Class Note for STAT 528 at OSU 66
Class Note for STAT 528 at OSU 66
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Date Created: 02/06/15
Statistics 528 Data Analysis I Overview of Today s Lecture Lecture 7 July 13 2006 Christopher Holloman The oimu State University Summer 2on5 o IPS Sections 52 61 Sampling Distribution of a Sample Mean Statistical Con dence Christopher Hoiiom an The omu State University Summer zuuo The Sampling Distribution of the Sample Mean 0 Imagine that we have an SRSof size n from a po ulation and measure a variable X on each In ividual in the sample 0 Each X is a measurement from the opulation and therefore has the distribution 0 the population 9 ux u and ox o o The sample mean of an SRS of size n is 21X1X2Xn 7t Christopher Holloman The omu State University Summer zuuo Question What are the mean and standard deviation of x The mean of the distribution of the sample mean 1 z t z n 1 quot39 ll 9 X is an unbiased estimate of u Christopher Hoiiom an The omu State University Summer zuuo The standard deviation of the distribution of the sample mean The observations are independent so we can use the addition rule for variances 1 71 7 71 71 72 n 7010101 n 1 1 2 no 2 0392 n n Christupher HDHDman The ohiu State University Summer zuu 0 Example The height X of a single randomly chosen young woman varies according to the N64S 25 distribution Suppose we randomly sample 100 young women What is mean and variance of the distribution of 2 Christupher huiium eh The ohiu State University Summer zuu 0 We know the mean and variance but that doesn t tell us everything we need to know about the distribution of the sample mean 0 First let s examine one special case the normal distribution Christupher HDHDman The ohiu State University Summer zuu Sampling Distribution of the Sample Mean If a population isgistributed NW0 then the sample mean c of n independent observations has the Nata r distribution 9 The sampling distribution of E depends on the sample size n the distribution is more spread out larger variance the smaller the sample size Christupher huiium eh The ohiu State University Summer zuu 10 observations l observation Christopher Holloman The Ohio State University Summer 2006 o More generally any linear combination of independent normal random variables is also normally distributed 0 The sample distribution is a special case of this it s a linear combination of n independent random variables Christopher Holloman The Ohio State University Summer 2006 Question What if we have an SRS from a population that is not normally distributed Answer Central Limit Theorem CLT Draw an SRS of size n from any population with finite mean u and finite standard deviation 6 When n is large the sampling distribution of the sample mean x is approximately normal E is approximately NamNZ Christopher Holloman The Ohio State University Summer 2006 Central Limit Theorem in action X N Exp1 Sam le size of 1 figure a 2 figure b 10 figure c 25 igure d 39 Distribution of a b sample means This even works for discrete random variables Christopher Holloman The Ohio State University Summer 2006 Question How large must the sample size be for the Central Limit Theorem to apply Answer It depends on the shape of the distribution we are samplin from More observations are required i the distribution of x is far from normal Rule of Thumb CLT is usually applicable for n gt 30 o Newly manufactured automobile radiators may have 0 1 2 or more leaks in them The number of leaks in radiators made by one sup lier has mean 015 and standard deviation 04 W at type of distribution is this 0 Suppose the supplier ships 400 radiators to an assembly lant What is the distribution of the mean number of eaks in this shipment 0 Over many years many of these shipments are made What range of va ues will contain the middle 95 of the mean number of leaks Chrlstupher Hullmn an The ome State Umverslty Summer znn Statistical Inference Confidence Intervals Idea Estimate parameters of the population distribution usmg data How Use the sampling distribution of sample statistics and methods based on what would happen if we used this inference procedure many timesquot 1 Confidence Intervals 2 Hypothesis Tests Note Be surethatfyou understand the meaning of these procedures In ad men to being able to use them Idea We use a sample statistic tgestimate a population parameter eg use x to estimate p A confidence interval tells us how confident we are in our estimate A confidence interval will have the form estimate margin of error The smaller the margin or error the higher our confidence in our estimate Chrlstupher Hullmn an The ome State Umverslty Summer znn Example Assume that the sampling distribution of E is Nu 45 x lies within 9 of u in 95 of all samples so u also lines within 9 of x in those samples Density curve of Probability 095 u 9 u unl nown uV II 9 Christopher Holloman The Ohio State University Summer 2006 9 In 95 oof samples E 9 lt u lt E 9 We say that x 9 x 9 is a 95 confidence interval for u Requirements of a Confidence Interval for an Unknown Parameter 1 an interval of the form a b where a and b are numbers computed from the data 2 a confidence level that gives the probability that an interval computed this way covers the parameter Usually confidence levels are 90 or 95 Christopher Holloman The Ohio State University Summer 2006 Definition of a Confidence Interval A level C confidence interval for a parameter is an interval computed from sample data by a method that has probability C of producing an interval containing the true value of the parameter Note The following statement is INCORRECT The probability that the unknown parameter is contained within a level C confidence interval is C Why is this wrong Christopher Holloman The Ohio State University Summer 2006 Density curve of Christopher Holloman The Ohio State University Summer 2006 Confidence Intervals for the Population Mean Recall E is approximately Nw xZ by the Central Limit Theorem To construct a level C confidence interval for u assuming we know a 0 Let 2 be the point such that the area under the NO1 curve between 2 and z is C Christopher Holloman The Ohio State University Summer 2006 Christopher Holloman The Ohio State University Summer 2006 0 Notice that any normal curve has probability C between the points 2 standard deviations below the mean and 2 standard deviations above the mean Why 0 So there is probability C that E lies between 039 n 1 z andMZJ Christopher Holloman The Ohio State University Summer 2006 o This is the same as saying that 95 percent of the time in repeated sampling from the population with mean u and standard deviation 6 u will lie between 0 a x z and xz J2 J 9 This is our level Cconfidence interval for u ie our estimate of u is x and our margin of error is z0J Christopher Holloman The Ohio State University Summer 2006 The most commonly used confidence levels are Example Scores on atest of quantitative skills range from 0 to 500 A Simple random sample of 840 men aged21 to 25 took the exam Their average score o o 0 was x 272 Su ose we know that the o ulation c 90 95 99 standard deviatiorepfor this test a is equaFtoJGO 2 1645 196 2576 What can we say about the population mean score it of all 95 million men in this age group 2 for other con dence levels can be found 5 Find a 90 confidence interval for the mean test score similarly from the Normal Table Table A from the bottom row of Table D t distribution critical values or using Minitab Christupher Hullnman The Ohm Christupher Hulluman The Ohm 5mg WWW Summer was We WWW 5mg zuns b Find a 99 confidence interval for the mean test Meaning of Confidence score Note We don t know if any of the above con dence Intervals contain u or not Then what do we mean by confidence c Find an 80 confidence interval for the mean The nleaning of Corlllfidence When we test score say 95 confident we mean that if you use 95 con dence intervals often in the long run 95 of your intervals Will contain the true value of u Christupher Hullnman The Ohm Christupher Hullnm an The Ohm 5mg WWW Summer was We WWW 5mg zuns Our con dence is in the process not in any one specific interval Remember that probability chance is associated only with a random phenomenon Alter you have constructed a confidence interval from a random sample there is no randomness left in it Hence it doesn t make sense to attach any probability statement to a speci c numerical confidence interval Chrlstopher Holloman The ohle State Unlverslty Summer znn Behavior of Confidence Intervals 0 Question What happens to the margin of error when sample size increases Does it increase decrease or stay the same 0 Question How does changing the sample size affect the size of the resulting confidence interval Chrlstopher hellem eh The ohle State Unlverslty Summer znn 0 Question What happens to the size of the con dence interval as we decrease the con dence level C Hint what happens to the value of z Chrlstopher Holloman The ohle State University Summer znn Note the tradeoff We would like to have a smaller margin of error narrower interval as well as high confidence but the interval gets wider as our confidence gets higher Chrlstopher hellem eh The ohle State University Summer znn 0 Question How does the size of 6 Thus we have 3 ways of reducing the affect the margin of error width of the con dence interval 1 Increase the sample size n 2 Decrease the con dence level C 3 Decrease the standard deviation 0 Chrlstnpher Hellemee The ome Chrlstnpher Hellem an The ome State Unlverslty Summer 2eee State Unlverslty Summ 2r zuue Choosing the Sampe Size Let n7 represent the desired margin of error Recall the formula of margin of error 0 We saw that we can have a high degree of 0 confidence as well as a small margin of error by m Z J using a large sample size Solving for n we get 0 Usually researchers will have a desired confidence level and margin of error they want to attain 2 z 039 n 0 So one aspect of designing any study is to decide m the number of observations needed Always round your answer up Chrlstnpher Hellemee The ome Chrlstnpher Hellem an The ome State Unlverslty Summer 2eee State Unlverslty Summ 2r 2eee Example Suppose the GSA at the Ohio State wanis to estimate the mean month income of OSU graduate studenis within 100 with 95 con idence How many studenis should the GSA sample Assume that the standard deviation of incomes of OSU graduate students is 421 Christopher Hoiioman The Ohio State University Summer znn 1O
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