假设检验属于离散趋势检验吗

假设检验属于统计推断的一部分。所谓“统计推断”，就是根据样本中的信息推断出总体的情况，而总体就是我们的目标。例如，总体体重是通过样本的体重来估计的，也就是说，使用样本统计量来估计总体参数。

而在使用样本推断总体的过程中，为什么常见的方式是使用假设检验呢？所隐含的实际上是一种妥协，因为我们永远无法直接证明。依然以体重为例，如果不对总体进行一一测量，就无法得到总体体重的真实平均值。

因此，一方面我们很难得到真实的值，另一方面又希望估计足够贴近真值，那么为什么不从假设开始，通过验证假设是否正确来获得呢？这便是假设检验的出发点。它非常简单，也非常符合我们的思维逻辑。这就像是在做代数题，直接推导一般情况很困难的时候，可以尝试带入具体的数值进行验算，说不定一下子就找到正确答案了。

按照这个思路，假设检验，便是首先对所研究的总体特征给出一个假设：不是不知道体重的总体平均值吗？我们直接假设它是65kg！做出假设后，我们用样本的数据来验证假设是否可信。这里的“可信不可信”用数学语言则表示为“概率高不高”。如果概率高，则说明可信；如果概率小，则值得怀疑。至于这里的概率是指什么，判断“高不高”的标准是什么，这便是假设检验的具体内容了

假设检验 Hypothesis Testing

四要素：

基本过程：

Note 1：关于 和

This is because hypothesis tests are designed to avoid rejecting when it is true. Therefore when the test rejects , one can be quite sure that is false. 这里涉及到下面要说的“假设检验中的两类错误”。

Note 2：关于统计量

Note 3：关于拒绝域

Type I error 和 Type II error 的关系：

We can always reduce the type I error by making the rejection region **aller. This will typically at the expense of larger type II error.

In practice，we want to have powerful tests with a given type I error.

The P-value is the **allest for which the given observed data (once you have done the random experiment) suggests rejection of

Smaller P-value indicates rejection of the null hypothesis.

are independently and identically distributed, with and known. Then

Note: If the variance (总体方差) is unknown, you can replace it by (样本方差), since is large.

小样本情况下，上述CLT中的正态分布可以用 分布近似，即

, then

过程同2.1.1 大样本均值检验

(1) for , the RR is

(2) for , the RR is

(3) for , the RR is

or

, then we have

(1) The likelihood of is

(2) Suppose ,

where are some sets of possible parameter values and .

Define generalized likelihood ratio as

where is the dimension of parameter space and is the dimension of parameter space

Note: 计算 时，涉及到 Maximum Likelihood Estimator.

Suppose each individual's category is a multinomial draw with probability .

Let be the number of observed individuals in each category. Then

Let be the simplex, i.e. .

The maximum likelihood estimator (MLE) over all is:

vs

Under and using MLE, we can get the expected number for each category as . Then

Note: While we could apply a likelihood ratio test here, Pearson's test has a bit more power.

检验两个分类变量是否相互独立。

Suppose we have observed an contingency table.

row and column variables are independent.

row and column variables are dependent.

Under we have following contingency table:

The MLEs for are

Then we can get expected number of individuals for each category.

Hypothesis Testing

Hypothesis: A statement about the value of a population parameter developed for the purpose of testing.

Hypothesis testing:假设检验 Based on sample evidence and probability theory; Used to determine whether the hypothesis is a reasonable statement and should not be rejected, or is unreasonable and should be rejected.

Step One: State the null and alternate hypotheses

Null Hypothesis H0 零假设或空假设:A statement about the value of a population parameter

Alternative Hypothesis H1备择假设: A statement that is accepted if the sample data provide evidence that the null hypothesis is false

Three possibilities regarding means(The null hypothesis always contains equality.)

H0:μ=0 H0:μ≤0 H0:μ≥0

H1:μ≠0 H1:μ0 H1:μ0

Step Two: Select a Level of Significance. 显著性水平

Level of Significance: The probability of rejecting the null hypothesis when it is actually true; the level of risk in so doing.

Type I Error: Rejecting the null hypothesis when it is actually true (α).

Type II Error: Accepting the null hypothesis when it is actually false (β).

Researcher

Null Hypothesis Rejects H0 Accepts H0

H0 is true Correct decision Type I error (a)

H0 is false Type II Error (b) Correct decision

Step Three: Select the test statistic.

Test statistic: A value, determined from sample information, used to determine whether or not to reject the null hypothesis.

z Distribution as a test statistic: The z value is based on the sampling distribution of X, which is normally distributed when the sample is reasonably large (recall Central Limit Theorem).

Step Four: Formulate the decision rule.

Critical value临界值: The dividing point between the region where the null hypothesis is rejected and the region where it is not rejected.

Sampling Distribution of the Statistic z, a

Right-Tailed Test, .05Level of Significance

Decision Rule决策规则: Reject the null hypothesis and accept the alternate hypothesis if

Computed -z Critical -z or Computed z Critical z

p-Value: The probability, assuming that the null hypothesis is true, of finding a value of the test statistic at least as extreme as the computed value for the test

Decision Rule: If the p-Value is larger than or equal to the significance level, a, H0 is not rejected. If the p-Value is **aller than the significance level, a, H0 is rejected.

Calculated from the probability distribution function or by computer

P-value level of significance: reject H0

P-value level of significance: fail to reject H0

Step Five: Make a decision.

One-Tailed Tests of Significance: The alternate hypothesis, H1, states a direction

Two-Tailed Tests of Significance: No direction is specified in the alternate hypothesis H1.

Test for the population mean from a large sample with population standard deviation known

Proportion: The sample proportion is p and p is the population proportion.

The fraction or percentage that indicates the part of the population or sample having a particular trait of interest.

Test Statistic for Testing a Single Population Proportion

假设检验（Hypothesis Testing） 到底是什么意思？

就是说，如果你认为世界上没有鬼这个假设是真的，那么你碰到鬼这个小概率事件基本不可能发生。如果某一天你碰到鬼了，那么你就有理由怀疑世界上没有鬼这个假设的真实性了。

假设检验名词解释是什么?

假设检验(hypothesis testing)，又称统计假设检验，是用来判断样本与样本、样本与总体的差异是由抽样误差引起还是本质差别造成的统计推断方法。

显著性检验是假设检验中最常用的一种方法，也是一种最基本的统计推断形式，其基本原理是先对总体的特征做出某种假设，然后通过抽样研究的统计推理，对此假设应该被拒绝还是接受做出推断。常用的假设检验方法有Z检验、t检验、卡方检验、F检验等。

假设检验注意问题：

1、作假设检验之前，应注意资料本身是否有可比性。

2、当差别有统计学意义时应注意这样的差别在实际应用中有无意义。

3、根据资料类型和特点选用正确的假设检验方法。

4、根据专业及经验确定是选用单侧检验还是双侧检验。

5、判断结论时不能绝对化，应注意无论接受或拒绝检验假设，都有判断错误的可能性