2.1 Simple Random Sampling

1. 2.1 Simple Random Sampling

2. Variable In statistics, a variable is a characteristic of the individuals to be measured or observed. The data that we observe for a variable are called observations. Definition

3. Example: Identifying Individuals, Variables, and Observations The five movies with the largest worldwide gross receipts of all time are shown in the table. 1. Identify the individuals. 2. Identify the variables. 3. Identify the observations for each variable.

4. Solution 1. The individuals are the movies: Avatar, Titanic, etc. 2. The variables are studio, worldwide gross receipts, and U.S. gross receipts. 3. The observations for the variable studio are Fox, Paramount Pictures, Buena Vista, Warner Brothers, and Buena Vista. The observations for the variable worldwide gross receipts, all in millions of dollars, are 2788, 2187, 1519, 1342, and 1215. The observations for the variable U.S. gross receipts, all in millions of dollars, are 761, 659, 623, 381, and 409.

5. Statistics Statistics is the practice of the following five steps: 1. Raise a precise question about one or more variables. The key word is precise. As Lewis Carroll once said, “If you don’t know where you are going, any road will get you there.” 2. Create a plan to answer the question. We must carefully design a plan so that our results are meaningful.

6. Statistics 3. Collect the data. After observing individuals, measuring individuals, or asking people questions, we usually enter the data in software and fix any errors we can catch. 4. Analyze the data. Constructing tables and graphs can help us look for patterns. There are many procedures that involve calculations that can deepen our understanding of the variable(s).

7. Statistics 5. Draw a conclusion about the question. Researchers often publish their findings. Learning something about one or more variables often raises new questions that can lead to more studies.

8. More Definitions A population is the entire group of individuals about which we want to learn. A sample is the part of a population from which data are collected.

9. Example: Identify the Variable, Sample, and Population of a Study Researchers wanted to estimate the percentage of all American adults who think it should be legal for same-sex couples to marry. Of 1009 adults surveyed, 56% said yes. The researchers then made a conclusion about the proportion of all American adults who think it should be legal for same-sex couples to marry (Sources: ABC News, Washington Post). 1. Define a variable for the study. 2. Identify the population. 3. Identify the sample.

10. Solution 1. We define the variable same-sex marriage to represent all the “yes” responses and “no” responses to the question “Do you think it should be legal for same-sex couples to marry?” 2. The population is all American adults because that was the group the researchers wanted to learn about. 3. The sample is the 1009 adults who were surveyed because that is the group the researchers observed.

11. Definitions Descriptive statistics is the practice of using tables, graphs, and calculations about a sample to draw conclusions about only the sample. Inferential statistics is the practice of using information from a sample to draw conclusions about the entire population. When we perform inferential statistics, we call the conclusions inferences.

12. Simple random sample A process of selecting a sample of size n is simple random sampling if every sample of size n has the same chance of being chosen. A sample selected by such a process is called a simple random sample. If we allow an individual to be selected more than once, then we are sampling with replacement. If we do not allow an individual to be selected more than once, then we are sampling without replacement.

13. Example: Estimating a Population Proportion with a Large Sample In fall semester 2013, there were 19,773 students enrolled in LaGuardia Community College in Long Island City, New York. There were 11,310 female students and 8463 male students (Source: LaGuardia Community College). 1. What proportion of the students were female? 2. By using the numbers 1 through 11,310 to represent the female students and the numbers 11,311 through 19,773 to represent the male students, the author used technology to randomly select 1000 students without replacement. The random sample consisted of 585 female students and 415 male students. What proportion of the sample was female students?

14. Example: Estimating a Population Proportion with a Large Sample 3. If we did not know the proportion of the population who are female students and used the result we found in Problem 2 to estimate it, would that be part of descriptive statistics or inferential statistics? Explain.

15. Solution 1. Because there were 11,310 female students out of 19,773 students, the proportion of the population who are female is 11,310/19,773 or approximately 0.572. This is called a population proportion. 2. Because there were 585 female students out of 1000 students, the proportion of the sample who are female is 585/1000 or approximately 0.585. This is called a sample proportion.

16. Solution 3. Using the sample proportion 0.585 to estimate the population proportion 0.572 is part of inferential statistics.

17. Definitions Sampling error is the error involved in using a sample to estimate information about a population due to randomness in the sample. A sampling method that consistently underemphasizes or overemphasizes some characteristic(s) of the population is said to be biased.

18. Types of Bias There are three types of bias in sampling: 1. Sampling bias 2. Nonresponse bias 3. Response bias

19. Guidelines for Constructing Survey Questions When constructing a survey question, Do not include judgmental words. Avoid asking a yes/no question. If the question includes two or more choices, switch the order of the choices for different respondents. Address just one issue.

20. Example: Identifying Bias For each study, identify possible forms of bias. Also, discuss sampling error. 1. Throughout a week, a group of students surveys other students studying in the library and finds that 855 of 950 students study every day. The group concludes that 90% of all students at the college study every day.

21. Example: Identifying Bias For each study, identify possible forms of bias. Also, discuss sampling error. 2. SNAP is the U.S. food program for low-income individuals and families. A conservative television news station conducts a call-in survey, asking callers, “Should federal funding for SNAP be decreased so the national budget can be balanced?” Of the 1420 callers, 994 answered yes. The station concludes that 70% of all Americans think that the budget for SNAP should be decreased.

22. Solution 1. The sampling method favors students who study in the library over those that don’t (sampling bias). Also, students busy studying for a test might refuse to take part in the survey (nonresponse bias). Finally, students who are surveyed might exaggerate how often they study (response bias). Even if the group of students had performed sampling without bias, there would still likely be sampling error, so the population percentage would probably not equal the sample percentage, although it would be close.

23. Solution 2. Because the station is conservative, its viewers will tend to be conservative, too. So, the survey favors Americans who are conservative (sampling bias). Also, the question is a yes/no question (response bias). Finally, the question includes two issues, reducing the SNAP budget and balancing the national budget (response bias). Even if the station had performed sampling without bias, there would still likely be sampling error, so the population percentage would probably not equal the sample percentage, although it would be close.

24. Definition Nonsampling error is error from using biased sampling, recording data incorrectly, and analyzing data incorrectly.

25. Solution There is no nonsampling error because the sampling is not biased, the data were recorded correctly, and the analysis was done correctly. The error 0.37 – 0.36 = 0.01 is due to sampling error.

26. Example: Identify Sampling and Nonsampling Errors Identify whether sampling error, nonsampling error, or both have likely occurred. 1. An employee in human resources randomly selects some of the employees, and by referring to accurate records, he correctly computes that the proportion of sampled employees who have worked at the company at least 10 years is 0.37. But the population proportion is actually 0.36. 2. An employee in human resources randomly selects some of the employees and asks them whether they have ever called in sick when they were healthy. The proportion of the sample who say yes is 0.05, but the population proportion is actually 0.28.