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# Statistics for Business 22S 008

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This 7 page Class Notes was uploaded by Cullen Conn on Friday October 23, 2015. The Class Notes belongs to 22S 008 at University of Iowa taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/228083/22s-008-university-of-iowa in Natural Sciences and Mathematics at University of Iowa.

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Date Created: 10/23/15

Chapter 14 Fundamental Concepts of Surveying Simple Random Sampling Element Universe Survey and Census Probability and Nonprobability Surveys Problems common to all surveys SKIP Chapter 15 Survey Designs Simple Random Samples Confidence Intervals for Means and Proportions Intervals Chap Hepagel Effect of Sample Size on Confidence Populations and Parameters Element Here or measurements 2 y My Chap 14 rpage 3 x z X xmeasurements lequot 2k The elements in a study are the basic units individuals or things about which information is sought page 371 The universe is the collection of elements about which we wish to be informed page 372 The set of all measurements on a variable in a universe is called a population page 406 Chap 14 7 page 2 and 1 N 2 0 ZXH X I X 1 N 2 0y KLEIN lly Any numerical characteristic of a population is called a parameter page 406 Most universes will contain many different populations Chap 14 7 page 4 Proportions When a variable is binary Y O or 1 then the population total I is just the number of ones or number of successes The population mean uy isjust the proportion of ones or successes othenNise written as uy TE Algebra shows that in this binary case 1 N 2 6y KLEIN7 A711711 This is the same as the standard deviation for a Bernoulli process Chap 14 rpage 5 Sampling Error the difference between the value of a sample estimate and the corresponding value in the population that is due only to the sampling process Precision means statements about the width of intervals within which predictions about characteristics of the universe are made The narrower the limits the more precise the predictions or estimates Chap 14 rpageT The Characteristics of Probability Surveyspage330 What do we get from a probability survey Clearly we do not get complete information about the characteristics of the universe Instead we get estimates or guesses about the characteristics of the population Because the elements in the sample are selected by randomization we can also provide quantitative statements about the precision of our estimates We can quantify the margin of sampling error Chap 14 7 page 6 Simple Random Sampling In general if a universe has N elements there are different samples of size n For example with a class of 480 students how many different samples of 6 could be selected N 480 nN7 n 6480 7 6 80l Lu 1646 x 1013 6 474 An extremely large number Therefore we illustrate with small examples Chap 14 7 page 8 Example page 382 A simple universe of size 9 Exhibit 146A Consider all possible samples of size 2 Exhibit 1468 List them and the associated values of several variables Now look at the sample means and their distributions and characteristics Exhibits 1468 and C Chap 14 rpage 9 Sampling Distribution The distribution of a sample statistic over all possible samples is called a sampling distribution Numerical characteristics of samples are called statistics Compare with population parameters All statistics are subject to sampling variation that is they will vary from sample to sample Chap 14 7 page 11 Notation for Data and Statistics Standard Devratl on Chap 147pag610 The mean II and standard deviation sy are statistics The sample mean II is an estimate of the population mean uy The sample standard deviation sy is an estimate of the population standard deviation 0 V All estimates are subject to sampling variation that is they will vary from sample to sample Chap 147page 12 Facts About the Sampling Distribution of the Mean First we have the simple result V y The standard deviation of a statistic is also called the standard error of the statistic Here we need to investigate the standard error ofthe mean Chap 14 7 page 13 How are the sampling fraction and finite population correction factor related If N 2000000 and n 1500 then f 15002000000 000075 and fpc 17 1 7000075 099962493 m 1 Unless we say otherwise we will assume for all of our work that N is much larger than n so that we take fpc 1 Chap 14 7 page 15 Standard Error of the Mean For the theoretical sampling distribution of the sample mean y it may be shown that the standard error is given by page 411 0 SE prc y Ar v where the nite population correction factor is fpc N A In Allef and f nN is called the sample fraction Chap 147page 14 Finally ifthe sample size n is moderately large and the sampling fraction fis small to moderate then the distribution of y is approximately normal This is the Central Limit Effectonce more page 411 The parameters ofthe normal distribution are of course Chap 147pag616 In practice since 03 and hence SE is unknown we must use the estimated standard error 5 5e l fpc n I For proportions o 4111711 and we use an estimated standard error of n Chap 14 7 page 17 Example New York TimesCBS News Poll reported Feb 28 1995 Are police searches without a warrant a good idea or a bad idea 69 said bad ideaquot 20 said good ideaquot What is the margin of error in that 69 What is the 95 confidence interval for the true proportion in the population 11 that think warrantess searches are a bad idea Chap 14 7 page 19 Interval Estimation Wm Proportions The sample proportion p is an estimate of the population or process proportion 11 The Confidence Interval for n is 1 7 i 2 LE P c n where 20 is chosen from a standard normal distribution to produce the desired confidence level Typically 20 2 for the usual 95 confidence This confidence interval is based on the Central Limit Effect for proportions and assumes that n is reasonably large Chap 147page 18 Here n 1190 and se M P n 06917069 069 031 1190 1190 0013 Since N is about 200000000 we set the fpc to 1 So with 95 confidence the margin of error is i20013 i0026 or about i3 percentage points Don t say 3 The 95 confidence interval for n is 069 i 0026 or 0664 to 0716 Chap 14 rpage 20 If we want 997 confidence we have 069 i 30013 069 i 0039 or 0651 to 0729 We have more confidence but the interval is wider For 99 confidence we look up the zvalue multiplier or 995 percentile and get 2 2575 So the margin of error is i25750013 0033 with 99 confidence The 99 confidence interval is 069i 0033 or 0657 to 00723 with 99 confidence Chap 14 7 page 21 Notice that both of these confidence intervals are ofthe form parameter estImate i zcseestimate where seesnmate is the standard error of the parameter estimate that is the estimate of the standard deviation of the parameter estimate Chap 14 7 page 23 Means The sample mean is an estimate ofthe population or process mean 11 Confidence interval for u Vi zci n where 26 is chosen from a standard normal distribution to produce the desired confidence level Typically 20 2 for the usual 95 confidence This confidence interval is based on the Central Limit Effect for means and assumes that n is reasonably large Chap 14 rpage 22 Confidence Levels page 412 The choice of the factor 3 in i 3sey was quite arbitrary but produced 997 confidence Other multiplers can be used to yield different confidence levels For example using a multiplier of 2 would give 95 confidence in the interval In general the intervals are of the form y zsey where z is chosen to acheive a desired confidence level Chap 14 rpage 24 The TradeOff Between High Confidence and Narrow Confidence Intervals page 412 To make precise statements about u we would like narrow confidence intervals and high confidence However with n and IV fixed the width of the confidence interval increases as the confidence increases Chap 14 7 page 25 Unfortunately p is not known at this point we have no data A conservative approach is to setp 12 as this is the worst case ie the one with the most variability Doing this and solving for n gives For example if we want a margin of error of about plus or minus 3 percentage points ie B003 then we need a sample of size n 1OO32 1111111 orabout1111 Chap 14 7 page 27 Choosing Sample Sizes page 417 When discussing parameter estimation the part zcseestimate is often called the margin of error or more correctly margin of sampling error It s the plus or minus part ofthe confidence interval In choosing a sample size for a study we might require that the margin of error be of a certain size say B Once B is specified we could attempt to choose a sample size n that will acheive the required margin of error For proportions and 95 confidence level this means we want to solve for n in the equa on B 2 M n Chap 14 rpage 26 For means 11 we have a somewhat more difficult situation Here we want 82 J77 buts is unknown We must have some idea of the variability within the population or process if we want to specify the margin of error when estimating the mean Sometimes we have some previous experience with a similar situation that will give us some guidance We may have to carry out a small pilot study to get a ballpark figurequot for the variability before we choose our sample size for the full study Chap 14 rpage 28

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