Using and understanding statistics and statistical procedures have become required skills in virtually every profession and academic discipline.
The purpose of this book is to help students master basic statistical concepts and techniques and to provide real-life opportunities for applying them. Although mathematically and statistically sound the author has also written books at the senior and graduate levels , the approach does not require students to examine complex concepts. Rather, the material is presented in a natural and intuitive way. Data exploration is emphasized as an integral prelude to statistical inference.
It includes hundreds of new and updated exercises with real data from journals, magazines, newspapers, and websites. Use real data. Stress conceptual understanding rather than mere knowledge of procedures. What is it? Does probability sampling always yield a representative sample?
Identify some advantages of probability sampling. What is simple random sampling? What is a simple random sample? Identify two forms of simple random sampling and explain the difference between the two.
What sampling procedure is that? List the 10 possible samples without replacement of size 3 from this population. If an SRS of size 3 is taken from the population, what are the chances of selecting 1, 3, and 5? Start at the single-digit number in line number 5 and column number 20, read down the column, up the next, and so on.
List the 6 possible samples without replacement of size 2 from this population. If an SRS of size 2 is taken from the population, what are the chances of selecting 2 and 3? Start at the single-digit number in line number 17 and column number 7, read down the column, up the next, and so on. If you have access to a random-number generator, use it to solve part a.
Applying the Concepts and Skills 1. In the year , an on-line poll was conducted over Memorial Day weekend that asked people what they were doing to observe the holiday.
The choices were: 1 stay home and relax, 2 vacation outdoors over the weekend, or 3 visit a military cemetery. Discuss this poll with regard to its suitability. Explain why a sample of 30 dentists from Seattle taken to estimate the median income of all Seattle residents is not representative. The five top Oklahoma state officials are displayed in Table 1. Use that table to solve the following problems.
List the possible samples of size 1 that can be obtained from the population of five officials. What is the difference between obtaining a simple random sample of size 1 and selecting one official at random? List the possible samples without replacement of size 5 that can be obtained from the population of five officials.
What is the difference between obtaining a simple random sample of size 5 and taking a census of the five officials? List the 10 possible samples without replacement of size 3 that can be obtained from the population of five officials. If a simple random sampling procedure is used to obtain a sample of three officials, what are the chances that it is the first sample on your list in part a?
The Recording Industry Association of America provides data on the best-selling albums of all time. As of May 28, , the top six best-selling albums of all time U. List the 15 possible samples without replacement of two artists that can be selected from the six. For brevity, use the initial provided. Describe a procedure for taking a simple random sample of two artists from the six.
If a simple random sampling procedure is used to obtain two artists, what are the chances of selecting P and A? M and E? Refer to Exercise 1. List the 15 possible samples without replacement of four artists that can be selected from the six. Describe a procedure for taking a simple random sample of four artists from the six. If a simple random sampling procedure is used to obtain four artists, what are the chances of selecting E, A, L, and B?
P, B, M, and A? List the 20 possible samples without replacement of three artists that can be selected from the six. Describe a procedure for taking a simple random sample of three artists from the six. If a simple random sampling procedure is used to obtain three artists, what are the chances of selecting M, A, and L? P, L, and E? From Wikipedia. List the 21 possible samples without replacement of two social media websites that can be selected from the seven.
If a simple random sampling procedure is used to obtain two of these social media websites, what are the chances of selecting B and A? T and G? In the game of keno, 20 balls are selected at random from 80 balls numbered 1— Use Table I in Appendix A to simulate one game of keno by obtaining 20 random numbers between 1 and Start at the twodigit number in line number 5 and column numbers 31—32, read down the column, up the next, and so on.
Suppose that you want to examine various characteristics of successful firms. Use Table I in Appendix A to obtain 10 random numbers that you can use to specify your sample. Start at the three-digit number in line number 14 and column numbers 10—12, read down the column, up the next, and so on. In an issue of Discover Vol. Casselman looked at the natural-hazards risk index of megacities to evaluate potential loss from catastrophes such as earthquakes, storms, and volcanic eruptions.
Urban areas have more to lose from natural perils, technological risks, and environmental hazards than rural areas. Suppose that you decide to take a simple random sample of five of these 10 megacities. Use Table I in Appendix A to obtain five random numbers that you can use to specify your sample. Dunn reports about the search for new undiscovered elements.
Since , scientists have been synthesizing elements one by one. The first was neptunium Np , element number There are, as of this writing, a total of 26 new synthetic elements. The following table provides their element numbers and symbols. Suppose that you decide to take a simple random sample of eight of these new elements. Use Table I in Appendix A to obtain eight random numbers that you can use to specify your sample.
As you know, probability sampling eliminates bias in choosing a sample from a list of the entire population. Nonetheless, many sources of bias often work their way into sample surveys of large human populations, such as those done 1.
Oftentimes, an accurate and complete list of the population is unavailable. In such cases, one or more groups will be omitted from the sampling process because they are not listed as part of the population. This type of bias is called undercoverage. Explain why a sample survey of households will generally suffer from undercoverage. Provide another example where bias due to undercoverage is likely to occur.
When responses are not obtained from some of the individuals in the sample because either those individuals cannot be reached or refuse to participate, we have nonresponse bias. Discuss some of the dangers of nonresponse.
Explain the consequences of such low response rates in trying to generalize the results to the entire population. When the behavior of the interviewer or respondent results in inaccurate responses, we have response bias. Provide some additional survey situations that might be conducive to response bias. Provide some additional factors that might lead to response bias. However, simple random sampling does have drawbacks.
For instance, it may fail to provide sufficient coverage when information about subpopulations is required and may be impractical when the members of the population are widely scattered geographically. In this section, we examine some commonly used sampling procedures that are often more appropriate than simple random sampling. Remember, however, that the inferential procedures discussed in this book must be modified before they can be applied to data that are obtained by sampling procedures other than simple random sampling.
Systematic Random Sampling One method that takes less effort to implement than simple random sampling is systematic random sampling. Procedure 1. Step 2 Use a random-number table or a similar device to obtain a number, k, between 1 and m.
Use systematic random sampling to obtain the sample. Solution We apply Procedure 1. Step 1 Divide the population size by the sample size and round the result down to the nearest whole number, m. The population size is the number of students in the class, which is , and the sample size is Referring to Step 1, we see that we need to randomly select a number between 1 and Using a random-number table, we obtained the number 22 but we could have conceivably gotten any number between 1 and 48, inclusive.
Hence, we need to list every 48th number, starting at 22, until we have 15 numbers. Doing so, we get the 15 numbers displayed in Table 1. Systematic random sampling is easier to execute than simple random sampling and usually provides comparable results. The exception is the presence of some kind of cyclical pattern in the listing of the members of the population e.
Cluster Sampling Another sampling method is cluster sampling, which is particularly useful when the members of the population are widely scattered geographically. Step 2 Obtain a simple random sample of the clusters. Step 3 Use all the members of the clusters obtained in Step 2 as the sample. Use cluster sampling to obtain a sample of size 60 from the population. Step 1 Divide the population into groups clusters. We know that the population size is and that the population has been divided into clusters of equal size We number the 20 members of the population in cluster 1 from 1—20, those in cluster 2 from 21—40, and so forth.
Using a random-number generator, we obtained clusters 3, 4, and Note: Of course, the 60 members sampled would be different if one or more of the three clusters that were randomly selected in Step 2 were different. Unfortunately, this method did not work very well.
Only The city council realized that the questionnaire generally had not been returned by the average homeowner. She was given two assistants to help her interview homeowners and 10 days to complete the project.
The planner first considered taking a simple random sample of homes: interviews for herself and for each of her two assistants. However, the city was so spread out that an interviewer of randomly scattered homeowners would have to drive an average of 18 minutes from one interview to the next. Doing so would require approximately 30 hours of driving time for each interviewer and could delay completion of the report.
The planner needed a different sampling design. The residential portion of the city was divided into blocks, each containing 20 homes, as shown in Fig. Explain how the planner used cluster sampling to obtain a sample of homes. The planner used the blocks as the clusters, thus dividing the population residential portion of the city into groups.
The planner numbered the blocks clusters from 1 to and then used a table of random numbers to obtain a simple random sample of 15 of the blocks. Each of the three interviewers was then assigned 5 of these 15 blocks. This method gave each interviewer homes to visit 5 blocks of 20 homes per block but saved much travel time because an interviewer could complete the interviews on an entire block before driving to another neighborhood.
The report was finished on time. Although cluster sampling can save time and money, it does have disadvantages. Ideally, each cluster should mirror the entire population. In practice, however, members of a cluster may be more homogeneous than the members of the entire population, which can cause problems. For instance, consider a simplified small town, as depicted in Fig.
The town council wants to build a town swimming pool. A town planner needs to sample homeowner opinion about using public funds to build the pool.
The clusters most strongly in favor of the pool would not have been included in the survey. In this hypothetical example, the town is so small that common sense indicates that a cluster sample may not be representative. However, in situations with hundreds of clusters, such problems may be difficult to detect. In other words, In stratified sampling, the population is first divided into subpopulations, called strata, and then sampling is done from each stratum.
Ideally, the members of each stratum should be homogeneous relative to the characteristic under consideration. In stratified sampling, the strata are often sampled in proportion to their size, which is called proportional allocation. Step 2 From each stratum, obtain a simple random sample of size proportional to the size of the stratum; that is, the sample size for a stratum equals the total sample size times the stratum size divided by the population size. Step 3 Use all the members obtained in Step 2 as the sample.
Use stratified sampling with proportional allocation to obtain a sample of size 10 from the population. Step 1 Divide the population into subpopulations strata. We know that the population size is and that the population has been divided into four strata of sizes , , , and Population size Similarly, we find that the sample sizes for strata 2, 3, and 4 are 3, 4, and 1, respectively.
Proceeding similarly, we obtained the required simple random samples shown in the final column of the following table. Note: Of course, the 10 members sampled would be different if one or more of the four samples in the last column of the preceding table were different.
The town has homeowners of which 25, , and 50 are upper income, middle income, and low income, respectively. Explain how we can obtain a sample of 20 homeowners, using stratified sampling with proportional allocation, stratifying by income group. We divide the homeowners in the town into three strata according to income group: upper income, middle income, and low income. Of the homeowners, 25 are upper income, are middle income, and 50 are lower income. Total number of homeowners Similarly, we find that the sample sizes for the middle-income and lower-income homeowners are 14 and 4, respectively.
Thus, we take a simple random sample of size 2 from the 25 upper-income homeowners, of size 14 from the middleincome homeowners, and of size 4 from the 50 lower-income homeowners. The sample consists of the 20 homeowners selected in Step 2, namely, the 2 upperincome, 14 middle-income, and 4 lower-income homeowners.
It also improves the precision of the statistical estimates because the homeowners within each income group tend to be homogeneous and makes it possible to estimate the separate opinions of each of the three strata income groups.
In both Examples 1. For instance, suppose that the population size is , the total sample size is 10, and the strata sizes are 29, 41, and Of course, sample sizes must be whole numbers. To remedy the situation here, we can simply round to get the strata sample sizes 3, 4, and 3. For instance, suppose that the population size is , the total sample size is 10, and the strata sizes are 26, 47, and Multistage Sampling Most large-scale surveys combine one or more of simple random sampling, systematic random sampling, cluster sampling, and stratified sampling.
Such multistage sampling is used frequently by pollsters and government agencies. For instance, the U. National Center for Health Statistics conducts surveys of the civilian noninstitutional U.
Data collection is by a multistage probability sample of approximately 42, households. Information obtained from the surveys is published in the National Health Interview Survey. The exin the listing ception is the presence of some kind of of the members of the population. A sample of size 5 is to be taken from the population, using systematic random sampling. Apply Procedure 1. Suppose that, in Step 2 of Procedure 1. Determine the sample.
A sample of size 9 is to be taken from the population, using systematic random sampling. A sample of size 20 is to be taken from the population, using cluster sampling. The clusters are of equal size 10, where cluster 1 consists of the members of the population numbered 1—10, cluster 2 consists of the members of the population numbered 11—20, and so forth. A sample of size 30 is to be taken from the population, using cluster sampling.
A sample of size 20 is to be taken from the population, using stratified random sampling with proportional allocation.
The strata are of sizes , , , and , where stratum 1 consists of the members of the population numbered 1—, stratum 2 consists of the members of the population numbered —, and so forth. Determine the sample sizes that will be taken from the strata.
A sample of size 10 is to be taken from the population, using stratified random sampling with proportional allocation. The strata are of sizes , , and , where stratum 1 consists of the members of the population numbered 1—, stratum 2 consists of the members of the population numbered —, and so forth.
Kocher looked at the origins of a diverse flock of cichlid fishes in the lakes of southeast Africa. Suppose that you wanted to select a sample from the hundreds of species of cichlid fishes that live in the lakes of southeast Africa. If you took a simple random sample from the species of each lake and combined all the simple random samples into one sample, which type of sampling design would you have used? Suppose that we divide the United States into the four census regions Northeast, North Central, South, and West , take a simple random sample of counties from each of the four regions, and combine all four simple random samples into one sample.
What type of sampling design have we used? Kennedy, presents an intimate biography of the extraordinary man. Published by Harper Perennial, c Jacoby and H. Handlin discussed the controversy about whether nonprobability samples are acceptable as evidence in litigation. The authors randomly selected 26 journals from a list of scholarly journals in the social and behavioral sciences.
They examined all articles published during one year in each of the 26 journals selected with regard to sampling methods.
What type of sampling design was used by these two authors in their investigation? Roberts et al. In the survey, 10 of the 46 schools participating in the immunization campaign were randomly chosen and then the parents of all the nonimmunized children at the 10 selected schools were sent a questionnaire. What type of sampling design was used by these authors in their survey?
Each of the people that used the parking facilities had a sticker with a unique number between 1 and The university committee on parking decided to sample 30 users of the parking facilities and obtain their views on those facilities.
The committee selected a number at random between 1 and and got the number The people interviewed were the ones whose stickers had numbers 10, , ,. What type of sampling design was used by the university committee on parking? In Exercise 1. Use systematic random sampling to accomplish that same task. Which method is easier: simple random sampling or systematic random sampling? Does it seem reasonable to use systematic random sampling to obtain a representative sample?
Use systematic random sampling to obtain a sample of 20 of the 80 balls. Does it seem reasonable to use systematic random sampling to simulate one game of keno? Students in the dormitories of a university in the state of New York live in clusters of four double rooms, called suites.
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Get BOOK. Introductory Statistics Books a la Carte Edition. Weiss received his Ph. Weiss has taught statistics, probability, and mathematics—from the freshman level to the advanced graduate level—for more than 30 years. In recognition of his excellence in teaching, Dr. In addition to his numerous research publications, Dr. He has also authored or coauthored books in finite mathematics, statistics, and real analysis, and is currently working on a new book on applied regression analysis and the analysis of variance.
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