When it comes to Poincare, people may first think of the famous "Poincare conjecture", but in this section we will tell a real and interesting Poincare story, which is very helpful to understand the hypothesis testing module in mathematical statistics.
When we buy some food, the weight of the food will fluctuate a little. For example, the weight label of bread packaging bag can be written as follows:
The weight of bread should be 1000g, but there may be an error of 50g for various reasons. Poincare is a man who eats bread every day. The same thing happened to him. A baker claimed that the average weight of bread sold to Poincare was 65,438+0,000 grams, with a difference of 50 grams. The baker sold a loaf of Poincare bread every day. In the face of this loyal customer, he was unprepared and sold 1 piece of bread of this mathematical genius every day according to his own way of buying and selling, but the baker's nightmare began.
In Poincare's eyes, the weight of bread should be 1000g, with a fluctuation range of 50 g. Expressed in mathematical language, the weight of bread obeys a normal distribution with an expected value of 1000g and a standard deviation of 50g. As a rigorous mathematician, Poincare weighs the bread he bought every day, and the recorded data (unit g) of the first nine days are as follows:
98 1 972 966 992 10 10 1008 954 952 969
The expected value (average) of this set of data is x=978.2. Although 1000g was not expected, there was also a fluctuation of 50 g. Although I felt a little uncomfortable, it was hard to say that there was a problem. However, as a mathematician, Poincare was 80% sure that the baker cut corners in the production process. But at this time, the evidence is hard to say. Poincare decided to stay put and continue to record 16 days, with a total of 25 data as follows:
The average recorded data in 25 days was 978.7g, which increased slightly, but at this time Poincare was 95% sure that the baker cut corners in the production process.
Poincare decisively reported to the quality inspection department. After the quality inspectors arrived, the baker denied it in every way, claiming that the bread he made was based on 1000g, with a maximum error of 50 g. All the data provided by Poincare conformed to the law he described, and the quality inspectors were at a loss for a moment. But the baker may not know his opponent, a mathematician who is proficient in hypothesis testing. The following is poincare's testimony:
First, if the baker's statement is correct, the mass x of each piece of bread obeys a normal distribution with 1000g as the expected value and 50g as the variance.
This is no problem. It seems that all these 25 cakes obey this law. But the average value of 25 breads also obeys the normal distribution, which is the second important point.
Second, the average of 25 breads also obeys the normal distribution, and the expectation is still 1000, but the variance has changed. The calculation formula is as follows:
In other words, the average weight of 25 breads follows the following normal distribution:
The baker and the quality inspector said they didn't understand. What does this mean? Poincare gave a popular explanation: the weight fluctuation of one bread is very small, and the average weight fluctuation of multiple breads is much smaller. Just like you roll the dice, if you roll the dice 1, the possible points are 1 to 6, but if you roll the dice 100 times, the average value of the total number of 100 times is basically a constant of 3.5. You can try if you don't believe me. Bakers and quality inspectors basically understand this truth and continue to listen to Poincare's third explanation.
Thirdly, since the average value of the total weight of 25 breads obeys the normal distribution with the expected value of 1000g and the variance of 10g, let's first look at the distribution characteristics of the normal distribution data, as shown in the following figure:
As can be seen from the above figure, 95.44% of the data falls within the range of [980, 1000g] with the expected1000 g as the center and 2 times variance as the fluctuation (20g). In other words, if the baker makes bread strictly according to 1000g, and 50g is floating, then the average value of 25 breads will have a 95.44% probability of falling within the range of [980, 1020]; conversely, the probability of falling below 980g or rising above 1020g is less than 5%.
After listening to Poincare's popular science about hypothesis testing, the quality inspector punished the baker, who admitted that he really made bread based on 980g and agreed to correct it.