1.2.1 Random Sampling INTRODUCTION In Lesson 1.1.3, we learned that in order to generalize results from a sample to the population, the sample must be representative of the population. 1 Explain what it means for a sample to be representative of the population. It might be helpful for you to think of some populations, like students at our college or professional basketball players. 2 Suppose our college is thinking of ways to raise money. Many students like parking spaces close to their classes. The administration is thinking of selling reserved parking spaces for $100. The college wants to know the percentage of students who would support this fee. One way to find the percentage of students that support this fee would be to conduct a census. A census is a survey of an entire population. The college would ask every student on campus if she or he would support the fee. Is this a reasonable plan? Why do you think so? 1.2.1 1
3 Read the following ways to sample students at our college. For each method: Tell whether the method would produce a sample that represents the student population. If the sample would not be representative, explain why. A Choose four 8:00 a.m. classes at random. Survey all the students in each class. B Put a poll on the front page of the college website. A poll is an opinion survey. Use the students who answer the question as the sample. C Talk to students as they enter the Student Center. 4 None of the sampling methods above will produce a sample that is representative of the college s population of students. Suggest a better method to gather a representative sample. NEXT STEPS When we sample, our goal is for every population member to have the same chance of being selected. One good way to do this is to select a simple random sample. In a simple random sample, all samples (of a given size) have the same chance of being chosen. 5 Biased samples result when sampling methods tends to leave out certain types of population members. The three sampling methods listed above all produced biased samples. There are different types of biased samples. 1.2.1 2
A One type of biased sample is a voluntary response sample. Good samples are chosen by researchers. In a voluntary response sample, the participants are self selected. In other words, each participant chooses to participate. Which sample from Question 3 above is a voluntary response sample? B Another biased sample is a convenience sample. Convenience sampling does not use random selection. It involves using an easily available or convenient group to form a sample. Many samples have convenience sampling problems. Which sample from Question 3 above is the best example of a convenience sample? 6 Suppose our college has 13,000 students. The college has the names and email addresses for all students in its database. Suggest a way that the administration could choose a simple random sample of 150 students to survey about the parking fee proposal. After the administration has chosen a sample, how could they actually conduct the survey? 7 When a researcher does not have a list of population members, it can be difficult or even impossible to get a simple random sample. In such cases researchers must still try to get a representative sample. Suppose that a writer for the student newspaper wants to survey students about the parking fee proposal. The writer does not have access to the college s student database. What is a method that the writer might use to get a sample that is representative of the student population? 1.2.1 3
Not What You Might Think Researchers are interested in the proportion of students who are registered to vote at a college. There are 5,000 students over the age of 18 who are enrolled at the college. Suppose 3,500 of these students are registered to vote and 1,500 are not registered. One way to graphically describe these data is to use a bar chart. Take a look at the bar chart, above. The bar on the left represents registered students. The bar on the right represents students who are not registered. The height of each bar represents the number of students in that category. Another common way to create a bar chart is to use proportions. In Lesson 1.1.1 we learned that a proportion is a number between 0 and 1, representing a portion of the total. We can talk about proportions as decimal numbers, percentages, or fractions. We compute proportions by dividing the number of individuals in a particular category by the total number of individuals. There are 3,500 students who are registered to vote out of 5,000 total students, so the proportion of students who are registered to vote is: 3500 = 0.70 = 70% 5000 The proportion for the not registered category is: 1500 = 0.30 = 30% 5000 1.2.1 4
Let s consider what happens when we take random samples from this population. Computer simulation was used to create a simple random sample of 50 students. In this sample, 32 students were registered to vote and 18 were not registered to vote. 8 Answer these questions about the proportion of registered students in this sample: A What proportion of students in the sample is registered to vote? B How does this proportion of registered students compare to the actual registered population proportion of 0.70? Are these values equal? C Does this surprise you? Why or why not? 1.2.1 5
We can create bar charts for both population data and for sample data. A bar chart for the simple random sample is shown below. 9 What differences do you notice between this bar chart and the bar chart from the population? NEXT STEPS When we take different samples from a population we often get different results. We can use computer simulation to explore how changing the sample size affects the results from the sample. Computer simulation was used to create two collections of random samples. For the first set, the computer simulated 100 simple random samples of 50 students each. The computer created these simple random samples from the population of 5,000 students in which 70% are registered to vote. For the second set, the computer created 100 simple random samples of 100 students each using the same population. For each sample the proportion of students who were registered to vote was computed. The two collections of sample proportions were used to construct the following two dotplots. 1.2.1 6
10 Which of the dotplots do you think is for the samples of size 50? Which do you think is for samples of size 100? Explain your answer. Think about which sample size, 50 or 100, might give you a better estimate of the population proportion. 11 Where do you find the population proportion of 0.70 in each dotplot? You have just seen that sample size affects precision of statistical estimates. The larger the sample size the closer the estimates will cluster around the population value. In the dotplots above, that proportions calculated from samples of size 100 were more tightly clustered around the true value of 0.70 than the proportions calculated from samples of size 50. Perhaps surprisingly, the precision of the estimate does not depend on the size of the population. It does not matter if the population size is 10,000 or 100,000 or 1,000,000. To understand this more, think about the following three populations: Population A: 10,000 people, 7,000 are registered to vote. Population B: 100,000 people, 70,000 are registered to vote. Population C: 1,000,000 people, 700,000 are registered to vote. 1.2.1 7
12 What proportion of people is registered to vote in each of the three populations? A Population A: Lesson 1.2.1: Random Sampling B Population B: C Population C: Computer simulation was used to select one hundred different simple random samples of size 500. The computer selected the samples from Population A and calculated the proportions. The researchers used the 100 sample proportions to make the dotplot labeled Population Size 10,000 in the following graph. The researchers repeated the process for the other two populations. One hundred simple random samples of size 500 were gathered from Populations B and C. 13 Label the population proportion of 0.70 on each of the graphs above. 1.2.1 8
14 We saw in the previous example that larger sample sizes give more precise estimates. All of these dotplots are based on the same sample size of 500. Does the population size seem to affect the precision of the estimates? Explain your reasoning. Think about how close the dots are to the population proportion of 0.70 in each of the three graphs. 1.2.1 9
STUDENT NAME DATE TAKE IT HOME 1 Imagine that you want to learn about the average number of hours, per day, that students at your college spend online. You want to select a simple random sample of 75 students from the full time students at your college. You have a list of all full time students, whose names are arranged in alphabetical order. How would you select a simple random sample of 75 students from this population? Describe your process. 2 You want to estimate the average amount of time, per week, that students at a particular college spend studying. For each method, determine if the method is reasonable and why. Which of the following sampling methods do you think would be best? A Method A: Select 50 students at random from the students at the college. B Method B: Select 100 students as they enter the library. C Method C: Select 200 students at random from the students at the college. D Method D: Select the 300 students enrolled in English literature at the college this semester. 1.2.1 10
3 The state of California allows people to vote on initiatives in elections. These initiatives become law if a majority of voters support them. In 1999, an initiative was proposed that would have required that, in any election, people should be allowed to vote for none of the above if they did not support any of the candidates. In order for the initiative to become law, more people needed to vote yes (in support of the initiative) than no (against it). A newspaper conducted a poll of California voters to see if more than half of people would vote yes. The results of the poll were: 55% were against the initiative and would vote no. 45% were for the initiative and would vote yes. A spokesperson for the group supporting the initiative questioned this result. He said a random sample of 1,000 registered voters was not large enough because there are about 14.6 million registered voters in California. The spokesperson said that 1,000 voters could not possibly represent all voters in 1 California. A Is his criticism of the sample size valid? Explain why or why not. Think about the type of sampling method that was used, the sample size and the population. B Would the criticism be valid if this had been a national initiative and 1,000 people were randomly selected from all registered voters in America? 1 Doug Willis, Pollsters: Sound Poll is about Quality, not Quantity, Ukiah Daily Journal, January 30, 2000, 7. 1.2.1 11