difference be inferred to the population, is there a difference in the overall situation? Will there be no difference between boys and girls in general, except that the values of the two samples are actually different? Therefore, we conduct t verification to calculate a t verification value. Compare it with the tdistribution of the random variable based on "No difference in population" established by the statistician to see how many % chances (that is, the significance SIG value) will get the current result. If the significance SIG value is very small, for example, <0.05 (less than 5% probability), that is, if there is no difference in "Overall" true ", then there is only a small chance (5%) in rare cases, this is the current situation. Although there is still a 5% chance error (1-0.05 = 5%), we can still say with "Confidence": this situation in the current sample (where there is a difference between male and female) it is not a coincidence. It is statistically significant. The imaginary assumption that there is no difference between male and female in general should be rejected. In short, there should be a difference in general.

The verification content of each statistical method is different. It is also a t-verification, which may be whether there is a difference in the overall verification, it can also be used to check whether a single value in the population is equal to 0 or a value.

As for F-verification, variance analysis (or translation variation analysis, analysis of variance), its principle is roughly described above, however, it is done by checking the variance of the variable. It is mainly used for the significance test of mean data difference, separation of relevant factors and estimation of its effect on total variation, interaction between analysis factors, and uniformity of variance) inspection.

3. Relationship between t-test and F-test

The T-test is to test the significance of the mean difference between the two samples. However, the t-test must know whether the two population variance (variances) is equal. The T-test value is calculated based on whether the variance is equal. That is to say, t-test depends on the variance of variances. Therefore, while performing t-test for equality of means, SPSS must also perform 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. is. 128 indicates that there is "no significant difference" in the variance Homogeneity test, that is, equal variances. Therefore, the data in the first row should be viewed in the result table of the t-test below, that is, the t-test result in the case of a normal variance.

2. in t-test for equality of means, the first row (variances = equal): t = 8.892, df = 84, 2-tail Sig =. 000, mean difference = 22.99 since Sig =. 000, that is, the differences between the two samples are significant!

3. Which Levene's test for equality of variances column does it look at Sig. (2-tailed) in T-test for equality of means? The answer is: Both of them should be viewed. Let's take a look at Levene's test for variance ity of variances. If there is no significant difference in the variance conformity test, that is, equal variances ), therefore, the result table of the subsequent t-test should look at the data in the first row, that is, the t-test result in the case of a correct variance. Otherwise, if there is a "significant difference" in the variance conformity test, that is, the two variance is not consistent (unequal variances), The result table of the subsequent t-test should look at the data in the second row, that is, the result of t-test in case of Different variance.

4. You did a t-test. Why is there a f value? This is because we need to evaluate whether the two population variance (variances) is equal. To test the variance, we need to make the Levene's test for variance of variances, so there is a f value.

Another explanation:

The T-test includes one-sample t-test, paired t-test, and two-sample t-test.

Single Sample t-test: Compare the average number of unknown samples and the average number of known samples to observe the differences between samples and the population.

Paired t-test: The paired design method is used to observe the following situations: 1. Two homogeneous subjects receive two different treatments respectively; 2. the same subject receives two different treatments; 3. Before and after the same subject is processed.

The F test is also called the variance Homogeneous Test. The F test is required in the Two-sample t-test.

The two samples are randomly selected from the two studies. To compare the two samples, we must first determine whether the two population variance is the same, that is, the variance uniformity. If the two population variance is equal, t-test is used directly. If the variance is not equal, t-test or variable transformation or rank sum test can be used.

To determine whether the two population variance is equal, use the ftest.

If a single group is designed, a standard value or an overall mean value must be provided, and a group of Quantitative observation results must be provided. The precondition for applying t-test is that the group of data must be normally distributed, the difference value of each pair of data must follow the normal distribution. For a group design, the individual is independent of each other, and the two groups of data are taken from the normal distribution population, and the variance is consistent. These prerequisites are required because the T-statistic calculated under these conditions is subject to the tdistribution, and t-test is based on the tdistribution.

In simple terms, the practical t-test is conditional. One of them is to conform to the uniformity of variance, which needs to be verified by the ftest.

1. Q: What is the degree of freedom? How to determine? A: (Definition) number of independent sample observations that constitute the sample statistic or the number of freely changed sample observations. In DF format. The degree of freedom is set for the following reason: when the overall mean is unknown, calculating the deviation using the sample mean (usually small s) will be subject to a limitation -- the standard deviation (small s) must be calculated) we must first know the average sample, and the sum of the data is a constant when both the average and N know. Therefore, the "last" sample data cannot be changed, because if it changes, the sum changes, and this is not allowed. As for some degrees of freedom is N-2 or something, is the same truth. When a statistic is calculated as an estimator, introducing a statistic will lose a degree of freedom. In layman's terms, there are 50 people in a class. We know that their average Chinese score is 80. Now we only need to know the scores of 49 people to deduce the scores of the remaining person. You can report scores of 49 people at will, but you can't say anything about the last person, because the average score has been fixed and the degree of freedom is less. Simply put, you have one hundred RMB, which is fixed and known. If you want to buy five items, you can buy the first four items as long as you have enough money, for example, you can eat KFC and buy a pen and buy clothes. The amount of money you spend varies. If you have only two yuan left, you may have to buy one bottle of cola at most, of course, you can also buy a meat slice, but no matter how you spend it, you only have two yuan, and it was settled when you spent 98 yuan. (This example is really good !!)

2. Question: degree of freedom in the X-side test answer: In the normal distribution test, the M (three statistics) Here is n (total), average, and standard deviation. Because we need to use the mean and standard deviation to determine the Normal Distribution Pattern During the Normality Test. In addition, we need to use N to calculate the number of theoretical times for each interval. So in the normal distribution test, the degree of freedom is K-3. (This one is special. Remember it !) In the Gini test of the overall distribution, degrees of freedom is a K-1. In the independence test and homogeneity test of the cross table, the degree of freedom is (r-1) × (c-1 ).

3. Q: What is the difference between t-test and variance analysis? A: T-test is applicable to the difference test between the mean numbers of two variables. Variance analysis is used to compare the mean numbers of more than two variables. The T-test for comparing mean values can be divided into three types. The first type is for designing quantitative data for a single group; the second type is for designing quantitative data for pairing; and the third type is for designing quantitative data for grouping. The difference between the two types of design is whether the two groups of research objects are matched to the child according to one or more features. No matter which type of t-test, it is reasonable to apply it under certain conditions. If a single group is designed, a standard value or an overall mean value must be provided, and a group of Quantitative observation results must be provided. The precondition for applying t-test is that the group of data must be normally distributed, the difference value of each pair of data must follow the normal distribution. For a group design, the individual is independent of each other, and the two groups of data are taken from the normal distribution population, and the variance is consistent. These prerequisites are required because the T-statistic calculated under these conditions is subject to the tdistribution, and t-test is based on the tdistribution. It is worth noting that the precondition for variance analysis is the same as that for group design t-test, that is, normality and variance uniformity. T-test is currently the most frequently used method in medical research. It is the most common hypothesis test method for processing quantitative data in medical papers. T-test has been widely used. The reason is as follows: the existing medical journals mostly require statistics, and the research conclusions must be statistically supported; in traditional medical statistics teaching, t-test is introduced as an entry-level method for hypothesis testing, making it the most familiar method for medical researchers. the t-test method is simple and the results are easy to explain. Simple, familiar with external requirements, contributed to the prevalence of t-test. However, some people do not fully understand this method, which leads to many problems in the application process. Some or even serious errors directly affect the reliability of the conclusion. These questions can be categorized into the following two situations: T-test is not considered, and t-test is used for comparison between the two groups; all experimental design types are regarded as multiple single-factor two-level designs, and the mean values are compared by T-test multiple times. The two cases above increase the risk of drawing incorrect conclusions to varying degrees. Moreover, when the number of experimental factors is greater than or equal to 2, the interaction between experimental factors cannot be studied.

Q: statistical significance (P value) A: the statistical significance of the results is an estimation method of the result's authenticity (which can represent the overall. Professionally, the P value is a decreasing indicator of the result credibility. The larger the P value, the less we can think that the association of variables in the sample is a reliable indicator of the association of variables in the population. The P value indicates the probability of making mistakes that are regarded as valid. For example, if P = 0.05 indicates that 5% of variables in the sample are associated may be caused by chance. That is, if there is no association between any variables in the population, we repeat similar experiments and we will find that about 20 experiments have one, and the association of variables we study will be equal to or better than our experimental results. (This does not mean that if there is an association between variables, we can get the same result of 5% or 95% times. When there is an association between variables in the population, the likelihood of repeated research and discovery associations is related to the Statistical effectiveness of the design .) In many research areas, a 0.05 P value is generally considered to be an unacceptable boundary level.

4. Q: How can I determine whether the result has a real significance? A: In the final conclusion, it is statistically significant to determine the significance level. It is inevitable that the area has a force outage. In other words, the selection of a level that deems the result invalid and is rejected is broken. In practice, the final decision usually depends on whether the result is first verified during dataset comparison and analysis, or only two-to-two> comparison between average numbers, dependent on the number of supportive evidence consistent conclusions in the overall data set, and on the practice in the previous research field. Generally, results of generating P values in many scientific fields less than or equal to 0.05 are considered to be the boundary of statistical significance, but this significance level also contains a fairly high possibility of making mistakes. Results 0.05 ≥ P> 0.01 was considered to be statistically significant, while 0.01 ≥ p ≥ 0.001 was considered to be highly statistically significant. However, it should be noted that this classification is only based on the informal judgment Convention on the basis of research.

5. Q: Are all inspection statistics normally distributed? A: This is not exactly the case. However, most tests are directly or indirectly related to them. They can be derived from normal distribution, such as t-test, F-test, or chi-square test. These tests generally require that the variables analyzed have a normal distribution in the population, that is, they satisfy the so-called normal hypothesis. Many observed variables are indeed normally distributed, which is also the reason that normal distribution is the basic feature of the real world. The problem arises when people use tests established on the basis of normal distribution to analyze data of non-normal distribution variables (see normality tests for non-parameter and variance analysis ). There are two methods under this condition: one is to use an alternative non-parameter test (that is, a non-distributed test), but this method is not convenient, because from the conclusions it provides, this method is inefficient and inflexible in statistics. Another method is: when the sample size is determined to be large enough, the test based on the normal distribution can still be used. The latter method is based on a very important principle, which plays an extremely important role in the overall test on the basis of normal equations. That is, as the sample size increases, the sample distribution tends to be normal, even if the variable distribution is not normal.

6. Question: hypothesis test connotation and steps answer: In hypothesis test, we may make two types of mistakes in decision-making due to randomness. One is that assumptions are correct, but we reject them, this type of error is a "discard truth" error, which is called the first type of error. The first type is an incorrect assumption, but we did not reject the assumption. This type of error is a "pseudo" error, it is called the second type of error. Generally, when the sample is determined, no decision can avoid two types of errors at the same time, that is, to avoid the occurrence rate of the first type of errors, this will increase the probability of a second type of error; or, while avoiding the probability of a second type of error, it will increase the probability of a first type of error. People often choose to control errors as needed to reduce the chance of such errors. In most cases, people control the probability of the first type of errors. The probability of occurrence of the first type of error is called the significance level, which is generally expressed by α. During the hypothesis test, is to control the probability of occurrence of the first type of errors by giving a value of significance level α in advance. On this premise, the hypothesis test is performed according to the following steps: 1) determine the hypothesis; 2) perform sampling to obtain certain data; 3) According to the hypothesis, construct the test statistic and calculate the specific value of the test statistic in this sample based on the data obtained from the sample; 4) Calculate the sample distribution based on the constructed test statistic and the given significance level, determine the rejection region and its critical value; 5) Compare the values of the test statistic and the critical value in this sampling. If the value of the test statistic is in the rejection region, the hypothesis is rejected. In this step, the hypothesis test has been basically completed, but because the test uses the method with a given significance level in advance to control the probability of making a mistake, the hypothesis test of the two data is similar, we cannot know that assumption is more prone to mistakes, that is, we can only know the maximum probability of making the first type of Errors Based on this sampling (that is, the given significance level ), but cannot know the specific probability level of making a mistake. Calculating the P value effectively solves this problem. The P value is actually a probability value calculated based on the sample distribution. This value is calculated based on the test statistic. By directly comparing the P value with the given significance level α, you can know whether to reject the hypothesis. Obviously, this replaces the method of comparing the value of the test statistic and the critical value. In addition, we can also know the actual probability of making the first type of error when the P value is smaller than α. P = 0.03 <α = 0.05, then we reject the hypothesis, the probability of making a mistake in this decision is 0.03. It should be noted that, if P> α, the assumption is not rejected. In this case, the first error will not occur.

7. Q: Is the bigger the value, the better, or the smaller the value, the better? A: Like other tests, the larger the calculated statistic, the closer the distribution to the tail end of the distribution, the smaller the probability value. If the test design is reasonable and the data is correct, the obvious or non-obvious results are objective responses. Nothing is better or worse.

8. Q: What is the difference between a paired sample t-test and a related sample test? A: The paired samples include same-source pairs (such as twins in animal experiments), condition pairs (such as the same environment), and self-matching (such as before and after individual Medication in medical experiments. (It seems that I have not explained it clearly. What is the difference between the two ?)

9. Q: Are the two groups of data with the same rate? What is the difference between the two-item distribution and the chi-square test? A: The chi-square distribution is mainly used to compare multiple groups of classes. It is used to check whether there is a significant difference between the total number of subjects and the observed frequency and expected frequency of a group, the number of intermediate frequencies per grid must be no less than 5. If the number is smaller than 5, adjacent groups are merged. This requirement is not required for two distributions. If there are only two types of classification, it is better to use the two-item test. If the table is 2*2, fisher can be used for exact test, which is better for small samples.

10. Q: How to compare the differences between the two groups of data answer: Answer from four aspects, 1 ). the design type is completely random design of two groups of data comparison, do not know whether the data is a continuity variable? 2 ). comparison Method: if the data is continuity data and the two groups of data are subject to normal distribution and variance uniformity (variance uniformity test), t-test can be used, if the preceding conditions are not met, Rank Sum test can be used. 3) Do you want to know if there are significant differences between the two groups of data? I don't know what this obvious difference means? Is the difference statistically significant (that is, the probability of the Difference) or the range in which the mean difference between the two populations fluctuates? For the former, the P value can be obtained in step 1. For the latter, the confidence interval of the mean number difference is used. Of course, the results of both can be obtained in SPSS.

11. Connection and difference between regression analysis and related analysis answer: regression analysis (regression): dependant variable is defined and can be forecasted by independent variable. correlation: The Relationship BTW two variables. --- a dose not define or determine B. regression is more useful for independent variables to explain the meaning of the dependent variable. There is a little causal relationship in it, and it can be a linear or non-linear relationship. Correlation is more inclined to explain the relationship between two or two, however, it generally refers to the linear relationship, especially the correlation index. Sometimes the image shows a particularly strong quadratic image, but the correlation index is still very low, just because the two are not linear, it does not mean that there is no relationship between the two. Therefore, pay special attention to how to interpret the value when making the relevant index. We recommend that you make image observation first. However, no matter the regression or correlation, we should pay special attention to the causal relationship. Not every significant regression factor or high correlation index means the causal relationship, it is possible that these factors are both subject to the third and fourth factors, and are the cause or result of other factors. For the differences between the two, I think it is easy to understand through the following example: for the relationship between two people, the relationship can only know that they are lovers. As for who are the leader and who are talking, if either the followers or the other one sneezes and the other responds, the correlation will not be competent. However, regression analysis can effectively solve this problem and there may not be a causal relationship between them. There are two main types of rollback: one is the solution, and the other is the solution. When using known auto-incremental regression, the unknown dependent data is obtained. The relational data is mainly used to understand the common changes of the two changes. If there is a causal relationship, it usually goes into the path analysis mode or the relational structure. In my opinion, we should look at regression analysis to find out whether there is a causal relationship through the quantitative relationship between independent variables and dependent variables under certain theories and intuition. The elder brother said, "regression may not have a causal relationship ...... If there is a causal relationship, we usually conduct path analysis or linear structure relationship model. "It is worth considering. In fact, regression analysis can be seen as a special case of the linear structure relationship model. I think it is true that regression is used to explore the causal relationship, because in fact, we did not judge the causal relationship completely based on the statistical results, this causal relationship can be affirmed only when the statistical results are consistent with the theoretical and practical results. Any statistical method is just a tool, but it cannot depend entirely on it. Even with SEM, we cannot fully determine its accuracy, because even if the method is good, the complex relationships of variables are displayed in a variety of ways, statistics may only tell you the optimal solution in one direction, but may not be the most practical. What's more, the quality of the sample data may make the results not match the facts, this causes people to doubt the accuracy of the statistical method. Statistics only indicate Statistical Association. Does not prove the relationship between factors. Regression has a causal relationship, and the correlation is not necessarily. Regression analysis is a statistical method for processing linear dependencies between two or more variables. Such problems are common. For example, the content of a metal element in human hair is related to the content of this element in blood, and the body surface area is related to height and weight. Regression analysis is used to illustrate the mathematical relationship between dependency changes. The existence of anything is not isolated, but interrelated and mutually restrictive. Height and weight, body temperature and pulse, age and blood pressure are all associated. Shows the closeness of objective things and presents them with appropriate statistical indicators. This process is related analysis.

The logic of random sampling and statistical inference the day before yesterday, McDull asked me how to construct a confidence interval. On the phone, I did not seem to have a clear explanation. Here I will re-prepare a note. At the same time, when I was working on a project in China, my colleagues experienced the embarrassment (they don't know how to check whether a variable conforms to the normal distribution, encourage yourself to think about what you have learned. People who have learned statistics know that they can perform random sampling on an unknown population (Population) by describing and calculating the sample (such as calculating the sample mean and sample variance ), then we can infer some general features (such as testing certain assumptions and constructing confidence intervals ). Of course, many modern inference methods are "cook books" and do not need to be detailed by non-professionals. For example, people engaged in economics often know how to test the F statistic and explain the conclusion without knowing what the f distribution is. They do not even need to calculate the F statistic. However, if you think carefully about the relationship, you can see that this random sampling-inference contains a certain philosophy, and in some places it does show human wisdom. In essence, this method uses a set of data (samples) that we have mastered 100% of the information. For a group of data that we have little or only some of the information (overall ), the process of fitting. In other words, push unknown processes with knowledge. Because it is an inference of "unknown", we cannot be 100% sure, but at the same time we use "Knowledgeable" data, so we are not sure at all. Perhaps we can see this kind of thinking behind it through a very simple example. Next we will start this exercise. [Example]: If a school has 20000 students, 1000 students are randomly selected from the school. Q: Is the average score of 20000 students in a course of this school 70/100? Note that there is no assumption about the overall distribution. [Question 1]: Can we use samples for Statistical Inference? It seems that this is not feasible yet, and there is a lack of logic. It is worth noting that the selected 1000 of students are "random samples "? The answer is no. The idea of random sampling is that the distribution of extracted samples is consistent with that of the population. This can be true for each observed value. However, random samples also require that each observation value be independent from each other. In this narrow sense, the probability of each observation value being obtained is the same. However, in the above example, this condition is obviously not met (many scholars often ignore this condition ). If you collect the transcript of 1000 students, the probability that the scores of these 1000 students are drawn depends on the order in which they are obtained. Because the scores of 1000 students are extracted at a time, the number of times each student receives is only one and cannot be repeatedly sampled. Therefore, the probability of the first student in the sample is 1/20000, that of the second student is 1/19999, and that of the Third student is 1/19998 ,...... 1,000th million students. That is to say, after some students are drawn, the probability of the next student being drawn is definitely not equal to 1/20000. Only when sampling with replacement is available can we say that the probability of each student being drawn is 1/20000, And we can ensure that random samples are taken ). However, in that case, we are likely to have less than 1000 samples, because the possibility of a student drawing more than one repetition is not zero. Fortunately, it is noted that the difference between the one thousand probability values is not big (because the overall value is large enough), so we can think that the score of these 1000 students is a random sample ). After such an approximation, we can make statistical inferences. This approximation is common in statistics. For example, central limit theorem describes that most distributions can be seen as normal distributions in an approximate way, which makes normal distributions play an extremely important role in statistics. An important feature of these distributions is that variables can be seen as the sum of observed values (functions). For example, two distributions can be seen as the sum of the results of a group of Bernolli tiral tests. In addition, for example, Poisson distribution can be considered as a binomial distribution, while the latter can be considered as a normal distribution. However, after completing this approximation, we still need more assumptions for Statistical Inference and test ). For example, we must understand the overall distribution, even if we do not know the specific values of all parameters. (Currently, we only discuss the parameter (parametric) method. [scenario 1]: We know exactly that the scores of these 20000 students conform to the (normal) distribution, and the mean is unknown (unkown mean) however, the variance is known (known variance ). [Question 2] What kind of Rule does the mean and variance of samples meet for any distribution? The following relationship can be obtained by using the simple expected properties of mathematics: (1) Expectation of sample mean = total mean. (2) variance of the sample mean = population variance/number of samples (fluctuation of the sample mean is not subject to a large variation of a single observed value) through these variables, we can construct the statistic Z: (3) z = (sample mean- Average Value)/Root number (population variance/number of samples ). According to (1) (2) and the central limit theorem AnyIn general, the Z statistic conforms to the standard normal distribution. It is worth noting that we have a lot of information about this Z statistic. For example, for any given a value, we can calculate the Z value that conforms to (4) Pr (z <| z |) = A %. However, because Z is a variable, we do not have 100% of the information. Note the meanings of formula (3) and (4). Since we can calculate the mean value of the sample, the variance of the sample, and the population variance, z is the only function determined by the mean of the population. Therefore, we can calculate the function expression of the total mean, because the total mean is only the inverse function of Z. Given a, we know the value range of Z and the change range of the average. This change range is what we call confidence interval. For example, Pr (C1 <overall mean <C2) = 90%, C1 is 5% percentile (percentile) c2 is a 95% percentile value. That is to say, the overall mean falls into C1, and the probability of the C2 range is 90%. So we can carry out the hypothesis test: H0: overall mean = 70 VSH1: Not H0. (Assume: size = 10% ). At this time, we know that PR (C1 <overall mean <C2) = 90%, so long as the overall mean <C1 or overall mean> C2, We can overturn H0 at the level of 10%. [Scenario 2] We do not know the population variance or the population mean. Let's look at formula (3). We know that we cannot use the normal distribution for inference, so we have to use a new method, that is, the tdistribution. According to the definition, sample variance = sum (observed I-sample mean) ^ 2; I = 1000. Sample Mean = sum (observed value I)/number of samples I = 1, 2 ,...... 1000. It can be proved that (the process is complex and orthogonal matrix operation is required), and (sample variance/population variance) conforms to (sample value-1) Chi-squared distribution of degrees of freedom. Sample variance and sample mean are independent variables. Then construct a new T variable: t = z/root number (chi-square/Degree of Freedom ). It is worth noting that each denominator is a fraction and each denominator has an unknown population variance. But fortunately, the two sell each other. Therefore, t is only a function determined by the mean of the population. Then we can build confidence intervals again. It should be noted that (I) for the population that conforms to any distribution, Z complies with the standard normal distribution, because the mean value of the sample is the sum of all observed values (multiplied by a constant ), as long as the number of samples is large enough, the central limit theorem ensures its approximation to the standard normal distribution. (Ii) However, if the population does not conform to the normal section, we cannot perform a t-test. Because the sample variance does not conform to the chi-square distribution, t does not conform to the tdistribution. Summarize the philosophy here. We used a sample with 100% information and calculated several values (Sample Mean and sample variance ). Then we constructed a statistic Z or t that we have mastered a lot of information. We can use this information to understand the general situation of a few pieces of information. The intriguing part is that the statistic Z, and T, which conform to the standard normal distribution, comes from the sample, and some from the population. In this case, we understand a part of the workload, which is just a bridge to our statistical inference. We don't have enough information because we analyze the overall information directly-We almost don't know anything. Even though we have sufficient information, this sample is not closely related to the overall mean. We only know the (1) and (2) types. As a result, the Z and T variables play the role of "saving the nation by curve. However, because of this, we can only say that we have a % confidence that the overall mean falls within the (C1, C2) range. In the absence of more information, we need to add more steps. For example, to construct a t variable, we need to prove that the sample mean and sample variance are independent variables. We also need to know the chi-square distribution. However, people have already discovered such distribution, paving the way for this method. In my opinion, these people have really promoted the development of the history of human thoughts. How do these people come up with a chi-square distribution? How to find the relationship between the normal distribution and the tdistribution? These are amazing. [Scenario 3] We do not know the overall distribution or any parameters. As mentioned above, if you do not know the population distribution, as long as you know the population variance, the Z variable conforms to the standard normal distribution. But now we don't know the population variance, so we can't even use the parameter method. We should use the nonparametic method or semi-parametric method ). However, the logic is still consistent, that is, we need to use an intermediate statistic to associate the sample with the population. For example, rank test and Size Test) create a new statistic.

Common statistical concepts: t-test, F-test, Chi-square test, P-value, degree of freedom

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