1、AP StatisticsSampling Distribution of a Sample Mean样本均值的抽样分布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北京二十一世纪国际学校Grade 12 高三年级高三年级 Yining Liu 刘亦柠刘亦柠“不明于计数，而欲举大事，犹无舟楫而欲经于水险也。不明于计数，而欲举大事，犹无舟楫而欲经于水险也。”-管子管子ObjectivesDistinguish Population Distribution,Sampling Distribution and Statis
2、tic DistributionKnow 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 distribution2Review:Mean and Variance of Discret
3、e 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:iixpxpxpxpxXE.)(332211Var(X)X2(x1X)2p1(x2X)2p2(x3X)2p3.(xiX)2piProblem SolvingQuality control of Nongfu Spring on t
4、he assembly line.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?Prologue In1802,Pierre-Simon Laplace,an influential French scholar first use the
5、 sample to estimate the number of people in a family in France.This is the earliest record of using sample inference.Sampling VariabilityDifferent random samples yield different statistics.This basic fact is called sampling variability:the value of a statistic varies in repeated random sampling.To m
6、ake sense of sampling variability,we ask,“What would happen if we took many many many samples?”PopulationSampleSampleSampleSampleSampleSampleSampleSample?Experiment 1:Population:1,1,2,2,3,3,4,4 Step1:Complete the Probability model and population distribution:x=the number you get from one random chos
7、en Population Distribution00.050.10.150.20.250.312342 Step2: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,Because all individual random phenomenon has equally likely outcome that the probability is 1/4,and t
8、he 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:1X2X3X4X5X6X7X8X9X10X11X12X13X14X15X16X1,31,42,12,22,32,43,13,23,33,44,14,24,34,4 =1/16 Step3:Complete the Probability model and Sampling distributi
9、on of sample mean(n=2):Probability model of the sample mean(sample size n=2):Sampling Distribution of sample mean(sample size n=2):2XXXX11611621621641631631600.050.10.150.20.250.311.522.533.54The distribution of values taken by the statistic in all possible samples of the same size from the same pop
10、ulation.Find the relationshipX22XnReally?Step4:Draw further sample(n=3):If we randomly select 3 numbers from the population(duplication is allowed),all the possible 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
11、,1,2,1,1,3,1,1,4,1,2,1,1,3,1,1,4,1,1,2,2,1,2,3,1,3,2,1,2,4,1,4,2,1,3,4,1,4,3,1,3,3,1,4,4,2,1,1,2,1,2,2,1,3,2,1,4,2,2,1,2,3,1,2,4,1,2,2,2,2,2,3,2,3,2,2,2,4,2,4,2,2,3,4,2,4,3,2,3,3,2,4,4,3,1,1,3,1,2,3,1,3,3,1,4,3,2,1,3,3,1,3,4,1,3,2,2,3,2,3,3,3,2,3,2,4,3,4,2,3,3,4,3,4,3,3,3,3,3,4,4,4,1,1,4,1,2,4,1,3,4
12、,1,4,4,2,1,4,3,1,4,4,1,4,2,2,4,2,3,4,3,2,4,2,4,4,4,2,4,3,4,4,4,3,4,3,3,4,4,4 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):0 1/50 1/25 3/50 2/25 1/10 3/25
13、7/50 4/25 9/50 1/5 1.0001.3331.6672.0002.3332.6673.0003.3333.6674.0002XXX164X1643643646646641064106412641264 Confirm the relationshipX22Xn14The Sampling 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
14、 mean and is less spread out than the population distribution.Here are the facts.Do these facts about the mean and standard deviation of x are true if we change the shape of the population distribution?Further Question:The Sampling Distribution of Sample Means The standard deviation of the sampling
15、distribution of x is x n The mean of the sampling distribution of x is x Suppose that x is the mean of an SRS of size n drawn from a large population with mean and standard deviation .Then:14Virtual Lab ExperimentStep1:Teachers DemoStep3:Peer discussion and conclusion.(4 mins)Challenge ZoneStep2:Stu
16、dents individual experiment(4 mins)16The Central Limit Theorem(in AP Statistics)Most population distributions are not Normal.What is the shape of the sampling distribution of sample means when the population distribution isnt Normal?It is a remarkable fact that as the sample size increases,the distr
17、ibution 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 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
18、 standard deviation.Draw an SRS of size from any population with mean and finitestandard deviation .The says that when is large,the sampling distribution of the sample mean innxcentral limit theorem(CLT)s approximatelyNormal:is approximately,xNnProblem Solving:Statistical EstimationThe process of st
19、atistical inference involves using 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.PopulationSamp
20、leCollect data from a representative 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 increas
21、e.It is less possible of drawing an extreme sample.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 examinati
22、on is impossible by access to every case.Sampling statistic inference makes it possible.19ExampleBased on service records from the past year,the time(in hours)that a technician requires to complete preventative maintenance on an air conditioner follows the distribution that is strongly right-skewed,
23、and whose most likely outcomes are close to 0.The mean time is =1 hour and the standard deviation is =1.Your company will service an SRS of 70 air conditioners.You have budgeted 1.1 hours per unit.Will this be enough?1xThe central limit theorem states that the sampling distribution of the mean time
24、spent working on the 70 units is:x n170 0.12The sampling distribution of the mean time spent working is approximately N(1,0.12)because n=70 30.z1.110.12 0.83P(x 1.1)P(Z 0.83)10.7967 0.2033If you budget 1.1 hours per unit,there is a 20%chance the technicians will not complete the work within the budgeted time.Assignment 作业作业:Page3055.10;5.11;5.12
