Hypothesis test is the most widely used statistical tool in six Hummer team projects. For example, it is necessary to judge whether the following conclusions are correct: "new employees complain more than old employees", "the average output has increased after the improvement work", "when the processing temperature is 180 degrees, the breaking strength of the gasket is higher than 160 degrees" and so on. Because we always observe the data with errors, we can't draw conclusions from the results of simple sample statistics, and we must use strict statistical hypothesis testing methods to draw accurate judgment conclusions.

I. Hypothesis test problem

Parameter estimation and hypothesis testing are two important aspects of statistical inference. Parameter estimation takes "number" as the output result and hypothesis testing takes "judgment" as the output result. To illustrate its basic idea, let's look at an example first.

Second, the hypothesis test steps

1, establish a hypothesis.

The first step of hypothesis testing is to establish hypotheses, which usually need to establish two hypotheses: the original hypothesis Ho and the alternative hypothesis H 1.

2. Select inspection statistics and determine the form of the rejected domain.

If we test the average of the total surplus, then we will use the sample average to derive the test statistics; If we test the variance of the normal population, we will get the test statistics from the sample variance.

According to the value of statistics, the whole sample space is divided into two parts: rejection domain W and non-rejection domain A. When the value of sample statistics falls within the rejection domain, the original hypothesis will be rejected, otherwise it will not be rejected. Therefore, we must find out the rejection domain in hypothesis testing.

According to different alternative assumptions; Denial domains can be bidirectional or unidirectional. After determining the type of denied domain, you should also determine the critical value. This should be determined according to the probability of allowing mistakes.

3. Give significance level A in the test.

When judging whether the original hypothesis is true or not, due to the randomness of the sample, two kinds of errors may appear in the judgment, and their meanings are shown in the following table. 1 error means that when the original hypothesis is true, due to the randomness of the sample, the observed value of the sample falls within the rejection domain w, thus making a decision to reject the original hypothesis. This kind of error is called 1 error, which is also called the probability of discarding truth.

The second wrong explanation: if the average tensile strength of steel bars is really higher than the original one, then the average tensile strength of steel bars is no longer 2 000kg, but we have not rejected the mistake that H0 has not improved, that is, we mistake "improved" for "not improved". Generally speaking, when H0 failed, we didn't reject H0, which is the second kind of mistake.

4. Give the critical value and determine the rejection domain.

When the significance level is a, the critical value can be obtained by looking up the table according to the distribution of the given test statistics, so as to determine the specific rejection domain. Under different substitution assumptions, the relationship among rejection domain, critical value and significance level A is different.

5, according to the sample observation value, calculate the test statistics. Collect sample data and calculate test statistics.

6. Judge according to whether the inspection statistics belong to the rejection domain.

① Compare the test statistics with the critical value of the rejection domain, and make a rejection when it falls into the rejection domain.

Draw a conclusion, or draw a conclusion that cannot reject the original hypothesis.

(2) Calculate the p value from the test statistics. The so-called p value is the probability of the status quo when the original hypothesis is established (strictly speaking, the status quo when the original hypothesis is established or the situation that is more unfavorable to the original hypothesis, that is, the probability of the situation that is more favorable to the alternative hypothesis). When this probability is small (for example, less than 0.05), this result should not appear in an experiment if the original hypothesis is established; But now it does appear, so we have reason to think that the premise of "the original hypothesis is established" is wrong, and we should reject the original hypothesis and accept the alternative hypothesis. Therefore, there can be a most general rule: if P

(3) According to the observed values of the samples, the confidence intervals of the overall parameters can be obtained. If the parameter value of the original hypothesis does not fall into this confidence interval, the conclusion that the original hypothesis is rejected is drawn, otherwise the conclusion that the original hypothesis cannot be rejected is drawn. At present, most statistical software provides corresponding confidence intervals, which is more convenient to judge by this method.