(Solution Manual) Sampling Design and Analysis

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Chapter 1 Introduction 1.1Target population: Unclear, but presumed to be readers of Timemagazine. Sampling frame: Persons who know about the online survey. Sampling unit =observation unit: One response to the survey. As noted in Section 1.3, samples that consist only of volunteers are suspect. This is especially true of surveys in which respondents must register and provide personal information to participate, as here. This survey is likely not intended to provide input to the editors, but rather to provide readers a way of feeling connected and involved with the magazine. It also is possible that persons with strong opinions about the band BTS organized and contributed multiple responses to the poll. 1.2Target population: All mutual funds. Sampling frame: Mutual funds in the list. Sampling unit =observation unit: One listing. As funds are listed alphabetically by company, there is no reason to believe there will be any selection bias from the sampling frame. There may be undercoverage, however, if smaller or new funds are not listed in the newspaper. 1.3Target population: Not specified, but a target population of interest would be persons who have read the book. Sampling frame: Persons who visit the website Sampling unit =observation unit: One review. The reviews are contributed by volunteers. They cannot be taken as representative of readers’ 1Solutions Manual for Sampling Design and Analysis, 3e by Sharon Lohr (All Chapters)2 CHAPTER 1. INTRODUCTION opinions. Indeed,therehavebeeninstanceswhereauthorsofcompetingbookshavewrittennegative reviews of a book, or a book’s author manages to flood the site with positive reviews. 1.4Target population: Persons eligible for jury duty in Pennsylvania. Sampling frame: State residents who are registered voters or licensed drivers over 18. Sampling unit =observation unit: One resident. Selection bias occurs largely because of undercoverage and nonresponse. Eligible jurors may not appear in the sampling frame because they are not registered to vote and they do not possess an Arizona driver’s license. Addresses on either list may not be up to date. In addition, jurors fail to appear or are excused; this is nonresponse. 1.5Target population: All persons experiencing homelessness in study area. Sampling frame: Clinics participating in the Health Care for the Homeless project. Sampling unit: Unclear. Depending on assumptions made about the survey design, one could say either a clinic or a person is the sampling unit. Observation unit: Person. Selection bias may be a serious problem for this survey. Even though the demographics for HCH patients are claimed to match those of the homeless population (but do we knowthey match?) and the clinics are readily accessible, the patients differ in two critical ways from non-patients: (1) they needed medical treatment, and (2) they went to a clinic to get medical treatment. One does not know the likely direction of selection bias, but there is no reason to believe that the same percentages of patients and non-patients are mentally ill. 1.6Target population: Likely voters in Iowa. Sampling frame: Persons who attend the State Fair and stop by the booth. Sampling unit =observation unit: One person. This is a convenience sample of volunteers, and we cannot trust any statistics from it. 1.7Target population: All cows in region. Sampling frame: List of all farms in region. Sampling unit: One farm. Observation unit: One cow. There is no reason to anticipate selection bias in this survey. The design is a single-stage cluster sample, discussed in Chapter 5. 1.8Target population: Licensed boarding homes for the elderly in Washington state. Sampling frame: List of 184 licensed homes. Sampling unit =observation unit: One home.3 Nonresponse is the obvious problem here, with only 43 of 184 administrators or food service man- agers responding. It may be that the respondents are the larger homes, or that their menus have better nutrition. The problem with nonresponse, though, is that we can only conjecture the direc- tion of the nonresponse bias. 1.9Target population: Wikipedia science articles. Sampling frame: Articles thought to be of interest by the sampler. Sampling unit: One article. This is a judgment sample, where the articles are deliberately chosen for review. The statistics apply only to the sample, and cannot be generalized to the population. There is additional possible selection bias because of the nonresponse; perhaps articles with many or few errors are harder to get reviewers for. 1.10Target population: The 32,000 subscribers to the magazine. Sampling frame: Magazine subscribers who saw the survey invitation. Sampling unit: One response. Since more than 32,000 responses were received, it appears that either non-subscribers responded to the survey or some subscribers responded multiple times. This is a convenience sample. The statistics have no validity because of the self-selected nature of the sample. A person who responded multiple times could unduly influence the survey results. In many surveys of this type, only persons with strong opinions respond. It’s possible in this case that persons who have been inconvenienced by a PC malfunction will be more likely to respond than persons who have experienced no problems with their PCs. 1.11Target population: Unclear. Presumably they want to generalize the results to adult women. Sampling frame: Persons who see and respond to the online invitation. Sampling unit: One response. It is possible that some persons responded multiple times. This is a convenience sample. The statistics have no validity because of the self-selected nature of the sample. We don’t even know if 4,000 different women responded, or even if the respondents are women; it could be just a handful of persons each responding multiple times. 1.12Target population: Faculty members at higher education institutions. Sampling frame: Faculty members in the purposively sampled institutions. The authors did not state how they selected the faculty in those institutions who were surveyed. Sampling unit: One faculty member. There are several sources of selection bias. The first is the purposive nature of the college sample. Although the authors stated that the institutions are “representative in terms of institutional type, geographic and demographic diversity, religious/nonreligious affiliation, and the public/private di- mension,” it is very easy for conscious or unconscious subjective biases to skew such a sample. A second source of potential bias comes from the selection of faculty to participate (if not every-4 CHAPTER 1. INTRODUCTION one or a random sample). Finally, only about 60% of the selected faculty members respond, so nonresponse is another potential source of bias. 1.13Target population: All ASA members. Sampling frame: ASA members with active e-mail addresses. Sampling unit: One member. ASA members could respond only if e-mailed the survey, and this ensured that each member responded at most once. But the low response rate raises concerns that the respondents may be persons who are strongly engaged in the topic. 1.14Target population: All married couples in the U.S. Sampled population: Married couples who were in therapy or counseling Sampling unit: One counselor Observation unit: One couple Couples in therapy may be experiencing more marital problems than other couples. 1.15Target population: All adults Sampling frame: Friends and relatives of American Cancer Society volunteers Sampling unit: One person This is a convenience sample, and the results cannot be applied to the general population. Here’s what I wrote about the survey elsewhere: “Although the sample contained Americans of diverse ages and backgrounds, and the sample may have provided valuable information for exploring factors associated with development of cancer, its validity for investigating the relationship between amount of sleep and mortality is questionable. The questions about amount of sleep and insomnia were not the focus of the original study, and the survey was not designed to obtain accurate responses to those questions. The design did not allow researchers to assess whether the sample was representative of the target population of all Americans. Because of the shortcomings in the survey design, it is impossible to know whether the conclusions in Kripke et al. (2002) about sleep and mortality are valid or not.” (Lohr, 2008, pp. 97–98) 1.16Target population: Readers of romance fiction Sampled population: Women who self-identified as readers of romance fiction Sampling unit: One woman Observation unit: One woman This was almost certainly a convenience sample. 1.17Target population: All statisticians5 Sampled population: Statisticians whose names are in the online directory of a statistical organi- zation (and whose e-mail address is correct) Sampling unit: One person Observation unit: One person There are several potential sources of bias. First, not all statisticians belong to one of the organi- zations and often the e-mail addresses in those listings are out of date. Then, there is nonresponse bias. There is also a potential problem that a statistician belonging to more than one organization may be sampled multiple times. 1.18Target population: All adults U.S. Sampled population: Adults with telephone access Sampling unit: Telephone number Observation unit: Adult There may be some undercoverage here, since not everyone has a telephone, but that is quite low. The main source of potential bias here is nonresponse. Fewer than 30% of the eligible numbers resulted in an interview. One can come up with potential reasons for the bias being in either direction. 1.19Target population: Members of Consumers Union Sampled population: Members with an e-mail address who would be willing to respond to a survey Sampling unit: One member (could be household) Observation unit: One member The big problem here is nonresponse. It is reasonable to assume that persons with strong opinions would be more likely to respond to the survey. 1.20Target population: All students in the state Sampled population: School districts in the state with internet access Sampling unit: One school Observation unit: One student This is a purposive sample of school districts, and a convenience sample of students within schools. 1.21Target population: All brains of deceased football players Sampled population: Brains that had been donated to the brain bank Sampling unit: One brain Observation unit: One brain This is not a representative sample because family members were probably more likely to donate their loved ones’ brains for scientific research if they had shown symptoms of post-concussive injury. You cannot conclude from this study that 99% of NFL players will get CTE, even though many6 CHAPTER 1. INTRODUCTION newspaper articles made that implication. The authors of the study, in Mez et al. (2017), were careful not to draw any implications about the population. One of their purposes was to correlate CTE with the amount of brain injury during life, and they found that men who had played football longer had, on average, worse brain damage. Again, thisfindingcannotnecessarilybegeneralizedtothepopulationofallfootballplayersbecause it’s possible that the relationship between time spent playing football and brain damage is stronger (or weaker) in this non-representative sample than in the population. But the results from the study are illuminating, and suggest that repeated brain trauma should be avoided. 1.22Target population: Not clear. All people? Readers? Sampled population: Persons willing to e-mail the columnist Sampling unit: One reader Observation unit: One reader This is a pure convenience sample, and worthless for generalizing to any population. And what was to stop a reader from sending multiple responses? 1.23Target population: All patients of the clinic Sampled population: All patients of the clinic Sampling unit: One patient Observation unit: One patient This likely gives a representative sample because patients are sampled as they enter. If patients arrive in random order, each has the same probability of being sampled. There might be some problems, if, say, patients arrive in busses that hold exactly 20 patients. Then it’s possible that the first person off every bus would be chosen, who might differ from others. 1.24Target population: All patients of the clinic Sampled population: All patients of the clinic Sampling unit: One patient Observation unit: One patient This procedure might give a biased sample because the next patient to leave is more likely to have had a longer appointment. Several patients with short appointments may leave while the interviewer is conducting the interview. Geldsetzer et al. (2018) discuss different options for taking a sample of patients for exit interviews. 1.25Target population: All watchers of the show Sampled population: Persons who want to express an opinion about the contestants. As long as they have the number to text or the website address, they do not need to have watched Sampling unit: One text or online submission7 Observation unit: One text or online submission This is a sample of convenience. According to dwts.vote.abc.go.com/#faq , accessed in January 2020, participants were allowed to submit up to 10 votes per couple online, and an additional 10 votes by text. The online votes had to be submitted through an ABC television account. But people with multiple cell phones could still submit multiple responses, and an organization could have members flood the line with responses. Before 2019, the couple with the lowest combined score, found by averaging the audience vote score with the average judges’ score, was eliminated from the competition. Because of complaints that determined fans were keeping weak dancers in the competition, the procedure changed in 2019. The judges chose one of the two lowest-scoring couples to eliminate. 1.26Target population: All watchers of the news broadcast Sampled population: Persons who go to the website to answer Sampling unit: One online submission Observation unit: One online submission Of course this survey is purely entertainment, and the television station is careful to say that the poll is unscientific—one cannot conclude from this poll that 10% of all Arizonans think the state should participate in daylight savings time. This is a convenience sample. 1.27Target population: All county election websites in Texas Sampled population: All county election websites in Texas Sampling unit: One county website Observation unit: One county website This is a census of all websites. Assuming there were no measurement errors, these statistics are accurate, with no sampling or nonsampling error. If the goal had been solely to estimate the percentage of county election websites that began with “http” rather than “https,” it might have been easier to take a sample. But with a goal of obtaining an estimate and also identifying counties that do not use https, a census is needed. 1.28Target population: All adults in the world who do not have children Sampled population: People who hear about the survey Sampling unit: One person This is a pure convenience sample, and worthless for measuring opinion about nonparenting adults. Those who visit the author’s website are likely to be persons who are already in agreement with the author’s views, and so will reinforce the author’s opinion. 1.29Target population: Every U.S. physician Sampled population: U.S. physicians with e-mail addresses in the file8 CHAPTER 1. INTRODUCTION Sampling unit: One physician Observation unit: One physician The low response rate for this survey is the biggest source of bias. 1.30Target population: Residents of Little Rock Sampled population: Residents of Little Rock who learn about the survey Sampling unit: One resident Observation unit: One resident This is a convenience sample, and it is impossible to say whether this group is representative of the vast majority of Little Rock residents who do not respond. For most surveys of this type, people with strong opinions are the ones who will be willing to take the time to fill out the survey. 1.32Target population: Residents of Guelph Sampled population: Residents of Guelph who visited one of the in-person sites or the web site Sampling unit: One resident (possibly with multiple visits?) Both of these are convenience samples. The in-person survey will miss those who do not visit a site, and the online survey will miss those who do not learn about the survey or who do not choose to participate. 1.32 (a) How many potatoes did you eat last year? How many people keep track of the number of potatoes they eat (unless they never eat potatoes)? Also, what counts as eating a potato? Do French fries count? How many potatoes are in a serving of scalloped or mashed potatoes? (b) According to the American Film Institute, Citizen Kane is the greatest American movie ever made. Have you seen this movie? This is a leading question; after being told that it is the greatest American movie ever made, respondents will be loath to say they have not seen it. (c) Do you favor or oppose changing environmental regulations so that while they still protect the public, they cost American businesses less and lower product costs? This leading question is from Kinsley (1981). (d) Agree or disagree: We should not oppose policies that reduce carbon emissions. Confusing because of double negative. (e) How many of your close friends use marijuana products? This is vaguely worded. What counts as a close friend? And what counts as a marijuana product?9 (f) Do you want a candidate for president who would make history based on race, gender, sexual orientation, or religion? This question appeared in The New York Times on January 30, 2020, see https://www. nytimes.com/interactive/2020/01/30/us/politics/democratic-presidential-candidates- quiz.html . Thequestionisnotjustdouble-barreled, itisquadruple-barreled. Anditisvague: What does it mean to make history? (g) Who was or is the worst president in the history of the United States? People don’t remember about James Buchanan, and tend to vote for the current or a recent president. (h) Agree or disagree: Health care is not a right, it’s a privilege. Asks two questions, not one. Answers will differ if the options are put in reverse order. And the agree/disagree format encourages people to agree. (i) The British Geological Survey has estimated that the UK has 1,300 trillion cubic feet of natural gas from shale. If just 10% of this could be recovered, it would be enough to meet the UK’s demand for natural gas for nearly 50 years or to heat the UK’s homes for over 100 years. From what you know, do you think the UK should produce natural gas from shale? Leading question. (j) What hours do you usually work each day? Presupposes that the person works. (k) Do you support or oppose gun control? A question such as “Do you support or oppose gun control?” will tell you little about public opinion because the term “gun control” can mean different things to different people. Some may think the term means requiring background checks before purchase, while others may thinkguncontrolprohibitsallgunownershipbyprivatecitizens. Astatisticfromsuchavague question cannot be interpreted without assuming how respondents interpret the question. (l) Looking back, do you regret having moved so rapidly to be divorced, and do you now feel that had you waited, the marriage might have been salvaged? Leading question. (m) Are the Patriots and the Packers good football teams? Asks two questions, not one. Also, what does it mean to be a good team? (n) Antarctic ice is now melting three times faster than it was before 2012. And the Arctic is warming at twice the rate of the rest of the world. On a scale from 1 to 5, with 5 being the highest, how alarmed are you by melting ice, the rise of global sea levels, and increased flooding of coastal cities and communities? This was a question from an Environmental Defense Fund survey that was mailed to the author. Who will not be alarmed after reading the introduction to the question? Not sur- prisingly, the last question on the survey asks for a donation of money.10 CHAPTER 1. INTRODUCTION (o) Overall, how strong was your emotional reaction to the performance of La Bohème ? (No emotional response, Weak response, Moderate response, Strong response, or Not applicable) What is an emotional reaction? Also, if this question is intended to measure satisfaction, someone could have a “strong emotional response” because the soprano was consistently off- key or the orchestra was terrible. (p) How often do you wish you hadn’t gotten into a relationship with your current partner? (All the time, Most of the time, More often than not, Occasionally, Rarely, Never) Presumes there is a current partner and is vague. This question is from Stasko (2015, p. 90). (q) Do you believe that for every dollar of tax increase there should be $2.00 in spending cuts with the savings earmarked for deficit and debt reduction? Double-barreled and leading (r) There probably won’t be a woman president of the U.S. for a long time and that’s probably just as well. This question is cited by Frankovic (2008); it is double-barreled. (s) How satisfied are you that your health insurance plan covers what you expect it to cover? (Very satisfied, Satisfied, Dissatisfied, Very dissatisfied, or Not applicable) Should ask what you expect the plan to cover as a separate question. This question was from a State of Arizona retirement survey. (t) Given the world situation, does the government protect too many documents by classifying them as secret and top secret? Is the question about the dire world situation, or about the government classifying too many documents? 1.34Students will have many different opinions on this issue. Of historical interest is this excerpt of a letter written by James Madison to Thomas Jefferson on February 14, 1790: A Bill for taking a census has passed the House of Representatives, and is with the Senate. It contained a schedule for ascertaining the component classes of the Society, a kind of information extremely requisite to the Legislator, and much wanted for the science of Political Economy. A repetition of it every ten years would hereafter afford a most curious and instructive assemblage of facts. It was thrown out by the Senate as a waste of trouble and supplying materials for idle people to make a book. Judge by this little experiment of the reception likely to be given to so great an idea as that explained in your letter of September. 1.35(a) There are several problems with the Landers survey. First, the sample is self-selected— only those who care enough to write a letter (and in 1975, this meant typing or handwriting it, and then putting it in an envelope with a stamp) would respond. Another problem is the leading question, which follows the story of the couple who regrets having children.11 Note that Landers continued to cite results from the survey long after it had been discredited. In a 1996 book, she recalled the survey as follows (note that the quotes from the letter are different here): On November 3, 1975, a young married couple wrote to say they were undecided as to whether to have a family. They asked me to solicit opinions from parents of young children as well as from older couples whose families were grown. “Was it worth it?” they wanted to know. “Were the rewards enough to make up for the grief?” The question, as I put it to my readers, was this: “If you had it to do over again, would you have children?” Well, dear friends, the responses were staggering. Much to my surprise, 70 percent of those who responded said no. (Landers, 1996, p. 107) Landers ran many surveys, as they stirred up reader interest and participation. For example, Landers (1996) described a survey on “divorce regret” which she took in 1993: In 1993, I asked my readers this question: “Looking back, do you regret having moved so rapidly to be divorced, and do you now feel that had you waited, the marriage might have been salvaged?” I asked for a yes or no answer on a postcard, but thousands of readers felt compelled to write long letters. I’m glad they did. I learned a lot. To my surprise, out of nearly 30,000 responses, almost 23,000 came from women. Nearly three times as many readers said they were glad they divorced, and most of them said they wished they had done it sooner. (pp. 14-15) For another example, on Valentine’s Day 1977, Landers asked readers to send a postcard answering the question: “If you had to do it over again, would you marry the person to whom you are now married?” She received more than 50,000 pieces of mail. 30% signed their cards; of those, 70% said they would marry the same person again. 80% of the signed cards came from women. The other 70% did not sign their cards; of those, 48% said yes and 70% were female. The problem with a self-selected survey such as the one taken by Landers (1976) is that it continues to be cited as reliable information when in fact the survey results represent only the persons who happened to respond to this survey. For example, Dvorak (2011) cited the survey in an article in theWashington Post . And many news organizations cite such surveys precisely because they have eye-catching results. Once an unreliable statistic is out there, it’s difficult to retract it. (b) Again, the GH survey is self-selected, but it may have elicited a different response from readers who wanted to overturn the previous survey. Also, the readership of GH may differ from that of newspapers.12 CHAPTER 1. INTRODUCTION (c) Main questions: Was the sample selected randomly? And what was the response rate? Note: I obtained the Newsday survey from Bryan Caplan’s blog at https://www.econlib.org/ archives/2009/02/parents_and_buy.html on February 2, 2020. Another poll, in 2013, asked adults aged 45 and over: “If you had it to do over again, how many children would you have, or would you have no children at all?” and reported that 11% would have no children (Newport and Wilke, 2013). This was based on a “random sample of 5100 adults” conducted by telephone, with both landline and cellular telephones included.Chapter 2 Simple Probability Samples 2.1(a)¯yU=98 + 102 + 154 + 133 + 190 + 175 6= 142 (b) For each plan, we first find the sampling distribution of ¯y. Plan 1: Sample number P(S) ¯yS 1 1/8 147.33 2 1/8 142.33 3 1/8 140.33 4 1/8 135.33 5 1/8 148.67 6 1/8 143.67 7 1/8 141.67 8 1/8 136.67 (i)E[¯y] =1 8(147.33) +1 8(142.33) +···+1 8(136.67) = 142 . (ii)V[¯y] =1 8(147.33−142)2+1 8(142.33−142)2+···+1 8(136.67−142)2= 18.94. (iii) Bias [¯y] =E[¯y]−¯yU= 142−142 = 0. (iv) Since Bias [¯y] = 0, MSE [¯y] =V[¯y] = 18.94 Plan 2: 1314 CHAPTER 2. SIMPLE PROBABILITY SAMPLES Sample number P(S) ¯yS 1 1/4 135.33 2 1/2 143.67 3 1/4 147.33 (i)E[¯y] =1 4(135.33) +1 2(143.67) +1 4(147.33) = 142.5. (ii) V[¯y] =1 4(135.33−142.5)2+1 2(143.67−142.5)2+1 4(147.33−142.5)2 = 12.84 + 0.68 + 5.84 = 19.36. (iii) Bias [¯y] =E[¯y]−¯yU= 142.5−142 = 0.5. (iv) MSE [¯y] =V[¯y] + (Bias[¯y])2= 19.61. (c) Clearly, Plan 1 is better. It has smaller variance and is unbiased as well. 2.2(a) Unit 1 appears in samples 1 and 3, so π1=P(S1) +P(S3) =1 8+1 8=1 4. Similarly, π2=1 4+3 8=5 8 π3=1 8+1 4=3 8 π4=1 8+3 8+1 8=5 8 π5=1 8+1 8=1 4 π6=1 8+1 8+3 8=5 8 π7=1 4+1 8=3 8 π8=1 4+1 8+3 8+1 8=7 8. Note that/summationtext8 i=1πi= 4 =n. (b)15 Sample,SP(S)ˆt {1,3,5,6}1/8 38 {2,3,7,8}1/4 42 {1,4,6,8}1/8 40 {2,4,6,8}3/8 42 {4,5,7,8}1/8 52 Thus the sampling distribution of ˆtis: kP(ˆt=k) 381/8 401/8 425/8 521/8 2.3No, because thick books have a higher inclusion probability than thin books. 2.4(a) A total of (8 3) = 56samples are possible, each with probability of selection1 56. The following is the sampling distribution of ¯y. k P (¯y=k) 21 32/56 3 1/56 31 34/56 32 31/56 4 6/56 41 38/56 42 32/56 5 6/56 51 37/56 52 33/56 6 6/56 61 36/56 7 1/56 71 33/56 Using the sampling distribution, E[¯y] =2 56/parenleftbigg 21 3/parenrightbigg +···+3 56/parenleftbigg 71 3/parenrightbigg = 5. The variance of ¯yfor an SRS without replacement of size 3 is V[¯y] =2 56/parenleftbigg 21 3−5/parenrightbigg2 +···+3 56/parenleftbigg 71 3−5/parenrightbigg2 = 1.429.16 CHAPTER 2. SIMPLE PROBABILITY SAMPLES Of course, this variance could have been more easily calculated using the formula in (2.12): V[¯y] =/parenleftbigg 1−n N/parenrightbiggS2 n=/parenleftbigg 1−3 8/parenrightbigg6.8571429 3= 1.429. (b) A total of 83= 512samples are possible when sampling with replacement. Fortunately, we need not list all of these to find the sampling distribution of ¯y. LetXibe the value of the ith unit drawn. Since sampling is done with replacement, X1,X2, andX3are independent; Xi(i= 1,2,3) has distribution k P (Xi=k) 1 1/8 2 1/8 4 2/8 7 3/8 8 1/8 Using the independence, then, we have the following probability distribution for ¯X, which serves as the sampling distribution of ¯y. k P (¯y=k) k P (¯y=k) 1 1/512 42 312/512 11 33/512 5 63/512 12 33/512 51 357/512 2 7/512 52 321/512 21 312/512 6 57/512 22 36/512 61 336/512 3 21/512 62 36/512 31 333/512 7 27/512 32 315/512 71 327/512 4 47/512 72 39/512 41 348/512 8 1/512 The with-replacement variance of ¯yis Vwr[¯y] =1 512(1−5)2+···+1 512(8−5)2= 2. Or, using the formula with population variance (see Exercise 2.30), Vwr[¯y] =1 nN/summationdisplay i=1(yi−¯yU)2 N=6 3= 2. 2.5(a) The sampling weight is 100/30 = 3.3333.17 (b)ˆt=/summationtext i∈Swiyi= 823.33. (c)ˆV(ˆt) =N2/parenleftbigg 1−n N/parenrightbiggs2 y n= 1002/parenleftbigg 1−30 100/parenrightbigg15.9781609 30= 3728.238, so SE(ˆt) =√ 3728.238 = 61.0593 and a 95% CI for tis 823.33±(2.045230)(61.0593) = 823 .33±124.8803 = [698 .45,948.21]. The fpc is (1−30/100) =.7, so it reduces the width of the CI. 2.6(a) Publications 0 2 4 6 8 100510152025 The data are quite skewed because 28 faculty have no publications. (b)¯y= 1.78;s= 2.682; SE[¯y] =2.682√ 50/radicalbigg 1−50 807= 0.367. (c) No; a sample of size 50 is probably not large enough for ¯yto be normally distributed, because of the skewness of the original data. The sample skewness of the data is (from SAS software) 1.593. This can be calculated by hand, finding 1 n/summationdisplay i∈S(yi−¯y)3= 28.924704018 CHAPTER 2. SIMPLE PROBABILITY SAMPLES so that the skewness is 28.9247040/(2.6823) = 1.499314. Note that this estimate differs from the estimate given by the UNIVARIATE procedure since the UNIVARIATE procedure adjusts for df using the formula skewness =n (n−1)(n−2)/summationdisplay i∈S(yi−¯y)3/s3. Whichever estimate is used, however, formula (2.29) says we need a minimum of 28 + 25(1.5)2= 84 observations to use the central limit theorem. (d)ˆp= 28/50 = 0.56. SE(ˆp) =/radicalBigg (0.56)(0.44) 49/parenleftbigg 1−50 807/parenrightbigg = 0.0687. A 95% confidence interval is 0.56±1.96(0.0687) = [0.425,0.695]. 2.7(a) A 95% confidence interval for the proportion of entries from the South is 175 1000±1.96/radicaltp/radicalvertex/radicalvertex/radicalbt175 1000/parenleftBig 1−175 1000/parenrightBig 1000= [0.151,0.199]. (b) As 0.309 is not in the confidence interval, there is evidence that the percentages differ. 2.8(a) A stratified sample would be a better choice because the sampling frame has a lot of information about the population that could improve the efficiency of the design. (b) A cluster sample would be a better choice. There is no list of patients, and they can only be accessed through their providers. First sample the allergists, then take a subsample of the patients of each sampled doctor. (c) This one depends on how the website is organized. If there is an index where topics are organized A-Z, an SRS might be appropriate. Some websites are organized by links, however, or with subpages. In that case, a cluster sample may be better. Or this may be a network sample, as discussed in Chapter 14. (d) For this sample, an SRS is likely the design of choice. The results of the audit need to be accepted by people on either side of the issue, and the only type of sample that many people think is fair is one that is selected completely randomly.19 2.9Ifn0≤N, then zα/2/radicalbigg 1−n NS√n=zα/2/radicaltp/radicalvertex/radicalvertex/radicalbt1−n0 N(1 +n0 N)S√n0/radicalbigg 1 +n0 N =zα/2/radicalbigg 1 +n0 N−n0 NS√n0 =zα/2S zα/2S e=e 2.10Design 3 gives the most precision because its sample size is largest, even though it is a small fraction of the population. Here are the variances of ¯yfor the three samples: Sample Number V(¯y) 1 (1−400/4000)S2/400 = 0.00225S2 2 (1−30/300)S2/30 = 0.03S2 3 (1−3000/300,000,000)S2/3000 = 0.00033333S2 2.11(a) The margin of error equals the tor normal critical value (approximately 2) times the standard error, SE (ˆp) =/radicalbig p(1−p)/n. For a fixed sample size n,p(1−p)is largest when p= 1/2. This can be shown through calculus since dp(1−p) dp= 1−2p and1−2p= 0whenp= 1/2. Alternatively, a graph of p(1−p)vs.pwill show the maximum at p= 1/2. (b) Table SM2.1 shows the margins of error for p= 0.5in the first column. The second and third columns give the margin of error when p= 0.2andp= 0.1, respectively. These values were calculated without the fpc. (c) Values are given in Table SM2.2. The fpc makes almost no difference for most of the sample sizes except when the sample size is more than 10% of the population. 2.12Assume the maximum value for the variance, with p= 0.5. Then use n0= 1.962(0.5)2/(.04)2, n=n0/(1 +n0/N). Table SM2.3 gives the results. The finite population correction makes a small difference for Tempe and Casa Grande, and a larger difference for Gila Bend and Jerome. 2.13The sampling weight is 20,000/200 = 1,000. Answers will vary for the remaining parts.20 CHAPTER 2. SIMPLE PROBABILITY SAMPLES Table 2.1: Margin of error (MOE) for an estimated proportion from an SRS, no fpc. MOE if Estimated Proportion is Sample Size 0.5 0.2 or 0.8 0.1 or 0.9 500.142098 0.113679 0.085259 1000.099211 0.079369 0.059527 5000.043933 0.035146 0.02636 10000.031027 0.024822 0.018616 50000.013862 0.01109 0.008317 100000.009801 0.007841 0.005881 500000.004383 0.003506 0.002630 100000 0.003099 0.002479 0.001859 1000000 0.00098 0.000784 0.000588 Table 2.2: Margin of error (MOE) for an estimated proportion from an SRS from a population withN=1 million. MOE if Estimated Proportion is Sample Size 0.5 0.2 or 0.8 0.1 or 0.9 500.142095 0.113676 0.085257 1000.099206 0.079365 0.059524 5000.043922 0.035137 0.026353 10000.031012 0.024809 0.018607 50000.013828 0.011062 0.008297 100000.009752 0.007802 0.005851 500000.004272 0.003417 0.002563 100000 0.00294 0.002352 0.001764 1000000 0 0 0 Table 2.3: Results for Exercise 2.12. City Population n0n Casa Grande 48,571 600.25 593 Gila Bend 1,922 600.25 458 Jerome 444 600.25 256 Phoenix 1,445,632 600.25 600 Tempe 161,719 600.25 598 2.14(a) The estimated proportion is 23/30 =0.767, or 76.7%. A 95% confidence interval for the21 proportion of database members with a third cousin or closer relative in the database is 23 30±2.045/radicalBigg 1 3023 30/parenleftbigg 1−23 30/parenrightbigg = [0.606,0.927] The confidence interval for the percentage is from 61% to 93%. Note that the fpc for this sample is negligible. (b) The confidence interval in (a) applies onlyto the persons in the database, because that is where the sample was selected. They may well have a different matching percentage than persons in the public at large. Perhaps relatives tend to sign up for genetic testing together, which would increase the chances that a sampled person would have a close relative in the database. 2.15(a) 10 12 14 16 18 20 age (months) The histogram appears skewed with tail on the right. With a mildly skewed distribution, though, a sample of size 240 is large enough that the sample mean should be normally distributed. (b)¯y= 12.07917;s2= 3.705003; SE[¯y] =/radicalbig s2/n= 0.12425. (Since we do not know the population size, we ignore the fpc, at the risk of a slightly-too-large standard error.) A 95% confidence interval is 12.08±1.96(0.12425) = [11 .84,12.32]. (c)n=(1.96)2(3.705) (0.5)2= 57.22 CHAPTER 2. SIMPLE PROBABILITY SAMPLES 2.16(a) Using (2.30) and choosing the maximum possible value of (0.5)2forS2, n0=(1.96)2S2 e2=(1.96)2(0.5)2 (0.1)2= 96.04. Then n=n0 1 +n0/N=96.04 1 + 96.04/580= 82.4. (b) Since sampling is with replacement, no fpc is used. An approximate 95% CI for the proportion of children overdue for vaccination is 93 120±1.96/radicaltp/radicalvertex/radicalvertex/radicalbt27 120/parenleftBig 1−93 120/parenrightBig 120= [0.70,0.85]. 2.17(a) We have ˆp= 0.2and ˆV(ˆp) =/parenleftbigg 1−745 2700/parenrightbigg(.2)(.8) 744= 0.0001557149 , so an approximate 95% CI is 0.2±1.96√ 0.0001557149 = [ .176,.224]. (b) The above analysis is valid only if the respondents are a random sample of the selected sample. If respondents differ from the nonrespondents—for example, if the nonrespondents are more likely to have been bullied—then the entire CI may be biased. 2.18(a) There is no selection bias from taking the sample, since it is an SRS from the directory. There may be undercoverage, though, if the online directory does not include everyone. (b) 30.7% female with 95% CI [23.9, 37.5] (c) 265 female members with 95% CI [206.3, 323.6] 2.19(a)¯y=301,953.7,s2=118,907,450,529. CI: 301953.7±1.96/radicalBigg s2 300/parenleftbigg 1−300 3078/parenrightbigg ,or[264883,339025] (b)¯y= 599.06,s2= 161795.4, CI: [556,642]23 (c)¯y= 56.593,s2= 5292.73, CI: [48.8,64.4] (d)¯y= 46.823,s2= 4398.199, CI: [39.7,54.0] 2.20(a) The data appear skewed with tail on right. (b)¯y= 20.15,s2= 321.357, SE[¯y] = 1.63 2.21(a) The data appear skewed with tail on left. 24 CHAPTER 2. SIMPLE PROBABILITY SAMPLES (b)¯y= 5309.8,s2= 3,274,784, SE[¯y] = 164.5 2.22 ˆp= 85/120 = 0.708 95%CI:85/120±1.96/radicalBigg 85/120 (1−85/120) 119/parenleftbigg 1−120 14938/parenrightbigg =.708±.081,or[0.627,0.790]. 2.23(a) The sampling weight is 7,048,107/5,000 = 1409.621. (b) 27.66%, with 95% CI [26.42, 28.90] (c) 415838 with 95% CI [369807.26, 461869.36] (d) 13.1% with 95% CI [12.16, 14.034] The CV is 0.036415. 2.24Sixty of the 70 samples yield confidence intervals, using this procedure, that include the true valuet= 40. The exact confidence level is 60/70 = 0.857, which is lower than 95%. 2.25(a) From (2.16), CV(¯y) =/radicalbig V(¯y) E(¯y)=/radicalbigg 1−n NS√n¯yU. Substituting ˆpfor¯y, andN N−1p(1−p)forS2, we have CV(ˆp) ==/radicalBigg/parenleftbigg 1−n N/parenrightbiggNp(1−p) (N−1)np2=/radicalBigg N−n N−11−p np. The CV for a sample of size 1 is/radicalbig (1−p)/p. The sample size in (2.32) will be z2 α/2CV2/r2. (b) I used a spreadsheet to calculate these values. p 0.001 0.005 0.01 0.05 0.1 0.3 0.5 Fixed error 4.3 21.2 42.3 202.8 384.2 896.4 1067.1 Relative error 4264176 849420 422576 81100 38416 9959.7 4268.4 p 0.7 0.9 0.95 0.99 0.995 0.999 Fixed error 896.4 384.2 202.8 42.3 21.2 4.3 Relative error 1829.3 474.3 224.7 43.1 21.4 4.3 2.26Decision theoretic approach for sample size estimation. g(n) =L(n) +C(n) =k/parenleftbigg 1−n N/parenrightbiggS2 n+c0+c1n.25 dg dn=−kS2 n2+c1 Setting the derivative equal to 0 and solving for ngives n=/radicalBigg kS2 c1. The sample size, in the decision theoretic approach, should be larger if the cost of a bad estimate, k, or the variance, S2, is larger; the sample size is smaller if the cost of sampling is larger. 2.27The supplementary books by Lohr (2022) and Lu and Lohr (2022) give code in SAS and R software, respectively, that may be used for this problem. The distribution from a sample of size 100 appears skewed. 2.28In a systematic sample, the population is partitioned into kclusters, each of size n. One of these clusters is selected with probability 1/k, soπi= 1/kfor eachi. But many of the samples that could be selected in an SRS cannot be selected in a systematic sample. For example, P(Z1= 1,...,Zn= 1) = 0 : since every kth unit is selected, the sample cannot consist of the first nunits in the population. 2.29(a) P(you are in sample) =/parenleftBigg 99,999,999 999/parenrightBigg/parenleftBigg 1 1/parenrightBigg /parenleftBigg 100,000,000 1000/parenrightBigg =99,999,999! 999! 99,999,000!1000! 99,999,000! 100,000,000! =1000 100,000,000=1 100,000. (b) P(you are not in any of the 2000 samples) =/parenleftbigg 1−1 100,000/parenrightbigg2000 = 0.9802 (c)P(you are not in any of xsamples) = (1−1/100,000)x. Solving for xin(1−1/100,000)x= 0.5 givesxlog(0.99999) = log(0 .5), orx= 69314.4. Almost 70,000 samples need to be taken! This problem provides an answer to the common question, “Why haven’t I been sampled in a poll?” 2.30(a) We can think of drawing a simple random sample with replacement as performing an ex- perimentnindependent times; on each trial, outcome i(fori∈{1,...,N}) occurs with probability pi= 1/N. This describes a multinomial experiment.