Hypothesis testing, as an important component of statistical inference, is a very useful method in statistics and is widely applied in quality improvement activities during the manufacturing process. Hypothesis testing is the process of making certain assumptions about the overall distribution parameters, then using sampling methods to extract sample data, and based on the observed values of the samples, using statistical analysis tools to test whether the previously made assumptions are correct, thereby deciding whether to accept or reject the assumptions.
In the manufacturing process of welded sealed load cells, the zero-point temperature characteristic of the load cell (hereinafter referred to as: ZTC) is an important influencing factor affecting the product qualification rate. There are many factors that affect the ZTC of the load cell, such as the temperature characteristics of the strain gauge, the surface mount process, the bridge circuit and the welding process, etc. During the production and manufacturing process, we collected a large amount of ZTC test data and found that there was a significant difference in the ZTC of the load cell before and after welding the sealing diaphragm. Subsequently, we applied the hypothesis testing method to verify the correctness of this idea.
The basic idea of hypothesis testing is what is called the proof by contradiction of probabilistic properties: To test whether the null hypothesis H0 is correct, we first assume that H0 is correct and see what results can be derived from it. If it leads to the occurrence of an unreasonable phenomenon, it indicates that "hypothesis H0 is correct" is wrong, that is, the null hypothesis H0 is incorrect. Therefore, we reject the null hypothesis H0. If it does not lead to the occurrence of an unreasonable phenomenon, Then it cannot be considered that the null hypothesis H0 is incorrect, and thus we do not reject H0. At this point, depending on the needs of the problem, further experimental investigations may be conducted or H0 may be accepted. It should be particularly noted that the basis of the proof by contradiction of probabilistic properties is the principle of low-probability events, which states that "low-probability events (and events with very low probabilities) are almost impossible to occur in a single experiment." In other words, in hypothesis testing, our decision to accept or reject H0 does not mean that we have proved the null hypothesis H0 correct or wrong. It merely indicates that H0 is correct or wrong based on the information provided by the sample with a certain degree of reliability.
The general steps of hypothesis testing are as follows (Note: The sequence of each step is sometimes not strict as follows) :
(1) According to the requirements of the actual problem, propose the null hypothesis H0 and the alternative hypothesis H1. Generally, hypotheses can be determined based on the following principles: A. The null hypothesis H0 represents a long-standing state (such as an existing production process), while the alternative hypothesis H1 reflects a change (such as a new production process that has not been fully verified); b. The sample observations show the supported conclusion and should be regarded as alternative H1; c. Try to classify mistakes with serious consequences as first-class mistakes.
(2) Determine the sample size and sampling plan to obtain the test data.
(3) Determine the type of test data and verify its validity. For instance, is the sample data a random number? Does it conform to a normal distribution or Poisson distribution, etc.? Then, different methods such as Z-test and t-test need to be selected based on the type and distribution characteristics of the data.
(4) Construct the form of the test statistic T and the rejection domain. The test statistic is to convert the sample statistic T (such as the mean) calculated from the sample data into a value under the assumption that the null hypothesis is correct. After determining the test statistic T, the form of the rejection domain is determined based on the null hypothesis H0 and the alternative hypothesis H1.
(5) Select an appropriate significance level α and determine the critical value. When judging whether the null hypothesis holds true, since the sample is used to infer the population, this determines that two types of incorrect decisions may be made: a. The null hypothesis H0 was originally correct, but we rejected H0. This type of error is generally called the first type of error, and its occurrence probability is usually denoted as α, that is: P {reject H0, H0 is true} =α. b. The null hypothesis H0 was originally incorrect, but we accepted H0. This type of error is generally called the second type of error, and its occurrence probability is usually denoted as β, that is: P {accepted H0; H0 is not true} =β. Although both α and β should be as small as possible, when the sample size is fixed, it is impossible to control both α and β to be very small at the same time. Therefore, the value of α should be reasonably determined based on the problem under study. Generally, two factors should be considered: Firstly, if there is considerable confidence in the null hypothesis, the significance level α should be set smaller; Secondly, it is necessary to consider the possible losses that may result from making a decision.
If the losses caused by the first type of mistake are significant, the significance level α should be set smaller; otherwise, α should be correspondingly larger. In practical applications, α is usually taken as some standardized values, such as 0.01, 0.05, 0.1, etc. Starting from P {rejection domain X: H0 is true}≤α, the probability of committing the first type of error is tested as close as possible to α. In particular, when the population is a continuous random variable, it is often made equal to α to determine the critical value, and thus the rejection field X is also determined.
(6) Determine whether to reject H0 based on the sample data. Calculate the statistic T from the sample data and compare it with the critical value to determine whether to reject H0.