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Solution Manual for Elementary Survey Sampling, 7th Edition

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    1 CHAPTER 2 ELEMENTS OF THE SAMPLING PROBLEM 2.1 An adequate frame listing individuals in a city is difficult to obtain. For that reason, and because data is desired on a family basis, it would be better to sample dwelling units. An adequate frame for dwelling units is also difficult to obtain, so a cluster sampling approach could be used by sampling city blocks and then measuring water consumption for the families living in the sampled blocks. 2.2 A common way to sample trees is to divide the farm into plots and then randomly or systematically sample plots on which trees would be counted. Unless trees are planted according to a regularly spaced design, it is difficult to use the trees themselves as sampling units. 2.3 The sampling design depends on a careful definition of the population of interest. As it would be almost impossible to get a listing of all cars owned by residents of a city, a better option would be to restrict the population of cars to something like “cars that use city parking lots on a working day” or “cars that belong to people visiting the malls on a weekend.” Then, a listing of parking lots or sections of parking lots could serve as frames for collections of cars. 2.4 If the number of plants is not too large, each could serve as a stratum from which employees would be sampled. In this case one would need a list of employees (a frame) for each plant. If the number of plants is large, then a sample of plants (clusters of employees) could be taken and a sample of employees (or all employees) could be interviewed in each sampled plant. 2.5 An area as large as a state is generally broken up into smaller areas, such as counties and farms within counties, for sampling. Each county may contain a number of farms, so there are various sampling options. Counties could be viewed as strata, with farms being sampled from each. If there are many counties, one might sample counties as clusters of farms and then sample farms from each sampled county. In either of these scenarios a list of farms by county would be needed as a frame. 2.6 Most polls of this type are done by telephone using random digit dialing, The state could be stratified by regions, with dialing taking place within each of these regions. Frames may be found for selected populations by using lists of registered voters or lists of property owners, but these frames do not cover the entire population of adults. Personal interviews generally produce the highest response rate, but they are expensive and require a list of individuals from which to sample. Mailed

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    2 questionnaires also require a list of individual addresses and have the lowest response rate, typically. Telephone interviews are probably the most viable choice for such a survey. 2.7 (a) A telephone survey would be the only way to cover the country with a well designed sampling plan in a reasonable time. (b) If the population is defined as subscribers to the paper, then a mailed questionnaire or interviews could be used. If the population is less well defined to include all readers or potential readers, than a telephone survey with random digit dialing may have to be used. (c) Homeowners are a well-defined group, and a sample could be contacted through either mailed questionnaires or personal interviews, although the latter would be time consuming. Telephone interviews could also be used, and random digit dialing would not be necessary. (d) Assuming dogs are registered, it should be relatively easy to sample from the list of registered owners and obtain the survey information by either telephone or mail. If there is no lost of dog owners, this would be a difficult problem probably best solved by random digit dialing. 2.8 “Do you consider yourself a political liberal or conservative?” “Do you favor an increase in the minimum wage?” Once the political label is decided, the options for the second question become more limited. Presented in the reverse order, the respondent has more freedom on the answer to the minimum wage question. 2.9 Closed questions limit options and nuances in answers, but are easier to analyze statistically. An open question could be of the form “What is your opinion on the school tax referendum?” A closed version could be “Do you pan to vote for or against the school tax referendum that is on the ballot in the next election?” This is an extremely closed version, other options could be offered. 2.10 “Do you favor an increase in the minimum wage to keep up with inflation?” “Do you favor an increase in the minimum wage so that many wage earners who are now living below the poverty line can afford to adequately feed and clothe their families?” 2.11 The no-opinion option should be used carefully and sparingly because it gives respondents an easy way out of questions on which they may well have a deeper opinion. 2.12 “I’m sure you are aware that most standardized tests contain multiple choice questions that favor students who memorize lots of facts as compared to those who learn to think deeply. Do you favor or oppose the increased use of standardized test scores to measure academic achievement?”

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    3 2.13 After errors of non-observation and errors of observation, the next most common source of errors in surveys is the mishandling of data in the data recording and analysis part of the survey. It is imperative that the data management process contains checks to see that data are recorded correctly, and that all recorded data are part of the analysis. 2.14 The pretest, which need not be on a randomly selected sample, is the best way to discover if a questionnaire contains questions that can be answered in reliable and valid ways. It also helps define issues that should be part of the training of field workers as well as issues of data collection, management and analysis. 2.15 The response rate is strongly related to the bias in survey results. A low response rate may imply that important segments of the population (such as retired people or single people) are under-represented in the survey data and., hence, in the reported results. 2.16 More people may well beat home at that hour, but they also do not like being interrupted at mealtime. One type of nonresponse is being traded for another, perhaps. 2.18 It is very difficult to get objective information on sensitive issues. The proportion who admitted to cheating is probably well below the actual proportion who cheated. (A technique in Chapter 11 will show how to improve the accuracy of responses in these situations.) 2.19 The results may be a bit biased because students regularly here that mathematics and English are the two subjects in which they need to do well in order to succeed in life. 2.20 The response rate is low and those who are most concerned about being able to pay for a college education are the ones most likely to respond. For others, this is not an important issue and they will tend not to respond. Thus, the result could suffer from a large bias. 2.21 The population being sampled here does not represent the population of the country and the responses are voluntary, not form a randomly selected sample. The question has an inherent bias toward favoring nuclear power plants. All aspects of the survey are directed toward obtaining a highly biased result. 2.22 This is a very low response rate and the GAO was justified in questioning the results. It is quite likely that some of the income groups (especially the low incomes) were greatly underrepresented. 2.23 (a) One rating point represents one percent of the viewing households, or 95.1 million × 0.01 = 951,000 households based on the fact that the sampled population is households.

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    4 (b) As a percentage, a share is larger than a rating because the denominator of the rating is the total number of sampled households, while the denominator of a share is the total number of sampled households that actually have a TV set turned on (viewing households). (c) 95.1 million × 0.217 = 20.64 million households could have been viewing this show (d) Much of the data collected by Nielsen depends upon people in the sampled households either pushing a button on a People Meter or writing in a diary to record what they are watching. This is far from a fool-proof system. 2.25 (a) Target Population 51% 12% 9% High risk cities 57.9% 33.8% 20.7% National 58.4% 13.5% 8.3% In the national survey, the sample percentages are quite close to those reported by the Census. Thus, randomization did a good job. In the survey of high-risk cities, the black and Hispanic percentages are much higher than those reported for the nation as a whole. (b) High-risk cities are not the typical cities of the population. One may expect that the randomization actually did a good job here as well. 2.26 (a) National: 100 - 84.9 = 15.1 % High Risk Cities : 100 - 80.4 = 19.6 % (b) National: 0.151 (1000) = 151 High Risk Cities: 0.196 (1000) = 196 2.28 Care about keeping weight down NS EX FS CS Total A lot 6297 3613 197 2114 12221 Somewhat 2882 1677 90 793 5442 A Little 1441 625 16 354 2436 Don’t care 1709 822 22 377 2930 Total 12329 6737 325 3638 23029 % A lot .51 .54 .61 .58 (a) It is anticipated that responses will be more honest when questions are asked about peers rather than about the respondent himself or herself.

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    5 (b) 12,329,000 (c) 12221 / 23029 = .53 (d) 3,638,000, 2114/3638 = .58 (e) 6297 / 12221 = .51, 2114 / 12221 = .17 (f) No, at least not strongly. As you see from the bottom row of calculated percentages, the percentage of students who care a lot about keeping their weight down is fairly constant across all categories of smoking. 2.29 Care about staying away from marijuana NS EX FS CS Total A lot 7213 2693 75 857 10838 Somewhat 2482 1861 109 1102 5554 A Little 744 542 27 298 1611 Don’t care 1878 1550 119 1312 4859 Total 12317 6646 330 3569 22862 % A lot .59 .41 .23 .24 (a) 7213 / 12317 = .59 (b) 857 / 3569 = .24 (c) 7213 / 10838 = .67 (d) 1878 / 4859 =.39 (e) Yes. Non smokers care more about staying away from marijuana than current smokers (59%, 24%, respectively). Also from (a), and (b), among those who care a lot about staying away from marijuana, 59% were non smokers, while 24% was current smokers. 2.30 (a) These are conditional proportions, calculated as the percentages of “yes”, “no” and “don’t know” responses within each smoking category. (b) Yes. Twelve percent of nonsmokers think smoking help reduce stress, while 46.5% of current smokers believe that. (c) No. Regardless of their smoking status, teenagers believe that almost all doctors are strongly against smoking.

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    6 2.31 In the actual study, 38% favored the law in the A1 form whereas only 29% favored the law in the B1 form. In the later study, 39% favored the law in the A2 form whereas only 26% favored it in the B2 form. There is a stronger counter argument in B2 as compared to B1.

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    7 CHAPTER 3 A REVIEW OF SOME BASIC CONCEPTS 3.1 Statistics has many definitions, but almost all involve the process of drawing conclusions from data. Data is subject to variability, so some would say that statistics is the study of variability with the objective of understanding its sources, measuring it, controlling whatever is controllable, and drawing conclusions in the face of it. For sample survey purposes, statistics involves a well-defined population, a sample selected according to an appropriate probabilistic design, and a methodology for making inferences from the sample to the population, usually in terms of estimation of population parameters. 3.2 A statistic is a function of (is calculated from) sample data whereas a parameter is a numerical characteristic of a population. In a common opinion poll, a sample of 500 residents may be asked whether or no they favor a certain candidate for office. The sample percentage is a statistic, but it is used to estimate the population percentage favoring that candidate, an unknown parameter. 3.3 An estimator is a statistic used to estimate a population parameter, like the sample proportion in Exercise 3.2. 3.4 A sampling distribution is a distribution of all possible values of a statistic. 3.5 The goodness of an estimator is usually measured by the standard deviation of its sampling distribution. The margin of error refers to two standard deviations of the sampling distribution of an estimator. Roughly speaking, the difference between an estimator and the true value of the parameter being estimated will be less than the margin of error with probability about .95. 3.6 An estimator should be unbiased (or nearly so) and have a small standard deviation of its sampling distribution. In other words, in repeated usage, an estimator’s values should pile up close to the value of the parameter being estimated. 3.7 An unbiased estimator is one for which the sampling distribution centers at the true value of the parameter being estimated.

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    8 3.8 The error of estimation refers to the difference between an estimator and the true value of a parameter being estimated. It is measured by the standard deviation of the sampling distribution of the estimator in question. 3.10 Summary Statistics Calories Cost in Dollars w/Hydra w/o Hydra w/ Hydra w/o Hydra Mean 64.78 64.62 .294 .27 Median 66.0 63.5 .260 .25 Stdev 8.51 9.09 .097 .05 Q 1 60 60 .230 .225 Q 3 70 70 .345 .320 Min 50 50 .220 .220 Max 80 80 .520 .350 Range 30 30 .300 .130 Scatterplot of cost vs. calories - * Hydra 0.50+ - cost - - - 0.40+ - - * - * - 0.30+ - - * * - * * - 2 0.20+ +---------+---------+---------+---------+---------+------ calories 48.0 54.0 60.0 66.0 72.0 78.0

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    9 Scatterplot of cost vs. calories (without Hydra) - 0.350+ * - * cost - - - 0.300+ - - - - * * 0.250+ - * * - - 2 - 0.200+ +---------+---------+---------+---------+---------+------ calories 48.0 54.0 60.0 66.0 72.0 78.0 (a) The mean is a good summary number for typical calories per serving. The standard deviation is a good summary number for the variation in the calories. Box plot of calories ------------------ -----------------I + I---------------- ------------------ +---------+---------+---------+---------+---------+------ 48.0 54.0 60.0 66.0 72.0 78.0 (b) Since there is an extreme value, the median is a good summary number for typical cost per serving, and IQR (Q 3 - Q 1 ) is a good summary for the variation in costs. Box plot of cost ------------------ Hydra ---I + I- * ------------------ ------+---------+---------+---------+---------+---------+ 0.240 0.300 0.360 0.420 0.480 0.540 (c) Because these drinks would not generally be combined by users, the totals have little practical value here. (d) On the average calories per serving; not much impact On the standard deviation: slight increase On the average cost per serving: decrease

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    10 On the standard deviation of the cost per serving: decreased Box plot of calories (without Hydra) ------------------ -----------------I + I---------------- ------------------ +---------+---------+---------+---------+---------+------ 48.0 54.0 60.0 66.0 72.0 78.0 Box plot of cost (without Hydra) ----------------------------- ----I + I-------------------- ----------------------------- ----+---------+---------+---------+---------+---------+-- 0.225 0.250 0.275 0.300 0.325 0.350 (e) There is no particularly influential drink on the average calories per saving, but Snappple (80 calories) has the most influence as it is furthest from the mean. 3.11 (a) Including the powdered drinks on the same list with the liquid drinks does not have much effect on the average calories per serving, as their calorie figures are within the range of the first data set. Including the powdered drinks lowers the average cost per serving and increases the standard deviation of cost because the new cost values are much lower than in the original set. calories cost mean stdev mean stdev w/o powder 64.78 8.51 .294 .097 w powder 64.82 7.93 .278 .102 Parallel box plots of calories ------------------ w/o powder -----------------I + I---------------- ------------------ ------------------ w/ powder -----------------I + I---------------- ------------------ +---------+---------+---------+---------+---------+------ 48.0 54.0 60.0 66.0 72.0 78.0

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    11 Parallel box plots of cost ------------- w/o powder ---I + I-- * ------------- ----------- w/ powder --------------I + I----- * ----------- --------+---------+---------+---------+---------+-------- 0.160 0.240 0.320 0.400 0.480 (b) Adding the light varieties to the list will not have much of an effect on the average cost and standard deviation of cost. Mean and standard deviation for two groups (cost) mean stdev w/o lites .294 .097 w/ lites .286 .088 Parallel box plots of cost ------------------ w/o lites ---I + I- * ------------------ ----------- w/ lites ---I + I-------- O ----------- ------+---------+---------+---------+---------+---------+ 0.240 0.300 0.360 0.420 0.480 0.540 (c) Adding the light varieties to the list will decrease the average calories per serving and increase the standard deviation of calories because the new cost figures are way below those of the original data set. Mean and standard deviation for two groups (calories) mean stdev w/o lites 64.78 8.51 w/ lites 55.45 22.69

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    12 Parallel box plots of calories -------- w/o lites -------I + I------ -------- ---------- w/ lites O * ----I + I------- ---------- +---------+---------+---------+---------+---------+------ 0 15 30 45 60 75 (d) Use the median because the median is not sensitive to extreme values. 3.12 Summary Statistics Area N mean median stdev Q 1 Q 3 Q 3 - Q 1 U.S 10 25.10 12.50 22.02 7.50 51.25 43.75 U.S.& Foreign 10 4.80 1.00 7.18 0.00 10.00 10.00 (a) As shown on the stem plot, these data are split into two groups and neither the mean nor the median are good measures of center. A more meaningful summary statistic is the total number of endangered species, 251 for those unique to the U.S. and 299 in the U.S. and foreign countries. Box plot of U.S. ------------------------------------------- -----I + I------- ------------------------------------------- --------+---------+---------+---------+---------+-------- 10 20 30 40 50 Stem plot of U.S. Stem-and-leaf of US Leaf Unit = 1.0 3 0 368 (3) 1 023 4 2 4 3 7 3 4 3 5 057 (b) Again, the total number of endangered species is a more meaningful statistic than either the mean or median. For the world, this total is 791 species.

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    13 Stem plot of U.S & Foreign Stem-and-leaf of US & Foreign Leaf Unit = 1.0 5 0 00000 5 0 23 3 0 3 0 3 0 8 2 1 2 1 2 1 2 1 6 1 1 9 Box plot of U.S. & Foreign ------------------------ I + I------------------------------- ------------------------ +---------+---------+---------+---------+---------+------ 0.0 3.5 7.0 10.5 14.0 17.5 (c) No. See part (a). 3.13 (a) 1 25 3 16 2 4 1 2 58 25 2 32 ( ) . × + × + × = = (b) μ = = × + × + × + × = ∑ xp x ( ) . . . . . 3 64 2 16 1 08 0 12 2 32 (c) V x x p x x p x x x ( ) ( ) ( ) ( ) = − = − ∑ ∑ μ μ 2 2 2 = + + + − = − = 3 64 2 16 1 08 0 12 2 32 6 48 5 3824 10976 2 2 2 2 2 (. ) (. ) (. ) (. ) . . . . σ = = V x ( ) . 105 3.14 (a) μ = = ∑ E x xp x ( ) ( ) = 2(.443) + 3(.229) + 4(.200) + 5(.086) + 6(.028) + 7(.014) = 3.069 (b) σ 2 = V ( x ) = ( x − μ ) 2 x ∑ p ( x ) = 1.458 σ = V ( x ) = 1.207

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    14 (c) The distribution of the sample data would reflect that of the population. Most of the data values would pile up around 2 and 3 , with a few larger values. The distribution of the sample would be skewed toward the larger values, with a center at approximately 3.07 and a standard deviation of approximately 1.21. (d) The sample mean x has approximately a normal distribution with mean μ x = μ x = 3.07 and standard deviation σ x = σ n = 1.21 20 = 0.0605 3.15 (a) The scatter plot shows that SAT and Percent are negatively correlated, with a curved pattern suggesting that the average score drops quickly as the percentages begin to increase and them levels off for higher percentages. The decreasing scores with increasing percentage taking the exam makes practical sense; in states with small percentages only the very best students are taking the exam. (b) The correlation coefficient is -0.877, but this is not a good measure to use here because of the curvature in the patter. Correlation measures the strength of a linear relationship between two variables. Scatter plot between Average Score and Percent 3.16 Tabled below are the new probabilities for samples of size 2 and estimates of the population total for the unequal probabilities of selection that favor the smaller population values. 1050 1200 Aver age Per cent 0 20 40 60 80 x x x x x x x 2 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

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    15 Sample Probability ˆ τ pps {1.2} 0.32 3.75 {1.3} 0.08 16.25 {1.4} 0.08 21.25 {2,3} 0.08 17.50 {2,4} 0.08 22.50 {3,4} 0.02 35.00 {1,1} 0.16 2.50 {2.2} 0.16 5.00 {3,3} 0.01 30.00 {4,4} 0.01 40.00 Calculation of expectations yields: E ˆ τ pps ( ) = 10 V ˆ τ pps ( ) = 81.25 3.17 The weights given in Section 3.3 for the four population values are w 1 = 4.0916, w 2 = 4.0916, w 3 = 1.3236 and w 4 =1.3236. The sum of the weights for each of the six possible samples, along with the probabilities of selecting each of these samples, are shown in the accompanying table. The expected value of the sum of the weights turns out to be 4.00, the number of values in the population. Sample Sum of weights Probability of sample, unequal weights {1,2} 8.1832 .0222 {1,3} 5.4152 .1111 {1,4} 5.4152 .1111 {2,3} 5.4152 .1111 {2,4} 5.4152 .1111 {3,4} 2.6472 .5333 3.18 For samples of size n=2 taken with probabilities proportional to the populations of the states, the pertinent data and the probabilities of selection with probabilities proportional to the population, both with and without replacement, are given in the first table that follows. With replacement probabilities of selection ( δ ) are directly proportional to the population sizes. Without replacement probabilities of selection ( π ) are found by first finding the probability for each possible sample, given on the

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