Through a typical case, the specific application of mathematical sampling method in real life is illustrated: in order to understand the acceptance of a new product by ordinary residents, it is necessary to select 1000 households in a city to conduct market research, and select 1 household members from each household as the survey object. Overall sampling design Because the number of households in a city may be as many as one million, most enterprises do not have such a huge list of households except some large market research institutions and national statistical departments. This situation determines that the sampling design can only adopt multi-stage sampling method. According to the survey requirements, sampling is divided into two stages. The first stage is to select 50 sample neighborhood committees from the list of neighborhood committees in the city, and the second stage is to select 20 households from each selected neighborhood Committee. A list of neighborhood committees in a city can be obtained from statistics or civil affairs departments. After the neighborhood committees are numbered, you can simply select the 50 neighborhood committees you need by generating random numbers by computer. If the list of neighborhood committees also includes information such as the number of households in the neighborhood committees, the method of unequal probability selection can be used in the selection. If the probability of a neighborhood Committee being drawn is directly proportional to the number of households in the neighborhood Committee, this method is called PPS (probability is directly proportional to scale) sampling method. PPS sampling is a "self-weighted" sampling method, which ensures that when 20 households are selected by neighborhood committees of different sizes, the representativeness of each household sample is the same, so that the final results can be averaged directly. Of course, if the data is not enough for PPS sampling, then the ex post weighting method can also effectively infer the survey results. After the sampling neighborhood committees in the neighborhood committees are selected, the neighborhood committees' maps will be used to select residents. At this time, it is necessary to send some samplers to the neighborhood Committee to draw the distribution map of households. Sampling personnel need to know the actual location and coverage of neighborhood committees and calculate the actual number of households in each building. Then, according to the requirements of sample size, the sampler selects several households as the final samples by equidistant or other methods. Ensure that the interviewer of the interviewee conducts home visits according to the sample families selected by the sampling personnel. Who is the actual respondent is the last question of sampling design. If the content of the survey involves the family situation of the respondents, the respondents' choices can be determined according to their status in family life, such as those who use computers the most, those with the highest income, those who are actually responsible for purchasing decisions, and so on. If the content of the survey involves personal behavior, every adult in the family can be the respondent. At this time, a second round of sampling is needed, because if the interviewer determines the interviewee artificially, the interviewee may eventually be biased towards a certain type of people, such as the elderly and women who are more accessible in the family. The second sampling method in the family was invented by Leslie Kish, a famous American sampling expert, and is generally called Kish table method. After the interviewer enters the household, first record the number of all family members who meet the survey conditions, and sort and number them according to their age. Subsequently, the interviewer finds a number in the table according to the intersection of the number of interviewed households and the number of family members, and takes the family members corresponding to this number as interviewees. The above case is a typical two-stage field sampling design of household survey. From the whole design process, it can be seen that the selection of neighborhood committees, households and respondents after entering the household embodies the principle of randomness. In any link, if the principle of randomness is violated, it may cause immeasurable deviation to the survey results. Sampling design in investigation is a complex technical link, and non-professional researchers need to pay special attention to this problem.