Three compulsory knowledge points of high school mathematics 1
Preliminary algorithm
1: the concept of algorithm
(1) Algorithm concept: Mathematically, an "algorithm" in the modern sense usually refers to a kind of problem that can be solved by a computer as a program or step, and these programs or steps must be clear and effective and can be completed in a limited number of steps.
(2) Features of the algorithm:
Picture finiteness: The sequence of steps of an algorithm is finite, and it must stop after a finite operation, not infinite.
Picture certainty: every step in the algorithm should be certain, and can be effectively executed to get a certain result, and there should be no ambiguity.
Image sequence and correctness: the algorithm starts from the initial step and is divided into several definite steps. Each step can only have one definite subsequent step, and the previous step is the premise of the next step. Only when the previous step is carried out can the next step be carried out and each step is accurate, and the problem can be completed.
Uniqueness of the picture: the solution of a problem is not necessarily unique, and there are different algorithms for a problem.
Universality of pictures: Many specific problems can be solved by designing reasonable algorithms, such as mental calculation and calculator calculation, which must be solved through limited and pre-designed steps.
2. Program block diagram
Basic concepts of (1) program block diagram:
The concept of picture program composition: program block diagram, also known as flow chart, is a graphic that accurately and intuitively represents the algorithm with specified graphics, pointing lines and text descriptions.
The program block diagram includes the following parts: program blocks representing corresponding operations; Streamlines with arrows; Necessary text description outside the program box.
Graphical symbols of image forming program box and their functions
Program box
name
function
draw
Start-stop frame
Indicating the beginning and end of the algorithm is essential for any flowchart.
draw
in-out box
The information representing the input and output of the algorithm can be used anywhere in the algorithm where input and output are needed.
draw
draw
Processing framework
In the algorithm, the assignment, calculation and formula required for data processing are written in different processing boxes for data processing.
Referee box
Judge whether a condition is established, and mark "Yes" or "Y" at the exit when it is established; If not, please mark "No" or "No".
3. Three basic logical structures of the algorithm: sequence structure, conditional structure and cyclic structure.
(1) sequence structure: the sequence structure is the simplest algorithm structure. Reports and boxes are made from top to bottom. It consists of several processing steps that are executed in turn. It is the basic algorithm structure that any algorithm can't do without.
(2) Conditional structure: Conditional structure refers to choosing different flow directions by judging whether the conditions in the algorithm are true or not.
Algorithm structure.
(3) Circular structure: In some algorithms, a processing step is often executed repeatedly from a certain place according to certain conditions. This is the cycle structure, and the repeated processing steps are the cycle body. Obviously, the loop structure must contain the conditional structure.
Three knowledge points in the second compulsory course of senior high school mathematics
statistics
2. 1. 1 simple random sampling
1. Population and sample
In statistics, the whole research object is called population, each research object is called individual, and the total number of individuals in the population is called overall capacity. In order to study the related properties of the population, we randomly select a part: research, which we call samples. The number of individuals is called sample size.
2. Simple random sampling, also called pure random sampling.
In other words, the survey units are randomly selected from the population, without any grouping, classification, queuing, etc. The characteristics are: the probability of each sample unit being extracted is the same (the probability is equal), and each unit of the sample is completely independent, and there is no certain correlation and exclusion between them. Simple random sampling is the basis of other sampling forms. This method is usually only used when the difference between the whole units is small and the number is small.
3. Common methods of simple random sampling:
(1) draw lots; (2) Random number table method; ⑶ Computer simulation method; ⑷ Direct extraction with statistical software.
In the sample size design of simple random sampling, the main considerations are: ① population variation; ② Allowable error range; ③ Degree of probability assurance.
4. Draw lots:
(1) Number each object in the investigation team;
(2) Prepare the lottery tool and implement the lottery.
(3) Measure or investigate each individual in the sample.
Please investigate the favorite sports activities of students in your school.
5. Random number table method:
Example: Select 10 students from the class to participate in an activity by using a random number table.
2. 1.2 systematic sampling
1. System sampling (equidistant sampling or mechanical sampling):
Sort the units of the population, then calculate the sampling distance, and then sample according to this fixed sampling distance. The first sample was selected by simple random sampling.
K (sampling distance) =N (overall size) /n (sample size)
Prerequisite: For the variables studied, the arrangement of individuals in the group should be random, that is, there is no regular distribution related to the variables studied. You can start sampling from different samples and compare the characteristics of several samples under the conditions allowed by the investigation. If there are obvious differences, it shows that the distribution of samples in the population follows a certain cycle law, and this cycle coincides with the sampling distance.
2. Systematic sampling, namely equidistant sampling, is one of the most commonly used sampling methods in practice. Because it has low requirements for sampling frames and simple implementation. More importantly, if there are some auxiliary variables related to the survey indicators available, and the whole unit is queued according to the size of the auxiliary variables, systematic sampling can greatly improve the estimation accuracy.
2. 1.3 stratified sampling
1. stratified sampling (type sampling):
First of all, according to some characteristics or signs (gender, age, etc.), all units in the group are divided into several types or levels. ), and then extract a sub-sample from each type or level by simple random sampling or systematic sampling. Finally, these sub-samples are combined to form a total sample.
Two methods:
1. Firstly, the population is divided into several layers by stratification variables, and then extracted from each layer according to the proportion of each layer in the population.
2. Firstly, the population is divided into several layers by stratification variables, and then the elements in each layer are arranged neatly in hierarchical order. Finally, samples are extracted by systematic sampling.
2. Stratified sampling is to divide people with strong heterogeneity into sub-populations with strong homogeneity, and then draw samples from different sub-populations to represent sub-populations, and all samples represent people again.
Stratification standard:
(1) Take the main variables or related variables to be analyzed and studied in the investigation as the standard of stratification.
(2) Ensure that the variables with strong homogeneity in each layer, strong interlayer heterogeneity and outstanding overall internal structure are used as stratified variables.
(3) Take those variables with obvious stratification as stratified variables.
3. The proportion of stratification:
(1) Proportional stratified sampling: a method to extract sub-samples according to the proportion of units of various types or levels to the total units.
(2) Non-proportional stratified sampling: If the proportion of some levels in the population is too small, the sample size will be very small. At this time, this method is mainly used to facilitate special research or comparison of different levels of subpopulations. If we want to infer the population from the sample data, we need to first weight the data of each layer, adjust the proportion of each layer in the sample, and restore the data to the actual proportion structure of each layer in the population.
2.2.2 Use the digital features of the sample to estimate the digital features of the population.
1, average:
2, sample standard deviation:
3. When estimating the population with samples, if the sampling method is reasonable, then the samples can reflect the information of the population, but the information obtained from the samples will be biased. In random sampling, this deviation is inevitable.
Although the distribution, mean and standard deviation we get from the sample data are not the real distribution, mean and standard deviation of the population, but only an estimate, this estimate is reasonable, especially when the sample size is large, and they do reflect the information of the population.
4.( 1) If the same constant is added or subtracted from each data in a set of data, the standard deviation remains unchanged.
(2) If each data in a set of data is multiplied by a constant k, the standard deviation becomes k times the original value.
(3) The influence of the maximum and minimum values in a set of data on the standard deviation, and the application of intervals;
The scientific truth of "removing a highest score and removing a lowest score"
2.3.2 Linear correlation of two variables
1, concept:
(1) regression linear equation
(2) Regression coefficient
2. Least square method
3. Application of linear regression equation
(1) describes the dependency between two variables; Linear regression equation can be used to quantitatively describe the quantitative relationship between two variables.
(2) Using regression equation to forecast; Substituting the predictor (independent variable x) into the regression equation to estimate the predictor (dependent variable y), the allowable interval of individual y value can be obtained.
(3) Use regression equation for statistical control, specify the change of Y value, and achieve the purpose of statistical control by controlling the range of X. If the regression equation between NO2 concentration in the air and traffic flow is obtained, the NO2 concentration in the air can be controlled by controlling traffic flow.
4. Matters needing attention in the application of linear regression
(1) regression analysis should have practical significance;
(2) Before regression analysis, it is best to make a scatter plot;
(3) Don't extend the tropic of cancer.
Senior high school mathematics compulsory three knowledge points 3
possibility
3.1.1-3.1.2 Probability of random events and its significance.
1, basic concept:
(1) inevitable event: the event that will happen under condition S is called the inevitable event relative to condition S;
(2) Impossible events: events that will not happen under condition S are called impossible events relative to condition S;
(3) Deterministic events: inevitable events and impossible events are collectively referred to as deterministic events relative to condition S;
(4) Random events: events that may or may not occur under condition S are called random events relative to condition S;
(5) Frequency and frequency: repeat the test for n times under the same condition S, and observe whether there is an event A, and call the frequency nA of the event A in the n tests as the frequency of the event A; The ratio fn(A) of event A = the probability of event A: for a given random event A, if the frequency fn (a) of event A is stable at a certain constant with the increase of test times, the constant is recorded as P(A) and called as the probability of event A. ..
(6) Difference and connection between frequency and probability: The frequency of a random event refers to the ratio of the number of times nA of the event to the total number of times n of testing, which has certain stability and always swings around a certain constant, and with the increase of testing times, the swing amplitude becomes smaller and smaller. We call this constant the probability of random events, which quantitatively reflects the probability of random events. Frequency can be approximated as the probability of the event under the premise of a large number of repeated experiments.
3. Basic Properties of1.3 Probability
1, basic concept:
The inclusion, union, intersection and equality of (1) events.
(2) If A∩B is an impossible event, that is, A ∩ B = Ф, then event A and event B are mutually exclusive;
(3) If A∩B is an impossible event and A∪B is an inevitable event, then event A and event B are mutually opposite events;
(4) When events A and B are mutually exclusive, the addition formula is satisfied: p (a ∪ b) = p (a)+p (b); If events A and B are opposite events, then A∪B is an inevitable event, so P(A∪B)= P(A)+ P(B)= 1, so there is P (A) = 1-P (B).
2, the basic nature of probability:
1) The probability of inevitable events is 1, and the probability of impossible events is 0, so 0 ≤ p (a) ≤1;
2) When events A and B are mutually exclusive, the addition formula is satisfied: p (a ∪ b) = p (a)+p (b);
3) If events A and B are opposite, then A∪B is inevitable, so P(A∪B)= P(A)+ P(B)= 1, so there is P (A) =1-P (B);
4) The difference and connection between mutually exclusive events and opposing events, mutually exclusive events means that in an experiment, event A and event B will not happen at the same time, including three different situations: (1) Event A happens and event B doesn't happen; (2) Event A does not occur, but Event B does; (3) Event A and Event B do not occur at the same time, while the opposite event means that there is only one event A and Event B, including two situations; (1) Event A occurs, but event B does not; (2) Event B happens and Event A doesn't, which is a special case of mutually exclusive events.
3.2. 1 —3.2.2 Generation of Classical Probability and Random Numbers
The use conditions of classical probability 1 and (1): the finiteness of test results and the equal possibility of all results.
(2) the solution steps of classical probability;
① Find the total number of basic events;
② Find the number of basic events contained in event A, and then use the formula P(A)= 1
3.3. 1—3.3.2 Generation of Geometric Probability and Uniform Random Numbers
1, basic concept:
(1) geometric probability model: If the probability of each event is only proportional to the length (area or volume) of the event area, such a probability model is called geometric probability model;
(2) Probability formula of geometric probability:
p(A)=；
(3) Characteristics of geometric probability: 1) There are infinitely many possible results (basic events) in the experiment; 2) The possibility of each basic event is equal.
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