Fundamentals of Surveying: Elements, Universes, Sampling and Confidence Intervals, Study notes of Business Statistics

Download Fundamentals of Surveying: Elements, Universes, Sampling and Confidence Intervals and more Study notes Business Statistics in PDF only on Docsity! Chap. 14 - page 1 Chapter 14: Fundamental Concepts of Surveying • Element, Universe, Survey, and Census • Probability and Nonprobability Surveys • Problems common to all surveys (SKIP) • Simple Random Sampling Chapter 15: Survey Designs • Simple Random Samples • Confidence Intervals for Means and Proportions • Effect of Sample Size on Confidence Intervals Chap. 14 - page 2 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 - page 3 Populations and Parameters Here Element y (0 or 1) measurements x measurements I1 Y1 X1 I2 Y2 X2 I3 Y3 X3 ... ... ... IN YN XN Total τy τx Mean µy µx Standard Deviation Sy Sx τx Xi i 1= N ∑= τy Yi i 1= N ∑= µx τx N ----= µy τy N ----= Chap. 14 - page 4 and Any numerical characteristic of a population is called a parameter. (page 406) Most universes will contain many different populations. σx 1 N --- Xi µx–( ) 2 i 1= N ∑= σy 1 N --- Yi µy–( ) 2 i 1= N ∑= Chap. 14 - page 5 Proportions When a variable is binary (Y = 0 or 1), then the population total, τy, is just the number of ones or number of “successes.” The population mean, µy, is just the proportion of ones or successes otherwise written as µy = π. Algebra shows that, in this binary case, This is the same as the standard deviation for a Bernoulli process. σy 1 N --- Yi µy–( ) 2 i 1= N ∑ π 1 π–( ) = = Chap. 14 - page 6 The Characteristics of Probability Surveys (page 380) 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 - page 7 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 - page 8 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? An extremely large number! Therefore we illustrate with small examples. N n    N! n! N n–( )! -------------------------= N! n! N n–( )! ------------------------- 480! 6! 480 6–( )! ----------------------------- 480! 6!( ) 474!( ) ------------------------- 1.646 1013×≈ = = Chap. 14 - page 17 In practice, since σy and hence SEy is unknown, we must use the estimated standard error For proportions and we use an estimated standard error of sey sy n ------ fpc= σ π 1 π–( )= sep p 1 p–( ) n --------------------= Chap. 14 - page 18 Interval Estimation (page 410) Proportions The sample proportion, p, is an estimate of the population (or process) proportion π. The Confidence Interval for π is where zc is chosen from a standard normal distribution to produce the desired confidence level. Typically, zc = 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. p zc p 1 p–( ) n --------------------± Chap. 14 - page 19 Example: New York Times/CBS News Poll reported Feb. 28 1995. “Are police searches without a warrant a good idea or a bad idea? 69% said “bad idea” (20% said “good idea”) What is the margin of error in that 69%? What is the 95% confidence interval for the “true” proportion in the population (π) that think warrantless searches are a bad idea? Chap. 14 - page 20 Here n = 1190 and (Since N is about 200,000,000 we set the fpc to 1.) So with 95% confidence the margin of error is ±2(0.013) = ±0.026 or about ±3 percentage points (Don’t say 3%) The 95% confidence interval for π is 0.69 ± 0.026 or 0.664 to 0.716 sep p 1 p–( ) n -------------------- 0.69 1 0.69–( ) 1190 ---------------------------------- 0.69 0.31( ) 1190 ------------------------- 0.013 = = = = Chap. 14 - page 21 If we want 99.7% confidence, we have 0.69 ± 3(0.013) = 0.69 ± 0.039 or 0.651 to 0.729 We have more confidence but the interval is wider. For 99% confidence we look up the z-value multiplier (or 99.5 percentile) and get z = 2.575 So the margin of error is ±2.575(0.013) = 0.033 with 99% confidence. The 99% confidence interval is 0.69 ± 0.033 or 0.657 to 0.0.723 with 99% confidence. Chap. 14 - page 22 Means The sample mean, y, is an estimate of the population (or process) mean µ. Confidence interval for µ where zc is chosen from a standard normal distribution to produce the desired confidence level. Typically, zc = 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. y zc s n ------± Chap. 14 - page 23 Notice that both of these confidence intervals are of the form where seestimate is the standard error of the parameter estimate, that is, the estimate of the standard deviation of the parameter estimate. parameter estimate zcseestimate± Chap. 14 - page 24 Confidence Levels (page 412) The choice of the factor 3 in ± 3sey was quite arbitrary but produced 99.7% 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 where z is chosen to acheive a desired confidence level. y zsey± Chap. 14 - page 25 The Trade-Off Between High Confidence and Narrow Confidence Intervals (page 412) To make precise statements about µ, we would like narrow confidence intervals and high confidence. However, with n (and N) fixed, the width of the confidence interval increases as the confidence increases. Chap. 14 - page 26 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 of the 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 equation B 2 p 1 p–( ) n --------------------= Chap. 14 - page 27 Unfortunately, p is not known at this point—we have no data! A conservative approach is to set p = 1/2 as this is the “worst” case, i.e., 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, i.e., B=0.03, then we need a sample of size n = 1/(0.03)2 = 1111.111 or about 1111. n 1 B2 ------= Chap. 14 - page 28 For means µ, we have a somewhat more difficult situation. Here we want but s 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 figure” for the variability before we choose our sample size for the full study. B 2 s n ------=