AP统计学-样本均值的抽样分布-课件

1、AP StatisticsSampling Distribution of a Sample Mean样本均值的抽样分布Beijing 21st Century International School北京二十一世纪国际学校Grade 12 高三年级 Yining Liu 刘亦柠“不明于计数，而欲举大事，犹无舟楫而欲经于水险也。” - 管子AP StatisticsSampling DistribuObjectivesDistinguish Population Distribution, Sampling Distribution and Statistic DistributionKnow

2、 the significance of mean and variance of Sampling Distribution of a Sample Mean, find the relationship from the parameters of population.Explore Central Limit Theorem (in Experiment)Summarize the advantages of sampling and statistic distribution2ObjectivesDistinguish PopulatiReview: Mean and Varian

3、ce of Discrete Random Variable Suppose that X is a discrete random variable whose probability distribution is Value:x1x2x3Probability:p1p2p3To find the mean (expected value) of X, The variance of X is:Review: Mean and Variance of Problem SolvingQuality control of Nongfu Spring on the assembly line.

4、Examine the average level of bacteria in each bottle.Question 1: How should we examine it? Difficulties? Question 2: How should we draw a sample? Question 3: Does the sample represent the population very well? Problem SolvingQuality controlPrologue In1802, Pierre-SimonLaplace ,an influential French

5、scholar first use the sample to estimate the number of people in a family in France. This is the earliest record of using sample inference.Prologue In1802, Pierre-SSampling VariabilityDifferent random samples yield different statistics. This basic fact is called sampling variability: the value of a

6、statistic varies in repeated random sampling.To make sense of sampling variability, we ask, “What would happen if we took many many many samples?”PopulationSampleSampleSampleSampleSampleSampleSampleSample?Sampling VariabilityDifferent Experiment 1:Population :1, 1, 2, 2, 3, 3, 4, 4Step1: Complete th

7、e Probability model and population distribution: x=the number you get from one random chosen Population DistributionExperiment 1:Population :1, 1Step2: Draw the sample (n=2):If we randomly select 2 numbers from the population (duplication is allowed), all the possible event sets are:A= 1 , 1 , 1 , 2

8、 , , , , , , , , , , , , , , Because all individual random phenomenon has equally likely outcome that the probability is 1/4, and the number selections are independent from each trail, so the probability of the occurrence of every possible event is .All sample mean of a possible SRS:1, 31, 42, 12, 2

9、2, 32, 43, 13, 23, 33, 44, 14, 24, 34, 4 = 1/16 Step2: Draw the sample (n=2):1Step3: Complete the Probability model and Sampling distribution of sample mean (n=2) :Probability model of the sample mean (sample size n=2): Sampling Distribution of sample mean (sample size n=2):The distribution of value

10、s taken by the statistic in all possible samples of the same size from the same population.Step3: Complete the ProbabilitFind the relationshipReally?Find the relationshipReally?Step4: Draw further sample (n=3):If we randomly select 3 numbers from the population (duplication is allowed), all the poss

11、ible event set is:The probability of the occurrence of every possible event is .All sample mean of a possible SRS: Group Work . = 1/64 A=1,1,1, 1,1,2, 1,1,3, 1,1,4, 1,2,1, 1,3,1, 1,4,1, 1,2,2, 1,2,3, 1,3,2, 1,2,4, 1,4,2, 1,3,4, 1,4,3, 1,3,3, 1,4,4, 2,1,1, 2,1,2, 2,1,3, 2,1,4, 2,2,1, 2,3,1, 2,4,1, 2,

12、2,2, 2,2,3, 2,3,2, 2,2,4, 2,4,2, 2,3,4, 2,4,3, 2,3,3, 2,4,4, 3,1,1, 3,1,2, 3,1,3, 3,1,4, 3,2,1, 3,3,1, 3,4,1, 3,2,2, 3,2,3, 3,3,2, 3,2,4, 3,4,2, 3,3,4, 3,4,3, 3,3,3, 3,4,4, 4,1,1, 4,1,2, 4,1,3, 4,1,4, 4,2,1, 4,3,1, 4,4,1, 4,2,2, 4,2,3, 4,3,2, 4,2,4, 4,4,2, 4,3,4, 4,4,3, 4,3,3, 4,4,4Step4: Draw furth

13、er sample (n=Step5: Complete the Probability model and Sampling distribution of sample mean (n=3) :Probability model of the sample mean (sample size n=3):Sampling Distribution of sample mean (sample size n=3):Step5: Complete the Probabilit Confirm the relationship Confirm the relationship14The Sampl

14、ing Distribution of a Sample Mean When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean and is less spread out than the population distribution. Here are the facts.The Sampling Distribution of Sample Means1414The Sampling Distribu

15、tion ofVirtual Lab ExperimentStep1: Teachers DemoStep3: Peer discussion and conclusion. (4 mins)Challenge ZoneStep2: Students individual experiment (4 mins)Virtual Lab ExperimentStep1: T16The Central Limit Theorem (in AP Statistics)Most population distributions are not Normal. What is the shape of t

16、he sampling distribution of sample means when the population distribution isnt Normal?It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution! When the sample is large

17、 enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.16The Central Limit Theorem (iProblem Solving : Statistical EstimationThe process of statistical inference involves using

18、 information from a sample to draw conclusions about a wider population.Different random samples yield different statistics. We need to be able to describe the sampling distribution of possible statistic values in order to perform statistical inference.PopulationSampleCollect data from a representat

19、ive Sample.Make an Inference about the Population.Why is a sample mean of the simple random sample more representative to the population mean if the sample size n is large enough?The variance of sampling distribution of a sample mean decrease as sample size n increase. It is less possible of drawing

20、 an extreme sample.Problem Solving : Statistical Advantages of Sampling Save time and money.Most population distributions are not Normal.When the sample is large enough, the distribution of sample means is very close to Normal which is easy to be analyzed.In certain context, the population examination is impossible by access to every case. Sampling statistic inference makes it possible.Advantages of Sampling Save ti19ExampleBased on service records from the past year, the time (in hours) that a technician requires to complete preventative maintenance on an air conditioner fol