STAB22 section 3.2. Figure 1: Randomly selected ringtones
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- Benedict Arnold
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1 STAB22 section Using table B: pick a place to start, without looking at the table, say row 128. Number the ringtones 0 9 (better than 01 10, because you ll end up discarding a lot that way), say reading along the rows. The first two digits on line 128 are 1 and 5, which correspond to ringtones Empire State of Mind and Down (remember that the first one was numbered 0). In software, type all the ring-tones into a column, and then randomly sample two of them (or, equivalently, shuffle all 10 and read off the first two). In Minitab, type the ringtone names into column C1 (or abbreviations for them, since some of them are a bit long). I used one-word abbreviations to save some typing. Select Calc, Random Data and Sample from Columns. Select 2 rows from column C1, and put the sampled rows into C2 (that is, make sure the boxes on the right-side of the dialog box say 2, C1 and C2 in that order). Click OK, and have a look at what appears in column 2: in my case Party in the USA and Thriller (yours will be different). Those are the two randomly sampled ringtones. In StatCrunch, again type the ringtone names into a column, select Data and Sample Columns, then type 2 into the Sample Size box, and leave Figure 1: Randomly selected ringtones Number of Samples at 1. Leave everything else at the default, and click on Sample Columns. Since some of the ringtone names are quite long, I made the columns wider to see them all (play with Edit, Columns, Widths) In (a), the population is all the individuals you want to know about, which is the ones that actually appear in your sample plus all the others that might have done but didn t. In (b), there s nothing wrong statistically with 1
2 the sampling procedure, just with the thing being surveyed: dihydrogen monoxide is H 2 O, or water! The survey as indicated might reveal to you the lack of respondents knowledge of chemistry, but not more than that. (Do the writers of our textbook perhaps have a sense of humour?) In (c), students who actually have cheated on an exam are very unlikely to raise their hands in a public forum like a lecture. (Also, there isn t really any random sampling happening here.) A way in which you can get information about this kind of issue is a randomized response survey (see page 296 of the text) in which you (in a randomized way) ask either the sensitive question have you cheated on an exam? or a safe question like did you ride the bus to school today?, and only keep a record of the responses given, not which question was asked. This, though, needs a very big sample to get much information For (a), the third chapter might be quite different from other chapters in the same book. You could select a chapter at random, or, better (I think), you can select a sample of pages at random and assess the reading level of the text on those pages. For (b), students who make it to class that early in the morning might be quite unlike the student body as a whole. You might imagine that such students are keener than average on their education, and might have a more informed opinion about this issue, compared to students who skip classes at this time in the morning (or who don t even enrol for such classes). For (c), as in 3.18(a), alphabetical order of names is not a random order (because of the way different ethnic groups have names that start with a few letters of the alphabet. For example, if I ve picked a Chan, the next name alphabetically is likely to be another Chan or a Chang or a Chen or something else of East Asian origin Population is all (local) businesses (the ones in the phone book). Sample is the 150 chosen at random. 73 of 150 responded, so 77, or 51.3%, failed to respond. (In a perfect world, you d get responses from all 150, so the 150 is at least the sample you d like to have. The 73 that responded is the sample you actually got, which was chosen by a mixture of randomness (being in the 150 in the first place) and self-selection (choosing to respond)) In (a), it looks as if the population should be all students currently attending this college, because these are the people who would be affected by changes in that college s core curriculum. ( Core curriculum means the classes that 2
3 everybody takes : things like an English or a Science requirement for everybody.) In (b), the population is all American residents (or maybe all American households ), the same people who would be enumerated on the census. In the past, all residents had to complete (or have an enumerator complete) a short form, with a randomly selected sample of residents having to complete a long form asking for more information. My suspicion is that this change is being made because some people are not filling in the long form correctly, and so useful information is not being obtained from it. The detailed followup procedure described ought to produce more accurate information, because if the ACS doesn t receive the survey back, or if it is incomplete, they have a procedure in place to complete it. In (c), the population looks as if it ought to be all people eligible to vote in a federal election, because the question (and answers given to the question) is not really of interest otherwise. This is a similar population to all residents of the country, but not the same, because people live here who are not entitled to vote (new immigrants, for example). In Canada, you have to be a citizen in order to vote (though, curiously, some provinces don t require you to be a Canadian citizen to vote in provincial elections) This is the same kind of procedure we used before in selecting groups for an experiment: number the items (in this case from 01 to 33), then choose two-digit groups from Table B starting at line 137, rejecting any giving a number bigger than 33 or that you ve chosen before. If you number down the columns (ie. alphabetically), you end up with 12 (Country View), 14 (Crestview), 11 (Country Squire), 16 (Fairington) and 08 (Burberry). You might think these are not very random, in that they all come from the top of the 2nd column or somewhere near there, but you can often rationalize apparent non-randomness after the fact. The key is that if you use Table B to take another sample of 5, you won t be able to guess ahead of time where in the list the sampled apartment complexes will come from. Also, you might have noticed that using Table B for this was hard work. because a lot of the digit pairs got rejected for being too large. Another way here is: reject 00; otherwise, keep subtracting 33 from the Table B number until you have 33 or less. Reject this if you ve had it before, otherwise it gives you the apartment complex to sample. For example, if Table B gives you 72, subtract 33 to give 39 (still too big), and subtract 33 to give 06. (Why does this work? How would you make it work if you had 36 apartment complexes 3
4 instead of 33?) You can also use software for this: enter the names into a column, and then (in Minitab) use Calc, Random Data and Sample. Instead of sampling all 33 rows into column C2, if you just sample 5, you ll get the 5 names you want. But if you shuffle them all, you can just read off the first 5 names in C2. Either way is good. In StatCrunch, use Data and Sample Columns as in 3.50 (choosing one sample of 5). Or enter the numbers 1 through 33 into a column, randomly select 5 of the numbers, and then convert them into names. Either way, this requires a good bit less thinking than using Table B. Contrast this with selecting treatment groups for an experiment, where you wan to assign all the subjects/units to a group. Here, all you want to do is to pick some of them for a group (your sample), so all you need to do is to pick the ones you want You need a simple random sample of 25 pixels out of 200. Since 200 has 3 digits, start from line 112 of Table B and pick 3-digit groups, discarding the ones you can t use: 596 (discard), 368, 880, 404, 711, 971, 935, 273 (discard them all), 089 (you can use this one) and so on. I also selected 064, 032, 117 and 003. To use software, get the numbers into a column. To do this quickly in Minitab, select Calc, Make Patterned Data and Simple Set of Numbers. In the dialog box, enter the place to store the results (say C1), and have the numbers go from first value 1 to last value 200. The rest you can leave as is. Click OK. You can check to see that C1 contains the right values. Then take a random sample of 25 of these values by selcting Calc, Random Data and Sample from Columns. In the boxes, fill in that you want to sample 25 values from C1 and put them in C2. Take a look at the pixels you sampled. Mine started with 109 and 116 and finished with 48 and 21, but yours will be different. In StatCrunch, the magic words are Data and Sequence Data; then fill in 1 through 200 by (steps of) 1. Then choose Data again and Sample Columns; select your column of 1 through 200 (called Sequence), change Sample Size to 25, and then click Sample Columns. My sampled pixels start with 51 and 13, and finish with 171 and 94. See how, using software, sampling 25 values is as easy as sampling 5? Imagine how much work it would be to use Table B for this You could select 4 digits at a time from Table B, and discard any of the ones that don t match a block number. This would be very inefficient, though, since you would be doing a lot of discard- 4
5 ing. More efficient is to select groups of 2 digits: 01 to 06 are the blocks in group 1, 11 to 22 are the blocks in group 2 (count them to be sure there are 12) and 31 to 56 are the blocks in group 3. I did it this way to make it easier to pick out the blocks that are selected. My random digits are 66, 92 (discard), 55 (group 3, block 25: 3024), 56 (group 3, block 26: 3025), 58 (discard), 39 (group 3, block 9: 3008), 10 (discard), 07 (discard), 11 (group 2, block 1: 2000), 20 (discard), 15 (group 2, block 5: 2004). Those are the 5 blocks. There are different ways of doing this: the answer in the textbook appears to have labelled the blocks 1000,..., 1005, 2000,..., 2011, 3000,..., 3025 and put the numbers 1 (01) to 44 next to those. Any way that makes it equally likely that each block would be chosen is OK. Using software, it would be much easier. The slowest part would be typing the block numbers into a column; once you ve done that, all you have to do is pick 5 of them at random The numbers you d get (for each sample/population size combination) would be completely predictable, which is hardly random. It s best to pick a random place in the table as well, which is what the instructions to start at row 117 (or whatever) are trying to do. Remember that random here really means unpredictable : you don t want to know ahead of time anything about which individuals are going to end up in the sample. For example, once you know that there are rows 101 through 200 in Table B, you can choose one of them before you open the book and start choosing digits. Unless you use Table B a lot, you won t have any idea of which digits are coming. (If you want to go a step further: there are 8 groups of digits on each line, so before you open your book, pick one of those at random as well. For example, say to yourself I m going to start with the 3rd group of digits on line 163.) 3.66 The aim is to end up sampling 200 students; if you sample every 45th name out of 9000, you ll end up with 9000/45 = 200 names. (You can try, say, every 50th name or every 30th name, and see that you ll end up with too few or too many names in your sample.) If you pick the first name at random, it s a random sample, but not a simple random sample like the ones we ve drawn so far. It s called, as the question says, a systematic random sample. The reason it s not a simple random sample is that, once you ve chosen the first name from the list of students, all the others are determined, whereas with a simple random sample, knowing the first student doesn t tell you anything about which other students are going to be in the sample. 5
6 Using Table B, line 125, we pick the random starting point as one of the students numbered 01 45, which turns out to be name 21. Then take every 45th name after that: 21st, 66th, 111th, etc. A systematic sample gives less accurate answers than a simple random sample of the same size, but it s much easier to draw. (Imagine the work involved in using Table B to take a simple random sample of 200 students out of 9000) Pick a name. If it s in the first 45, then it has the same 1-in-45 chance of being selected as any other name in the first 45. After that, whether a name gets selected depends on what was selected before. Say we look at the 114th name. This will be selected if the = 69th name was. And the 69th name will be selected if the = 24th name was. There s a 1-in-45 chance of picking the 24th name, so there s also a 1-in-45 chance of picking the 69th or 114th name. The same logic applies to any other name on the list, so every name (individual) has the same chance to be chosen. What makes a simple random sample a simple random sample is that, not only does each individual have the same chance to be chosen, but this is also true regardless of which other individuals are in the sample. But this independence doesn t work for a systematic sample: name 47 is in the sample if name 2 is, and isn t if name 2 isn t. If you know whether name 2 is in the sample, you also know whether name 47 is. (Another way to see this is that there are only 45 different possible systematic samples, depending which one of the first 45 names you choose, but for a simple random sample, any selection of 200 names out of 9000 has to be equally likely, and there are a lot more than 45 possible simple random samples.) 3.68 This is a stratified random sample. The sample is chosen in two stages, with selection at the first stage (area codes) done according to the size of the population in each area code. This would mean that an area code containing a lot of people has a greater chance of being chosen (so that each individual has the same chance to appear in the sample regardless of whether they live in an area code with many people or few). It s easy to confuse this with a multistage sample. A stratified sample chooses a sample of numbers within every area code, whereas a multistage sample (which is what you re starting to do below) chooses only some area codes and then samples within only those area codes you selected. Number the area codes from 01 to 25 (say, in the order given). Table B, line 122, gives 13, 15, 05, 6
7 09, 08, 07, 10, 25, 23, 18 (ignoring some repeats), which gives area codes 760, 818, 415, 408, 650, 510, 831, 619, 909, 562. Fascinating fact (or not): telephone systems switching machinery only used to be able to handle area codes whose middle digit was 0 or 1 (back in the days before 10-digit dialling, the machinery had to know whether you were dialling an area code or a 7-digit local number, whose first 3 digits do not have a 0 or a 1 in the middle). So area codes with a 0 or 1 (416, 905) are old, while area codes like 647 are new (it came in with the advance of cellphones and the resulting high demand for new numbers). It used to be the case that a number in a different area code from yours was automatically a long-distance call, but no longer, because area codes cover much smaller areas than they used to Nothing much new here: label the students and choose 4 of them with the aid of Table B, then number the faculty members 0 9 and choose 2 of them using Table B. (This way of numbering the faculty members is more efficient than using 01 10, because to do the latter way you have to take the digits in twos from Table B and discard a lot of them; to do it the suggested way, you only have to take single digits and discard (almost) none of them. (Why almost?) Of course, if you use software, there s no thinking required: enter the names into columns, and select a random sample from each. The point here is to take the two samples separately; it s tempting to think that there has to be some way of taking them together, but there isn t The reason for stratifying like this is that for the bigger accounts, the same percentage of error is a larger amount in dollar terms, so you want to make sure that the bigger accounts are checked. It doesn t matter so much about the smaller accounts. A stratified sample like this is really three separate simple random samples: of the 100 accounts over $50,000, sample them all of the 500 accounts between $1000 and $50,000, sample 5% = 25. Label these accounts , and select 3-digit groups from Table B starting at line 115, discarding any that don t work. This leaves you with 417, 494, 322, 247, 097 for the labels of the first five sampled accounts. of the 4400 accounts under $1000, we want to sample 1% = 44. Number the accounts 7
8 , and take 4-digit groups from Table B, discarding any you can t use. I m continuing from where I finished with the midsize accounts, rather than starting again from the beginning of row 115. This way, the first five sampled accounts are numbered 3698, 1452, 6318 (discard), 9332 (discard), 5921 (discard), 4459 (discard), 2605, 6314 (discard), 2480, Such a sample would omit people who are not in the phone book. This would include people who do not have a phone at all (ie. poor people), people who have unlisted numbers, and people who have a cellphone rather than a landline. These sets of people don t have much in common, but they are all people you would want to have in your sample and would otherwise miss. Random-digit dialling would catch those people with unlisted numbers, and might catch those who use a cellphone only (depending on how cellphone numbers are handed out and how the area codes and exchanges are chosen for the randomdigit dialling). Then of course there s the matter of don t call lists. People on these lists cannot be in the sample because they have opted out. This is kind of a self-selection in reverse (a) is slanted towards a yes response. Many other things may be associated with brain cancer: should they have a warning too? (For example: eating carrots, driving cars, or whatever else you can think of.) (b) lists two benefits of national health insurance, and no drawbacks, so it will also tend to produce a yes response. (c) is unclear (imagine listening to someone reading this to you over the phone): both in the words and sentence structure used, and the use of the vague economic incentives. An improvement: would you be willing to pay more for the things you buy if this money was used for recycling? 3.76 Think about families who have no children at all: they cannot possibly be in this survey because there are no children to be in this teacher s class. In the same way, the more children a family has, the greater that family s chance of being in the survey : for example, with 4 children, there are 4 chances for that family to be in the survey and say The two statements are logically opposites: everyone who agrees with the first statement ought to disagree with the second, and vice versa. But in practice, people like to agree with statements in surveys (as here), and you might argue that both statements are designed to favour an Agree answer. To generalize this, opinion polls often 8
9 fail to reveal public attitudes clearly because the statements with which people are invited to agree or disagree tend to favour (accidentally or on purpose) a certain answer. 9
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