Statistics, 8th Edition, Jia Junping, Chapter 6 Summary of knowledge points of statistics and sampling distribution and answers to exercises after class

One 、 Knowledge framework

Two 、 Exercises
Adjust a bottling machine so that the average filling volume of each bottle is μ Oz. , By observing that the filling volume of each bottle by this bottling machine obeys the standard deviation σ＝1.0 The normal distribution of ounces . Randomly select this machine for filling 9 Bottles make up a sample , And measure the filling volume of each bottle . Try to make sure that the sample mean deviates from the overall mean by no more than 0.3 Ounce probability . Explain ： Set the filling volume of each bottle as X,X Is the sample mean , Sample size is n. As a whole X It follows a normal distribution , Sample mean X It also follows a normal distribution , And the mean value is the same , The standard deviation is

therefore

3、 ... and 、 Brief introduction

1 What are statistics ？ Why introduce statistics ？ Why don't statistics contain any unknown parameters ？

answer ：（1） The definition of statistics ： set up X1,X2,…,Xn It's from the whole X The extracted capacity is n A sample of , If a function is constructed from this sample T（X1,X2,…,Xn）, Does not depend on any unknown parameters , It's called a function T（X1,X2,…,Xn） It's a statistic .
（2） The reason for introducing statistics ： in application , When a sample is taken from a population , It cannot be directly used to infer the relevant properties and characteristics of the population , This is because although the sample is representative of the population , Contains information of a general nature , But it is still scattered . To make statistical inference possible , First of all, we must concentrate the information we are concerned about scattered in the sample , For different research purposes , Construct different sample functions .
（3） The reason why the statistics do not contain unknown parameters ： Statistics are a function of samples . Construct specific statistics from samples , In fact, it is processing the overall information contained in the sample according to certain requirements , Concentrate the information scattered in the sample on the value of statistics , Different statistical inference problems require the construction of different statistics , So the statistics do not contain unknown parameters .2 Determine which of the following sample functions are statistics ？ Which are not statistics ？

T1＝（X1＋X2＋…＋X10）/10
T2＝min（X1,X2,…,X10）
T3＝X10－μ
T4＝（X10－μ）/σ

answer ： Statistics cannot contain unknown parameters , so T1、T2 It's statistics ,T3、T4 Not statistics .

3 sketch χ2 Distribution 、t Distribution 、F The relationship between distribution and normal distribution .

answer ：（1）χ2 The relationship between distribution and normal distribution ： A random variable X1,X2,…Xn Are independent of each other , And all obey the standard normal distribution , Then their square sum is subject to the degree of freedom n Of χ2 Distribution .
（2）χ2 Distribution 、t The relationship between distribution and normal distribution ： A random variable X It obeys the standard normal distribution ,Y Obey the degree of freedom as n Of χ2 Distribution , And X And Y Independent , Then the degree of freedom of obedience is n Of t Distribution .
（3）χ2 Distribution and F The relationship between distributions ： A random variable Y and Z The degrees of freedom are m and n Of χ2 Distributed and independent , Then the first degree of freedom of obedience is m, The second degree of freedom is n Of F Distribution .

4 What is sampling distribution ？

answer ： The concept of sampling distribution ： In general X When the distribution type of is known , If for any natural number n, Can derive statistics T＝T（X1,X2,…,Xn） The mathematical expression of the distribution of , This distribution is called an exact sampling distribution . The exact sampling distribution is mostly obtained under the normal population . Under the condition of normal population , There are mainly χ2 Distribution 、t Distribution 、F Distribution , It is often called three statistical distributions .

5 Briefly describe the significance of the central limit theorem .

answer ： The significance of the central limit theorem is embodied in ：
（1） It proposes that the sum of a large number of independent random variables has an approximate normal distribution , It not only provides a simple method to calculate the approximate probability of the sum of independent random variables , It also helps to explain why the empirical frequencies of many natural groups appear bell shaped （ Normal ） The fact that curves
（2） The conclusion of central limit theorem makes normal distribution play a very important role in mathematical statistics , It also makes the normal distribution widely used .