Survey Sampling. Situation: Population of N individuals (or items) e.g. students at this university light bulbs produced by a company on one day

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1 Survey Sampling Situation: Population of N individuals (or items) e.g. students at this university light bulbs produced by a company on one day Seek information about population e.g. amount of money students spent on books this quarter percentage of students who bought more than 10 books in this quarter lifetime of light bulbs Full data collection is often not possible because it is e.g. too expensive too time consuming not sensible (e.g. testing every produced light bulb for its lifetime) Statistical approach: collect information from part of the population (sample) use information on sample to draw conclusions on whole population Questions: How to choose a sample? What conclusions can be drawn? Survey Sampling, Jan 19,

2 Survey Sampling Objective of a sample survey: Gather information on some variable for population of N individuals: x i x 1,..., x N value of interest for ith individual values for population Sample of length n: x 1,..., x n values obtained from sampling Parameter - number that describes the population, e.g. µ pop = 1 N σ 2 pop = 1 N N j=1 x j N ( x j µ pop ) 2 j=1 population mean population variance Estimate population parameter from sampled values: ˆµ pop = x = 1 n n x i i=1 ˆσ 2 pop = s 2 = 1 n 1 sample mean N (x j x) 2 sample variance j=1 A function of the sample x 1,..., x n is called a statistic. Survey Sampling, Jan 19,

3 Sampling Distribution Suppose we are interested in the amount of money students at this university have spent on books this quarter. Idea: Ask 20 students about the amount they have spent and take the average. The value we obtain will vary from sample to sample, that is, if we asked another 20 students we would get a different answer. Sampling distribution The sampling distribution of a statistic is the distribution of all values taken by the statistic if evaluated for all possible samples of size n taken from the same population. In our example, the sampling distribution of the average amount obtained from the sample depends on the way we choose the sample from the population: Ask 20 students in this class. Ask 20 students in your department. Ask 20 students in the University bookshop. Select randomly 20 students from the register of the university. The design of a sample refers to the method used to choose the sample from the population. Survey Sampling, Jan 19,

4 Example: Sampling Distribution Consider a population of 20 students who spent the following amounts on books: x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 x Frequency (%) Frequency (%) (a) (b) σ = σ = x Sampling distribution of n x = 1 x n i i=1 for sample sizes (a) n = 2 (b) n = 3 (c) n = (c) 12 σ = x Frequency (%) x Survey Sampling, Jan 19,

5 Bias Example: Suppose we are interested in the amount of money students at this university have spent on books last quarter. Sample: 20 students in the University bookshop Do we get a good estimate for the average amount spent on books last quarter by UofC students? Students who buy more books and spend more money on books are more likely to be found in bookshops than students who buy less books. The sample mean might overestimate the true amount spent on books. The sample is not representative for the population of all students. Careful: A poor sample design can produce misleading conclusions. The design of a study is biased if it systematically favors some parts of the population over others. A statistic is unbiased if the mean of its sampling distribution is equal to the parameter being estimated. Otherwise we say the statistic is biased. Survey Sampling, Jan 19,

6 Bias Examples: Biased Sampling Midway Airlines Ads in the New York Times and the Wall Street Journal stated that 84 percent of frequent business travelers to Chicago prefer Midway Metrolink to American, United, and TWA. The survey was conducted among Midway Metrolink passengers between New York and Chicago. A 1992 Roper poll asked Does it seem possible or does it seem impossible to you that the Nazi extermination of Jews never happened? 22% of the American respondents said seems possible. A reworded poll 1994 asked Does it seem possible to you that the Nazi extermination of Jews never happened, or do you feel certain that it happened? This time only 1% of the respondents said it was possible it never happened. ABC network program Nightline once asked whether the United Nations should continue to have its headquarters in the United States. More than 186,000 callers responded, and 67% said No. A properly designed sample survey showed that 72% of adults want the UN to stay. A call-in poll conducted by USA Today concluded that Americans love Donald Trump. USA Today later reported that 5,640 of the 7,800 calls for the poll came from the offices owned by one man, Cincinnati financier Carl Lindner. Survey Sampling, Jan 19,

7 Caution about Sample Surveys Undercoverage occurs when same groups in the population are left out of the process of choosing the sample no accurate list of the population results in bias if this group differs from the rest of the population Nonresponse occurs when a chosen individual cannot be contacted or does not cooperate results in bias if this group differs from the rest of the population Response bias subjects may not want to admit illegal or unpopular behaviour subjects may be affected by the interviewers appearance or tone subjects may not remember correctly Question wording confusing or leading questions can introduce strong bias do not trust sample survey results unless you have read the exact questions posed Survey Sampling, Jan 19,

8 Simple Random Sampling A simple random sample (SRS) of size n consists of n individuals chosen from the population in such a way that every set of n individuals is equally likely to be selected. Every possible sample has an equal chance of being selected. Every individual has an equal chance of being selected. Random selection eliminates bias in sampling. SRS or Not? Is each of the following samples an SRS or not? A deck of cards if shuffled, and the top five dealt. A sample of Illinois residents is drawn by choosing all the residents in each of 100 census blocks (in such a way that each set of 100 blocks is equally likely to be chosen) A telephone survey is conducted by dialing telephone numbers at random (i.e. each valid phone number is equally likely). A sample of 10%of all student at the University of Chicago is chosen by numbering the students 1,..., N, drawing a random integer i from 1 to 10, and drawing every tenth student beginning with i. (E.g. if i = 5, students 5, 15, 25,... are chosen.) Survey Sampling, Jan 19,

9 Stratified Sampling Example: Population: Students at this university Objective: Amount of money spent on books this quarter Knowledge: Students in e.g. humanities spend more money on books Use knowledge to build sample: divide sample into groups of similar individuals, called strata choose simply random sample within each group size of samples in each groups e.g. proportional to size of groups Can reduce variability of estimate significantly. Survey Sampling, Jan 19,

10 Summary A number which describes a population is a parameter. A number computed from the data is a statistic. Use statistics to make inferences about unknown population parameters. A Simple random sample (SRS) of size n consists of n individuals from the population sampled without replacement, that is, every set of n individuals has an equal chance to be the sample actually selected. A statistic from a random sample has a sampling distribution that describes how the statistic varies in repeated data production. A statistic as an estimator of a parameter may suffer from bias or from high variability. Bias means that the mean of the sampling distribution is not equal to the true value of the parameter. The variability of the statistic is described by the spread of its sampling distribution. Survey Sampling, Jan 19,