Problems for Chapter 8: Producing Data: Sampling. STAT Fall 2015.

Transcription

1 Population and Sample Researchers often want to answer questions about some large group of individuals (this group is called the population). Often the researchers cannot measure (or survey) all individuals in the population, so they measure a subset of individuals that is chosen to represent the entire population (this subset is called a sample). The researchers then use statistical techniques to make conclusions about the population based on the sample. Sampling Designs Describes exactly how to choose a sample from the population. good = unbiased bad = biased design of a study is biased if it is systematically. Simple Random Sampling Each individual in the population has the same chance of being chosen for the sample. Each group of individuals (in the population) of the required size (n) has the same chance of being the sample actually selected. Random selection: - table of random digits (Table B) - computer software Random selection eliminates bias in the choice of a sample from a list of the population. Nevertheless we need to reduce all sources of error. Stratified Random Sample To select a stratified random sample, first divide the population into groups of similar individuals, called strata. Then choose a separate SRS in each stratum and combine these SRSs to form the full sample. Both of these techniques are biased systematically favor certain outcomes: Voluntary response sampling Allowing individuals to choose to be in the sample. Example: Advice columnist Ann Landers asked her readers, "If you had it to do over again, would you have children?" A few weeks later, her column was headlined: 70% OF PARENTS SAY KIDS NOT WORTH IT. * The people who responded felt strongly enough to take the trouble to write Ann Landers. * Their letters showed that many of them were angry at their children. * These people don't fairly represent all parents. A statistically designed opinion poll on the same issue a few months later found that 91% of parents would have children again. Convenience sampling Selecting individuals that are easiest to reach. Example: A group wants to know how much exercise the average adult New Mexican receives each week. A booth was set-up in front of a local gym, and 27 of the 45 adults asked, agreed to fill out a small survey. 1

2 Cautions about Sampling Surveys: A survey conducted on a sample from the population of all individuals about which we desire information. Undercoverage some individuals or groups in the population are left out of the process of choosing the sample Nonresponse individuals chosen for the sample cannot be contacted or refuse to cooperate/respond Response bias behavior of respondent or interviewer may lead to inaccurate answers or measurements Wording of questions confusing or leading (biased) questions; words with different meanings Examples: Undercoverage A classic example of undercoverage is the Literary Digest voter survey, which predicted that Alfred Landon would beat Franklin Roosevelt in the 1936 presidential election. The survey sample suffered from undercoverage of low-income voters, who tended to be Democrats. How did this happen? The survey relied on a convenience sample, drawn from telephone directories and car registration lists. In 1936, people who owned cars and telephones tended to be more affluent. Undercoverage is often a problem with convenience samples. Nonresponse To prepare for her book Women and Love, Shere Hite sent questionnaires to 100,000 women asking about love, sex, and relationships % responded - Hite used those responses to write her book - angry women are more likely to respond Response bias For this survey on drug use, would it matter if a police officer is conducting the interview? (bias from interviewer). Answers to questions that ask respondents to recall past events are often inaccurate because of faulty memory. For example, many people telescope events in the past, bringing them forward in memory to more recent time periods. Have you visited a dentist in the last 6 months? will often draw a Yes from someone who last visited a dentist 8 months ago. Wording effects If you found a wallet with $20 in it, would you return the money? vs. If you found a wallet with $20 in it, would you do the right thing and return the money? 2

3 Problem 1. A political scientist wants to know how college students feel about the Social Security system. She obtains a list of the 3456 undergraduates at her college and mails a questionnaire to 250 students selected at random. Only 104 questionnaires are returned. (a)what is the population in this study? Be careful: about what group does she want information? (b)what is the sample? Be careful: from what group does she actually obtain information? The important message in this problem is that the sample can redefine the population about which information is obtained. Problem 2. The city council of a suburb of Columbus is interested in the level of public support for a new recreation center. A marketing research firm is selected which then selects a simple random sample of 50 adult residents and contacts each to determine whether the resident would be interested in joining this recreation center if it were built. Of these, 35 indicated they would be interested in joining the recreation center and 15 were against it. 1. The sample is a. the 35 residents interested in joining the recreation center. b. the 15 residents not interested in joining the recreation center. c. the 50 residents selected. d. all residents interested in joining the recreation center. 2. The population of interest is a. the residents in the suburb that support the new recreation center. b. the 50 residents contacted. c. all adult residents in the suburb. d. all household in the suburb. 3. The chance that all 50 residents in a neighborhood end up being the sample of residents selected is a. the population of the suburb divided by 50. b. the same as for any other set of 50 residents. c. smaller than average due to the cluster effect. d. smaller than average due to stratification. 3

4 Table B table of random digits a long string of the digits. The digits appear in groups of five and numbed rows just made the table easier to read. How to choose an SRS using Table B: Give each member of the population a numerical label of the same length (same number of digits). You can label members in any order (preferred to follow by rows or by columns). Read consecutive groups of digits of the appropriate length from Table B: - choose a row to start, - read the list from the left to right by digit groups, - ignore any group of digits that was not used as a label or that duplicates a label already in the sample, - read digit groups of needed length from selected line of Table B until you have chosen required number of random subjects, - your sample contains the individuals whose labels you found. Problem 3. Use simple random sampling to select an employee group of four employees from the following listing of 12 employees. An excerpt of line 110 from Table B follows. Label each employee and use line 110 to select a simple random sample of 4 for an employee group. Circle the selected employees. Berliner Wolfe Verducci Taylor Blumenthal Stasny Lin Montoya MacEachern Santner Critchlow Ito Line Problem 4. Sickle cell disease is an inherited disorder of the red blood cells that can cause severe pain and many complications. Fifty subjects with sickle cell disease volunteer to participate in a study of the drug hydroxurea. Use the excerpt from the table of random digits below to choose the first three subjects. Line Problem 5. A group wants to know how much exercise the average adult New Mexican receives each week. A booth was set-up in front of a local gym, and 27 of the 45 adults asked, agreed to fill out a small survey. a) What us the population? - Adults of New Mexico b) What type of sampling is this? - Convenience c) What is the non-response rate? ((45 27)/45) 100=(18/45) 100=40 % 4

5 Examples of Stratified Random Sample 1) A company has a total of 360 employees in four different categories: Managers 36 Drivers 54 Administrative Staff 90 Production Staff 180 How many from each category should be included in a stratified random sample of size 20? In general the size of the sample in each stratum is taken in proportion to the size of the stratum. This is called proportional allocation. 2) A labor organization wants to study the attitudes of college faculty members toward collective bargaining. These attitudes appear to differ depending on the type of college. The American Association of University Professors classifies colleges as follows: Class I. Offer doctorate degrees and award at least 15 per year. Class IIA. Award degrees above the bachelor's but are not in Class I. Class IIB. Award no degrees beyond the bachelor's. Class III. Two-year colleges. Discuss the design of a sample of faculty from colleges in your state, with total sample size about 200. I would use a stratified sample (by class of college) with an equal allocation (i.e. 50 in each class). Obtain a list of faculty from the AAUP and select a random sample from each stratum. 5